# Landis-type results for discrete equations

Aingeru Fernández-Bertolin  Facultad de Ciencia y Tecnología, Universidad del
País Vasco / Euskal Herriko Unibertsitatea (UPV/EHU), Departamento de
Matemáticas, UPV/EHU, Apartado 644, 48080 Bilbao, Spain
<EMAIL_ADDRESS>,  Luz Roncal  BCAM - Basque Center for Applied
Mathematics, 48009 Bilbao, Spain and Ikerbasque, Basque Foundation for
Science, 48011 Bilbao, Spain and Universidad del País Vasco / Euskal Herriko
Unibertsitatea, Apartado 644, 48080 Bilbao, Spain<EMAIL_ADDRESS>and
Diana Stan  University of Cantabria, Department of Mathematics, Statistics and
Computation, Avd. Los Castros 44, 39005 Santander, Spain<EMAIL_ADDRESS>
###### Abstract.

We prove Landis-type results for both the semidiscrete heat and the stationary
discrete Schrödinger equations. For the semidiscrete heat equation we show
that, under the assumption of two-time spatial decay conditions on the
solution $u$, then necessarily $u\equiv 0$. For the stationary discrete
Schrödinger equation we deduce that, under a vanishing condition at infinity
on the solution $u$, then $u\equiv 0$. In order to obtain such results, we
demonstrate suitable quantitative upper and lower estimates for the
$L^{2}$-norm of the solution within a spatial lattice $(h\mathbb{Z})^{d}$.
These estimates manifest an interpolation phenomenon between continuum and
discrete scales, showing that close-to-continuum and purely discrete regimes
are different in nature.

###### Key words and phrases:

Landis-type results, discrete equations, Carleman estimates

###### 2010 Mathematics Subject Classification:

Primary: 39A12; Secondary: 35B05.

A. Fernández-Bertolin is partially supported by project PID2021-122156NB-I00
(AEI/FEDER, UE) and acronym HAMIP, and the Basque Government through the
project IT1615-22.

L. Roncal is partially supported by the projects CEX2021-001142-S,
RYC2018-025477-I and PID2020-113156GB-I00/AEI/10.13039/501100011033 with
acronym “HAPDE” funded by Agencia Estatal de Investigación, grant BERC
2022-2025 of the Basque Government and IKERBASQUE

D. Stan is supported by the project PID2020-114593GA-I00 financed by
MCIN/AEI/10.13039/501100011033

##  1\. Introduction and main results

Let us consider the stationary Schrödinger equation

(1.1) $(-\Delta+V)u=0\quad\text{ in }\mathbb{R}^{d},$

with $|V(x)|\leq 1$. The question about the maximal rate of decay of a
solution to (1.1) was raised by Landis in 1960 (see [KL88]). The so-called
Landis’ conjecture states that if a solution of the equation (1.1) satisfies
$|u(x)|\leq\exp{(-C|x|^{1+})}$, then $u\equiv 0$. Meshkov [Mesh91] disproved
the conjecture by constructing a complex-valued potential $V$ and a nontrivial
solution $u$ to (1.1) with $|u(x)|\leq\exp{(-C|x|^{\frac{4}{3}})}$. Moreover,
he showed that if $|u(x)|\leq\exp{(-C|x|^{\frac{4}{3}+})}$, then $u\equiv 0$.
In words, the exponent $4/3$ is optimal in the complex case. Later on,
Bourgain and Kenig [BK05] showed a quantitative form of Meskhov’s result which
was in connection with the resolution of Anderson localization for the
Bernoulli model in higher dimensions. We also mention the complex
counterexample by Cruz-Sampedro [CS99] that improves Meskhov’s result.

A refinement of Landis’ conjecture was presented by Kenig [K05], who raised
the question about the validity of the conjecture for real-valued potentials
and solutions. A partial positive answer was given by Kenig, Silvestre, and
Wang [KSW15], who proved that a quantitative form of the conjecture is true in
the plane ($d=2$) for real-valued $V\geq 0$. Landis’ conjecture in the real
case with $d=1$ was studied by Rossi [R21]. Recently, Logunov, Malinnikova,
Nadirashvili, and Nazarov [LMNN20] showed that Landis’ conjecture is true for
$d=2$ and real potentials, and Davey [Davey23] extended the result also for
$d=2$ and real potentials, by allowing the potentials to grow. Up to our
knowledge, the Landis conjecture in its general form in higher dimensions
$d\geq 3$ remains still an open question.

Let us consider now the time-dependent Schrödinger equation

(1.2) $\partial_{t}u(t,x)=i(\Delta u+Vu).$

In a series of works, Escauriaza, Kenig, Ponce, and Vega [EKPV06, EKPV08,
EKPV10, EKPV16, EKPV12] proved that if $V$ satisfies one of the following
conditions

  * (i)

$\lim_{R\to\infty}\int_{0}^{T}\sup_{|x|>R}|V(t,x)|\,dt=0$

  * (ii)

$V(t,x)=V_{1}(x)+V_{2}(t,x)$, where $V_{1}$ is real-valued and for some
positive $\alpha$ and $\beta$,

$\sup_{[0,T]}\|e^{\alpha\beta
T^{2}|x|^{2}/(\sqrt{\alpha}t+\sqrt{\beta}(T-t))^{2}}V_{2}(t)\|_{L^{\infty}(\mathbb{R}^{d})}<+\infty,$

and we assume that $u$ is a solution to (1.2) which fulfills the decay
conditions $|u(0,x)|\leq Ce^{-\alpha|x|^{2}}$, $|u(T,x)|\leq
Ce^{-\beta|x|^{2}}$ with $\alpha\beta>1/(16T^{2})$, then $u\equiv 0$.
Similarly, for the heat equation with potential, the following result can be
deduced: let $V(t,x)\in L^{\infty}(\mathbb{R}\times\mathbb{R}^{d})$ and $u$ be
a solution to $\partial_{t}u=\Delta u+Vu$; if $|u(T,x)|\leq
Ce^{-\delta|x|^{2}}$ and $\delta>1/(4T)$, then $u\equiv 0$. The conditions on
$\alpha\beta$ and $\delta$ are sharp. These results can be understood as
dispersive and parabolic analogues of Landis’ conjecture.

In this paper, we are interested in: $(1)$ studying analogous Landis-type
results to the ones described above when we consider equations involving
discrete Laplacians in a mesh of size $h>0$, and $(2)$ analysing the different
behaviour of the solutions as long as the size of the mesh is shrinking to
zero. In general, we refer to Landis-type results when we are interested in
the maximum vanishing rate of solutions to equations with potentials, namely:

  * •

In the case of elliptic equations, we are concerned about the maximal rate at
which nontrivial solutions vanish at infinity.

  * •

Concerning dispersive or parabolic equations, we are concerned about the
maximal spatial decay rate of nontrivial solutions when time varies whithin a
bounded interval.

Let us consider the discrete lattice $(h\mathbb{Z})^{d}$ with
$h\in\mathbb{R}_{+}$, $d\geq 1$. Given a function
$f:(h\mathbb{Z})^{d}\to\mathbb{R}$, we define the discrete Laplacian as

(1.3)
$\Delta_{\operatorname{\operatorname{d}}}f_{j}:=\frac{1}{h^{2}}\sum_{k=1}^{d}\big{(}f_{j+e_{k}}-2f_{j}+f_{j-e_{k}}\big{)},\quad
j\in\mathbb{Z}^{d},$

where we denote $f_{j}:=f(hj)$ and $e_{k}$ is the unit vector in the $k$-th
direction. Landis-type uniqueness theorems for the time-dependent Schrödinger
equation with a discrete Laplacian were studied in [FB19, FB18, FBV17,
JLMP18]. We consider the equation, in a mesh of size $h=1$,

(1.4)
$\partial_{t}u_{j}=i(\Delta_{\operatorname{\operatorname{d}}}u_{j}+V_{j}u_{j})\quad\text{in
}\mathbb{Z}^{d}\times\mathbb{R}_{+},$

where $V:\mathbb{Z}^{d}\times\mathbb{R}_{+}\to\mathbb{R}$ is a bounded
potential and we use the notation $u_{j}=u_{j}(t):=u(j,t)$. It was proved in
[JLMP18] (for $d=1$) and in [FBV17] (for arbitrary dimensions) that, if $u$ is
a solution to (1.4) and there exists a constant $\gamma$ such that

$|u(j,0)|+|u(j,1)|\leq C\exp(-\gamma|j|\log|j|),\quad
j\in\mathbb{Z}^{d}\setminus\\{0\\},$

then $u=0$.

Estimates on the decay of stationary solutions of discrete Schrödinger
operators

$\Delta_{\operatorname{\operatorname{d}}}u_{j}+V_{j}u_{j}=0\quad\text{in
}\mathbb{Z}^{d},$

where $V:\mathbb{Z}^{d}\to\mathbb{R}$ is a bounded potential and
$u_{j}:=u(j)$, and sharp uniqueness results for this equation, were obtained
in [LM18] by Lyubarskii and Malinnikova: if $u(x)$ satisfies the following
decay estimate

$\liminf\limits_{N\rightarrow\infty}\frac{\ln(\max_{|n|_{\infty}\in\\{N,N+1\\}}|u(n)|)}{N}<-\|V\|_{\infty}-4d+1$

where $|n|_{\infty}=\max\\{n_{1},\ldots,n_{d}\\}$, for $n\in\mathbb{Z}^{d}$,
then $u\equiv 0$.

Comparing these results to the continuum versions, we see that the decay
assumption is much weaker in the discrete results, which, intuitively, points
out to the fact that the discretization process deteriorates the fast decay of
the solutions at large scales. The goal of this paper is to show Landis-type
results for the semidiscrete heat equation and the stationary discrete
Schrödinger equation with bounded potentials, in a mesh $(h\mathbb{Z})^{d}$,
for $h>0$ and $d\geq 1$, and through our results we want to connect the
different behaviors above mentioned, studying the discrete solutions at
different scales. More concretely, we will deal with the following problems:

  1. (1)

the semidiscrete heat equation

(1.5)
$\partial_{t}u_{j}=\Delta_{\operatorname{\operatorname{d}}}u_{j}+V_{j}u_{j}\quad\text{in
}(h\mathbb{Z})^{d}\times\mathbb{R}_{+}\qquad u_{j}(0)=\psi_{j}\quad\text{ in
}(h\mathbb{Z})^{d},$

where $V:(h\mathbb{Z})^{d}\times\mathbb{R}_{+}\to\mathbb{R}$ is a bounded
potential, $u:(h\mathbb{Z})^{d}\times\mathbb{R}_{+}\to\mathbb{R}$, and
$\psi:(h\mathbb{Z})^{d}\to\mathbb{R}$.

  2. (2)

the stationary discrete Schrödinger equation

(1.6) $\Delta_{\operatorname{\operatorname{d}}}u_{j}+V_{j}u_{j}=0\quad\text{in
}(h\mathbb{Z})^{d},$

where $V:(h\mathbb{Z})^{d}\times\mathbb{R}_{+}\to\mathbb{R}$ is a bounded
potential and $u:(h\mathbb{Z})^{d}\to\mathbb{R}$.

Notice that for the equation (1.5) we use the notation
$u_{j}=u_{j}(t):=u(hj,t)$, whereas for (1.6) the notation is $u_{j}:=u(hj)$.
The different meaning with the notation involving only the spatial lattice
component will be clear from the context.

The main results of the paper are Theorems A and B. An interesting, new
contribution is that the results will be obtained through quantitative
estimates within a spatial lattice $(h\mathbb{Z})^{d}$ which manifest an
interpolation phenomenon between continuum and discrete scales. In the case of
the semidiscrete heat equation, under assumption of a two-time spatial decay
condition on the solution, we conclude that the solution is trivial, see
Theorem A; for the stationary discrete Schrödinger equation, we prove that if
a solution decays at a certain rate, then the solution is trivial, see Theorem
B.

###  1.1. Qualitative estimates

Our first main result will show qualitative behaviour of solutions to (1.5),
assuming a two-time decaying condition. Such a condition is given in terms of
Macdonald’s functions $K_{\nu}(x)$ (see Appendix B for the definition). We
denote111Along the paper, we will be considering an interval of time
$t\in[0,1]$. All the results can be adapted to an interval $[0,T]$ for a fixed
$T>0$; in that case there will be constants which will depend on $T$.

$\|V\|_{\infty}:=\sup_{j\in\mathbb{Z}^{d},\,t\in[0,1]}\\{|V_{j}(t)|\\},\qquad\|u\|_{2}:=\Big{(}h^{d}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}|u_{j}(t)|^{2}\,dt\Big{)}^{1/2}.$

######  Theorem A (Landis-type result for the semidiscrete heat equation).

Let $h>0$ and $u\in C^{1}([0,1]:\ell^{2}((h\mathbb{Z})^{d}))$ be a solution to

$\partial_{t}u_{j}=\Delta_{\operatorname{\operatorname{d}}}u_{j}+V_{j}u_{j}\quad\text{in
}(h\mathbb{Z})^{d}\times\mathbb{R}_{+},$

where $\|V\|_{\infty}$, $\|u\|_{2}$ are finite and independent of $h$.

  1. (1)

Let $\gamma$ be a positive constant and assume that

$h^{d}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}\Big{(}\frac{K_{j_{k}}^{2}(\frac{\gamma}{h^{2}})}{K_{0}^{2}(\frac{\gamma}{h^{2}})}\Big{)}\big{(}|u_{j}(0)|^{2}+|u_{j}(1)|^{2}\big{)}<\infty$

uniformly with respect to $h$. Then there exist $C,h_{0}>0$ such that if
$\gamma<C$ and $h\in(0,h_{0})$, $u\equiv 0$.

  2. (2)

There exists $\mu_{0}=\mu_{0}(d)$ such that if, for $\mu>\mu_{0}$

$h^{d}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}\Big{(}\frac{K_{j_{k}\mu}(\frac{2}{eh^{2}})}{K_{0}(\frac{2}{eh^{2}})}\Big{)}\big{(}|u_{j}(0)|^{2}+|u_{j}(1)|^{2}\big{)}<\infty,$

then $u\equiv 0$.

The uniqueness property in $(1)$ of Theorem A corresponds with a situation in
which the mesh is shrinking to the continuum Euclidean setting. Up to our
knowledge, this situation had not been explored so far. On the other hand,
property $(2)$ is linked to a purely discrete regime, where we should
understand $h$ as a fixed parameter. The parameter $\gamma$ involved in $(1)$
is not playing a relevant role in the purely discrete regime in the sense that
there cannot be a suitable choice of $\gamma$ which produces a contradiction
in the reasoning, leading to the desired conclusion (this will become clear in
Subsection 3.3, see also Remark 3.6). This forces to consider a different
condition in $(2)$ for the purely discrete regime. This result in $(2)$ is
actually a slightly improved and scaled version of [FBV17, Theorem 1.2] (see
also [JLMP18]).

We want to point out that if $\mu_{0}$ in $(2)$ of Theorem A, a quantity that
we are not quantifying, is seen to be small enough, then one could prove the
first part of the theorem, for any $\gamma$, as a consequence of the second
part. However, we have decided to state and prove the two parts of Theorem A
separately to stress out the results one can get by analyzing, on the one
hand, the close-to-continuum setting, which resembles the decay condition
given by Escauriaza, Kenig, Ponce and Vega, and, on the other hand, the purely
discrete setting.

The second main result concerns the elliptic equation (1.6).

######  Theorem B (Landis-type result for the stationary discrete Schrödinger
equation).

Let $h>0$ and $u\in\ell^{2}((h\mathbb{Z})^{d})$ be a solution to

$\Delta_{\operatorname{\operatorname{d}}}u_{j}+V_{j}u_{j}=0$

where $\|V\|_{\infty}$, $\|u\|_{2}$ are finite and independent of $h$.

  1. (1)

There exists $\mu_{0}=\mu_{0}(d)$ such that if

$h^{d}\sum_{j\in\mathbb{Z}^{d}}e^{\mu_{0}|jh|^{4/3}}|u_{j}|^{2}<\infty,$

uniformly with respect to $h$, then there exists $h_{0}>0$ such that if
$h\in(0,h_{0})$, $u\equiv 0$.

  2. (2)

There exists $\mu_{0}=\mu_{0}(d)$ such that if, for some $\beta>3$

$h^{d}\sum_{j\in\mathbb{Z}^{d}}e^{\mu_{0}|jh|^{1+\frac{1}{\beta}}\log|j|h}|u_{j}|^{2}<\infty,$

uniformly with respect to $h$, then there exists $h_{0}>0$ such that
$h\in(0,h_{0})$ implies $u\equiv 0$.

  3. (3)

There exists $\mu_{0}=\mu_{0}(d)$ such that if,

$h^{d}\sum_{j\in\mathbb{Z}^{d}}e^{\mu_{0}|j|\log|j|h}|u_{j}|^{2}<\infty,$

then $u\equiv 0$.

Let us discuss the different statements in Theorem B. The first part concerns
the close-to-continuum setting and it will be derived from the study of
nontrivial solutions close to the continuum, posing our problem in a mesh of
size $h$ which shrinks to zero. We observe that we get the same exponent as in
[BK05, Mesh91]. The third part, for which $h$ is considered a fixed parameter,
concerns the purely discrete setting, and the result can be compared with the
uniqueness result in [LM18]. The second part lies in between these two regimes
as we will derive these result by understanding the behavior of nontrivial
solutions at scales $h^{-\beta}$ for $\beta$ large. While we are still posing
our problem in a lattice that shrinks to zero, we are looking at our solutions
in a region that is far from the close-to-continuum setting. As we can see in
the result, what we deduce is that an interpolating decay between the
continuum decay (exponent $4/3$) and the purely discrete decay (exponent 1 up
to a logarithm) provides unique continuation. As it happens in the parabolic
setting, we can see these theorems as a cascade of results, since $(3)$
implies $(2)$ and $(2)$ implies $(1)$. Since our goal is to understand the
effect of the discretization on the continuum result, we have decided to state
and prove each part separately because it clearly shows how the largest decay
possible for a nontrivial solution deteriorates as we escape the close-to-
continuum region.

The strategy of the proofs follows the scheme of the continuum results in
[EKPV12]; similar approach was also exploited in [FBV17]. In the case of
Theorem A, the first step (inspired in the classical approach used by Agmon
for elliptic equations [A65]) is to show logarithmic convexity estimates for
certain weighted norm of the solution to (1.5). With this logarithmic
convexity at hand, the decay conditions at times $t=0$ and $t=1$ imply upper
bounds for the $\ell^{2}$-norm of the solution localized to a region of size
$|hj|\simeq R$ with $R$ large. By distinguishing two regimes, we deduce upper
bounds, under slightly different conditions, within the close-to-continuum and
the purely discrete regime. The two scenarios arise naturally when we study
the second ingredient, which are suitable lower bounds obtained through a
Carleman inequality.

Indeed, the second step is to prove a Carleman-type inequality, whose proof
relies on the computation of a commutator between a symmetric and
antisymmetric conjugate operator involving a parameter (Carleman parameter).
Thanks to this inequality, one can deduce lower bounds for nontrivial
solutions of (1.5) with a general bounded potential. In order to do that, one
considers suitable cut-off functions and defines a function to which the
action of the commutator is computed. The only assumption to obtain the lower
bound is that the solution is nontrivial. It is at this point where the
quantitative nature of the problem plays a role and it forces to the
appearance of two cases: the lower bounds involve hyperbolic functions with
arguments depending on the mesh size. We linearize these functions by using
the corresponding asymptotics, which lead to the dichotomy of conditions and
to the two regimes: the close-to-continuum scenario and the purely discrete
regime.

Finally, with an appropriate choice of the Carleman parameter and comparing
with the corresponding upper bound, a contradiction is reached in each case,
so that it is deduced that $u\equiv 0$.

The assumptions on the decay of the solutions in Theorem A are stated in terms
of Macdonald’s Bessel functions. We remark that in the present paper, since we
are interested in the role of the size of the mesh $h>0$ and the interpolation
phenomenon between the discrete and the continuum regimes, we have to be very
careful in the quantification of the arguments; this translates into a fine
use of asymptotics of Bessel and hyperbolic functions at different scales.

The Landis-type result for the semidiscrete heat equation is a blueprint for
scaled versions of the results in [FBV17, JLMP18], namely for the discrete
time-dependent Schrödinger equation in the mesh $(h\mathbb{Z})^{d}$. It is
expected that the same strategy will go through, although it is likely more
technically delicate. We do not tackle this problem in the present paper. On
the other hand, the approach used to study the semidiscrete heat equation
facilitates the investigation of the elliptic equation, where a similar idea
to obtain the corresponding lower bounds is used and, moreover, the estimate
of the commutator is automatically obtained from the proof of the Carleman
estimate for the semidiscrete heat equation. It is interesting to notice that
Theorem B is indeed a discrete version of the original Landis’ conjecture.
Observe also that we are not assuming that the potential is either complex-
valued or real-valued. Further, the problem is in connection with the
Anderson–Bernoulli model on the lattice, which is the random Schrödinger
operator on $\ell^{2}(\mathbb{Z}^{d})$ given by

$H=-\Delta_{\operatorname{\operatorname{d}}}+\delta V,$

where $V$ is a random potential whose values $V_{j}\in\\{0,1\\}$ for
$j\in\mathbb{Z}^{d}$ are independent and satisfy
$\mathbb{P}[V=0]=\mathbb{P}[V=1]=1/2$, with $\delta>0$ being the strength of
the noise, see [BK05, DS20, L22, LZ22].

###  1.2. Quantitative estimates: semidiscrete heat equation

The proof of Theorem A will rely on precise quantitative upper and lower
bounds for the solutions to (1.5). We start showing the upper bound, which
holds under the two-time decaying assumption for the solutions.

######  Theorem 1 (Upper bound, close-to-continuum regime).

Let $d\geq 1$ and let $u\in C^{1}([0,1]:\ell^{2}((h\mathbb{Z})^{d}))$ be a
solution to (1.5). Assume that for some $\gamma>0$ there exists a finite
positive constant $c$ independent of $h$ such that

(1.7)
$h^{d}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}\Big{(}\frac{K_{j_{k}}^{2}(\frac{\gamma}{h^{2}})}{K_{0}^{2}(\frac{\gamma}{h^{2}})}\Big{)}\big{(}|u_{j}(0)|^{2}+|u_{j}(1)|^{2}\big{)}<c.$

Then there exists $h_{0}>0$, with $\frac{\gamma}{h_{0}^{2}}\geq M=100$, such
that for $0<h<h_{0}$, if we choose $R>1$ with $Rh<\frac{\gamma}{2}$ and
$\frac{R}{h}\geq M$, then

$h^{d}\int_{0}^{1}\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|hj|<R+1\end{subarray}}(|u_{j}(0)|^{2}+|u_{j}(t)|^{2})\,dt\leq
C_{\gamma}e^{-dR^{2}/\gamma},$

where $C_{\gamma}$ is a positive constant independent of $h$ and $R$.

Theorem 1 will follow from a log-convexity argument which will be shown in
Section 3 and hence it may be compared with the convexity result [FB18,
Theorem 2.2]. In the current situation, in order to justify the computations
of the formal log-convexity property, the introduction of the scale $h$ makes
things more delicate. Indeed, when dealing with a general $h$, the bounds
blow-up as the scale shrinks to $0$ and we have to use an approximation
argument to get uniform bounds in $h$.

######  Remark 1.1.

The motivation for the condition (1.7) in Theorem 1 (and hence in Theorem A
$(1)$) relies on the fact that it can be seen as a discrete version of the
decay of the inverse continuous Gaussian function. Indeed, this is reasonable
in view that the fundamental solution of the semidiscrete heat equation is
given in terms of a modified Bessel function $I_{\nu}(x)$ defined in (B.1)
(see [CGRTV]) which asymptotically behaves, essentially, as the inverse of a
Macdonald’s function. On the other hand, it could be thought that a suitable
condition might consist of discretizing the continuum weight. Nevertheless,
the fundamental solution to the semidiscrete equation is not a mere
discretization of the solution to the continuum equation, although the latter
converges in certain sense to the former as $h$ shrinks to zero, see Appendix
A.

Observe that Theorem 1 is stated for a regime in which $Rh$ is small, so it
can be understood as a close-to-continuum regime. If we study the
corresponding upper bound in the regime in which $Rh$ is large, which can be
understood as a purely discrete regime, under the same decay condition (1.7)
we see that the obtained estimate is not enough to conclude a Landis-type
result as in Theorem A, see Remark 3.6. The reason, essentially, is that the
relevant parameter $\gamma$ which should be exploited to produce a
contradiction is not playing any significant role in the upper and lower
bounds in this regime.

Nevertheless, as it was shown in [FBV17, Theorem 1.2] in the case $h=1$, it is
possible to impose a weaker condition (although not sharp) which produces an
appropriate upper bound which gives rise to a contradiction. Motivated by this
fact, we state an upper bound in the purely discrete regime which will help us
prove the Landis-type result in Theorem A (2).

######  Theorem 2 (Upper bound, purely discrete regime).

Let $u\in C^{1}([0,1]:\ell^{2}((h\mathbb{Z})^{d}))$ be a solution to (1.5) and
assume that, for a fixed $\mu>0$,

(1.8)
$h^{d}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}\frac{K_{j_{k}\mu}(\frac{2}{eh^{2}})}{K_{0}(\frac{2}{eh^{2}})}\big{(}|u_{j}(0)|^{2}+|u_{j}(1)|^{2}\big{)}<\infty.$

Then there exists $h_{0}>0$ such that for $h\in(0,h_{0})$ and $R>1$ satisfying
$Rh\geq\frac{2}{e\mu}$, the following holds

$h^{d}\int_{0}^{1}\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|hj|<R+1\end{subarray}}|u_{j}(t)|^{2}dt\leq Ce^{-\mu
c_{0}\big{(}\frac{R}{h}\log(Rh)+\frac{R}{h}\log\mu\big{)}}$

for some positive constants $c_{0}$ and $C$ independent of $h$ and $R$.

######  Remark 1.2.

It can be checked that the function
$u_{j}(t)=\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}e^{-2t/h^{2}}\frac{I_{j_{k}}(2t/h^{2}+1/h^{2})}{I_{0}(1/h^{2})}$
is a solution to equation (1.5) so that (1.7) and (1.8) are satisfied, see
Section 6.

######  Remark 1.3.

The upper bound in Theorem 2 essentially reduces to the one in [FBV17,
Corollary 2.1] when $h=1$. It is possible to adjust the argument of the
Macdonald’s functions in condition (1.8) to allow the case $h=1$.

The change of regimes in $Rh$ exhibited in Theorems 1 and 2 will be also
explicit in the lower bound (and actually this change of regimes will become
apparent in the proof of the lower bound).

######  Theorem 3 (Lower bound).

Let $u\in C^{1}([0,1]:\ell^{2}((h\mathbb{Z})^{d}))$ be a nontrivial solution
to (1.5). Then there exist $h_{0}>0$, $R_{0}=R_{0}(d,u(0),\|V\|_{\infty})>0$,
and a positive constant $C$, such that for $R\geq R_{0}$ and $h\in(0,h_{0})$
it follows that

(1.9) $h^{d}\int_{0}^{1}\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|hj|<R+1\end{subarray}}(|u_{j}(0)|^{2}+|u_{j}(t)|^{2})\,dt\gtrsim\begin{cases}e^{-CR^{2}}\qquad&\mbox{
if }Rh\leq 1\\\ e^{-C\frac{R}{h}\log(Rh)}\qquad&\mbox{ if }Rh>1.\end{cases}$

We emphasize that in both of our quantitative results, the identified critical
regime $Rh$ being very small corresponds to the case in which the mesh is
shrinking to the continuum Euclidean setting. The estimates in the regime in
which $Rh$ is large concern a purely discrete situation and for this we are
also giving a uniqueness result in Theorem A (2).

###  1.3. Quantitative estimates: stationary discrete Schrödinger equation

The proof of Theorem B will be a consequence of lower bounds for the solutions
to (1.5). The strategy to obtain such lower estimates relies on a Carleman
estimate (see Theorem 6) which is nothing but a simplified version of the
Carleman estimate for the semidiscrete heat equation in Theorem 5. Indeed,
there are no conditions between the parameters (unlike the case for the
semidiscrete heat equation) thanks to the positivity of the commutator.

######  Theorem 4 (Lower bound).

Let $u\in\ell^{2}(h\mathbb{Z})^{d}$ be a non-trivial solution to

$\Delta_{\operatorname{\operatorname{d}}}u_{j}+V_{j}u_{j}=0$

for a bounded potential $V$.

  1. (1)

(Case $hR^{1/3}\leq 1$) There exists $R_{0}=R_{0}(d,\|V\|_{\infty})>0$ and a
positive constant $C$ such that for $h>0$ and $R\geq R_{0}$ satisfying
$hR^{1/3}\leq 1$, it follows that

(1.10) $e^{-CR^{4/3}}\leq h^{d}\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|hj|<R+1\end{subarray}}|u_{j}|^{2}.$

  2. (2)

(Case $hR^{1/3}>1$) There exists $R_{0}=R_{0}(d,\|V\|_{\infty})>0$ and a
positive constant $C=C(d)$ such that for $h>0$ and $R\geq R_{0}$ satisfying
$hR^{1/3}>1$, it follows that $R$ can be written as $R=h^{-\beta}$ for some
$\beta>3$, and

$e^{-CR^{1+\frac{1}{\beta}}\log(R^{1-1/\beta})}\leq
h^{d}\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|hj|<R+1\end{subarray}}|u_{j}|^{2}.$

Moreover, if we look at what happens when $R$ is large, not depending on $h$,
we get $e^{-C\frac{R}{h}\log(Rh)}$ as a lower bound, where $C=C(d)$.

######  Remark 1.4.

The exponent in (1.10) in Theorem 4 is the same as in the continuum problem
and, as shown by Meshkov [Mesh91] and by Bourgain and Kenig [BK05], it is
sharp if the potential is allowed to be complex.

###  1.4. On sharpness and improvement of the results

Concerning the sharpness of the results, in Section 6 we will show an example
that illustrates the optimality of the quantitative estimates in Theorems 1,
2, and 3 and hence of Theorem A. On the other hand, in view of the results in
[LM18, Subsection 4.4], and the examples in Subsection 7.3 (see Corollary
7.3), it is reasonable to expect that $(2)$ in Theorems B and 4, while being
analogs of the analysis carried out in the parabolic case to prove Theorem A,
are not sharp. At present, it remains uncertain whether $(1)$ in Theorem 4,
and consequently in Theorem B, particularly concerning the close-to-continuum
regime, achieves optimality. This issue stands as an interesting problem
deserving further exploration.

Furthermore, another interesting open question is whether the two-time
decaying condition in Theorems 1 and 2 can be relaxed to a one-time decaying
condition, as it is done in [EKPV16] for parabolic evolutions in the Euclidean
case. See [EKPV08], where this result is proved for the Schrödinger equation
in $\mathbb{R}^{d}$, and also [EKPV08, Theorem 4], where a non-optimal version
for the heat equation in the continuum case is shown. Finally, it would be
also interesting to extend the results presented in this paper, or improved
versions of them, to the non-stationary Schrödinger equation in the lattice
$(h\mathbb{Z})^{d}$.

###  1.5. Organization of the paper

The paper is organised as follows. In Section 2 we introduce notational
conventions and technical results which will be used later. Section 3 contains
a weighted log-convexity property of the solutions of (1.5) and the proofs of
Theorem 1 and 2. Section 4 is devoted to the proof of a Carleman estimate for
the semidiscrete heat operator, which will be a fundamental tool in the
subsequent proof of Theorem 3 in the same section. After that, in Section 5 we
present the proof of the Landis-type results in Theorem A. Optimality of the
results concerning the semidiscrete heat equation are discussed in Section 6.
Finally, in Section 7, we present the results within the setting of discrete
Schrödinger equation, namely the proofs of Theorems 4 and B.

Acknowledgments. The authors are greatly indebted to Angkana Rüland for very
helpful discussions at various stages of the project.

Part of this work was carried out during L. Roncal and D. Stan’s stay at Isaac
Newton Institute (INI) for Mathematical Sciences in Cambridge, during the
programme Fractional Differential Equations; this work was supported by EPSRC
grant no EP/R014604/1.&#34. The authors gratefully acknowledge INI’s support
and hospitality.

##  2\. Preliminaries and auxiliary results

###  2.1. Notational conventions

For $f:(h\mathbb{Z})^{d}\to\mathbb{R}$, the $\ell^{p}$ norms on the lattice
$(h\mathbb{Z})^{d}$ are defined as

$\|f\|_{\ell^{p}((h\mathbb{Z})^{d})}:=\Big{(}h^{d}\sum_{j\in\mathbb{Z}^{d}}|f_{j}|^{p}\Big{)}^{1/p},\quad
1\leq
p<\infty,\qquad\|f\|_{\ell^{\infty}((h\mathbb{Z})^{d})}:=\sup_{j\in\mathbb{Z}^{d}}|f_{j}|,$

and the scalar product is given by $\langle
f,g\rangle_{\ell^{2}((h\mathbb{Z})^{d})}:=h^{d}\sum_{j\in\mathbb{Z}^{d}}f_{j}g_{j}$.
We will sometimes use just
$\|f\|_{\ell^{p}},\|f\|_{p},\|f\|_{\ell^{\infty}},\|f\|_{\infty}$ without
further comment. We will also define the left/right difference operators at
scale $h$ as $D_{\pm,k}f_{j}:=\pm h^{-1}(f_{j\pm e_{k}}-f_{j})$. Observe that
the (normalised) discrete Laplacian in (1.3) can be expressed as
$\Delta_{\operatorname{\operatorname{d}}}:=\sum_{k=1}^{d}\Delta_{\operatorname{\operatorname{d}},k}:=\sum_{k=1}^{d}D_{-,k}D_{+,k}$.
We also write $D_{k}^{s}f_{j}:=\frac{1}{2h}(f_{j+e_{k}}-f_{j-e_{k}})$ to
denote the symmetric difference operator in the $k$-th direction and
$D^{s}f_{j}:=\sum_{k=1}^{d}D_{k}^{s}f_{j}$. Summation rule has the form

(2.1)
$\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}D_{+,k}f_{j}\,g_{j}=-\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}f_{j}\,D_{-,k}g_{j}.$

With the letters $c,C,\ldots$ we denote structural constants that depend only
on the dimension and on parameters that are not relevant. We shall write
$X\lesssim Y$ to indicate that $X\leq CY$ with a positive constant $C$
independent of significant quantities and we denote $X\simeq Y$ when
simultaneously $X\lesssim Y$ and $Y\lesssim X$. We write $f(x)\sim g(x)$ as
$x\to c$ to indicate that $f(x)/g(x)\to 1$ in the limit point $c$. We will
sometimes use the notation $X\lll Y$ to denote that $X$ is much smaller than
$Y$, i.e., there exists a big constant, say $C\geq 10^{3}$, such that $X\lll
C^{-1}Y$. Analogously, $X\ggg Y$ means that $X\geq CY$, for instance.

###  2.2. Auxiliary results

In this subsection we collect some technical results which will be used later
on. First, we prove the following energy estimate.

######  Lemma 2.1 (Energy estimate).

Let $u\in C^{1}([0,1]:\ell^{2}((h\mathbb{Z})^{d}))$ be a solution to the
initial value problem (1.5) with $V\in
L^{\infty}((h\mathbb{Z})^{d}\times[0,1])$ and
$u(0)\in\ell^{2}((h\mathbb{Z})^{d})$. Then

(2.2)
$\|u(\cdot,t)\|_{\ell^{2}((h\mathbb{Z})^{d})}^{2}+2h^{d}\int_{0}^{t}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}|D_{-,k}u_{j}(\tau)|^{2}\,d\tau\leq
e^{2t\|V\|_{L^{\infty}((h\mathbb{Z})^{d}\times[0,1])}}\|u(\cdot,0)\|_{\ell^{2}((h\mathbb{Z})^{d})}^{2},\quad\forall
t\in(0,1].$

###### Proof.

Let $u$ be a solution to (1.5). Then, by summation rule (2.1),

$\displaystyle\frac{d}{dt}\sum_{j\in\mathbb{Z}^{d}}u^{2}_{j}$
$\displaystyle=-2\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}|D_{-,k}u_{j}|^{2}+2\sum_{j\in\mathbb{Z}^{d}}V_{j}u^{2}_{j}\leq-2\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}|D_{-,k}u_{j}|^{2}+2\|V\|_{L^{\infty}((h\mathbb{Z})^{d}\times[0,1])}\sum_{j\in\mathbb{Z}^{d}}u^{2}_{j}.$

We multiply by $e^{-2t\|V\|_{L^{\infty}((h\mathbb{Z})^{d}\times[0,1])}}$ so
that

$\frac{d}{dt}\Big{(}e^{-2t\|V\|_{L^{\infty}((h\mathbb{Z})^{d}\times[0,1])}}\sum_{j\in\mathbb{Z}^{d}}u^{2}_{j}\Big{)}\leq-2e^{-2t\|V\|_{L^{\infty}((h\mathbb{Z})^{d}\times[0,1])}}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}|D_{-,k}u_{j}|^{2}$

and we integrate in time from $0$ to $t$ to obtain

$e^{-2t\|V\|_{L^{\infty}((h\mathbb{Z})^{d}\times[0,1])}}\sum_{j\in\mathbb{Z}^{d}}u^{2}_{j}-\sum_{j\in\mathbb{Z}^{d}}u^{2}_{j}(0)\leq-2\int_{0}^{t}e^{-2\tau\|V\|_{L^{\infty}((h\mathbb{Z})^{d}\times[0,1])}}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}|D_{-,k}u_{j}(\tau)|^{2}\,d\tau,$

whence the desired result is deduced. ∎

The second technical result is a smoothing estimate, or Caccioppoli-type
inequality for solutions to (1.5), providing interior parabolic regularity. We
will denote by $A(r_{1},r_{2})$ the region
$\\{hj\in(h\mathbb{Z})^{d}:r_{1}<|hj|<r_{2}\\}=B_{r_{2}}\setminus\overline{B_{r_{1}}}$,
with $B_{r}=B_{r}(0)\cap(h\mathbb{Z})^{d}$.

######  Lemma 2.2 (Interior parabolic regularity).

Let $u\in C^{1}([0,1]:\ell^{2}((h\mathbb{Z})^{d}))$ be a solution to the
initial value problem (1.5) with
$V:(h\mathbb{Z})^{d}\times[0,1]\rightarrow\mathbb{R}$ uniformly bounded in $h$
and $u(0)\in\ell^{2}((h\mathbb{Z})^{d})$.

Then there exist constants $C_{1}>0$ and $C_{2}>1$ depending on
$\|V\|_{L^{\infty}}$ such that, for $0<t<1$, we have

(2.3)
$C_{1}\sum\limits_{k=1}^{d}\int_{0}^{1}\|h^{-1}(u(h(\cdot+e_{j}),t)-u(h\cdot,t))\|_{\ell^{2}(A(R-1,R))}^{2}\,dt\leq
C_{2}\int_{0}^{1}\|u(h\cdot,t)\|_{\ell^{2}(A(R-2,R+1))}^{2}\,dt\\\
+\|u(\cdot,0)\|^{2}_{\ell^{2}(A(R-2,R+1))}.$

###### Proof.

Let us denote $A_{1}:=A(R-1,R)$, $A_{2}:=A(R-2,R+1)$. Let
$\eta:(h\mathbb{Z})^{d}\to[0,1]$ be a cut-off function given by $\eta=1$ in
$A_{1}$, and $\eta=0$ in $A_{2}^{c}$. We multiply equation (1.5) by
$(u\eta^{2})_{j}(t)$, apply the summation by parts formula, and subsequently
integrate it over time

(2.4)
$\displaystyle\int_{0}^{1}\sum\limits_{j\in\mathbb{Z}^{d}}\Big{[}\partial_{t}u_{j}(t)\cdot(u\eta^{2})_{j}(t)+h^{-2}\sum\limits_{k,n=1}^{d}(u_{j+e_{k}}(t)-u_{j}(t))\big{(}(u\eta^{2})_{j+e_{n}}(t)-(u\eta^{2})_{j}(t)\big{)}$
$\displaystyle\qquad-V_{j}(t)u_{j}(t)(u\eta^{2})_{j}(t)\Big{]}\,dt=0.$

Integrating by parts in the first term above, we have

$\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\partial_{t}u_{j}\cdot(u\eta^{2})_{j}\,dt=-\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}u_{j}(t)\partial_{t}(u\eta^{2})_{j}(t)\,dt+\sum_{j\in\mathbb{Z}^{d}}u_{j}(1)(u\eta^{2})_{j}(1)-\sum_{j\in\mathbb{Z}^{d}}u_{j}(0)(u\eta^{2})_{j}(0).$

Moreover, since $\eta$ does not depend on time, we compute directly the above
integral

(2.5)
$-\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}u_{j}(t)\partial_{t}(u\eta^{2})_{j}(t)\,dt=-\frac{1}{2}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\partial_{t}u_{j}^{2}(t)\cdot\eta_{j}^{2}\,dt=\frac{1}{2}\sum_{j\in\mathbb{Z}^{d}}u_{j}^{2}(0)\eta_{j}^{2}-\frac{1}{2}\sum_{j\in\mathbb{Z}^{d}}u_{j}^{2}(1)\eta_{j}^{2}.$

Hence, (2.4) reads as

$\displaystyle-\frac{1}{2}\sum_{j\in\mathbb{Z}^{d}}u^{2}_{j}(0)\eta_{j}^{2}+\frac{1}{2}\sum_{j\in\mathbb{Z}^{d}}u_{j}^{2}(1)\eta_{j}^{2}+\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\Big{[}h^{-2}\sum\limits_{k,n=1}^{d}(u_{j+e_{k}}(t)-u_{j}(t))((u\eta^{2})_{j+e_{n}}(t)-(u\eta^{2})_{j}(t))$
$\displaystyle\qquad-V_{j}(t)u_{j}(t)(u\eta^{2})_{j}(t)\Big{]}\,dt=0.$

Multiplying by $2$, expanding and rearranging, we have

$\displaystyle\sum_{j\in\mathbb{Z}^{d}}u^{2}_{j}(1)\eta_{j}^{2}+2\int_{0}^{1}h^{-2}\sum_{j\in\mathbb{Z}^{d}}\sum\limits_{k,n=1}^{d}(u_{j+e_{k}}(t)-u_{j}(t))(u_{j+e_{n}}(t)-u_{j}(t))\eta^{2}_{j}(t)\,dt$
$\displaystyle=\sum_{j\in\mathbb{Z}^{d}}u^{2}_{j}(0)\eta_{j}^{2}-2\int_{0}^{1}h^{-2}\sum_{j\in\mathbb{Z}^{d}}\sum\limits_{k,n=1}^{d}(u_{j+e_{k}}(t)-u_{j}(t))u_{j+e_{n}}(t)(\eta_{j+e_{n}}^{2}(t)-\eta_{j}^{2}(t))\,dt$
$\displaystyle\quad+2\int_{0}^{1}h^{-1}\sum_{j\in\mathbb{Z}^{d}}V_{j}(t)u_{j}(t)(u\eta^{2})_{j}(t)\,dt.$

We multiply by $h^{d}$ and observe that we are boiled down to handling the
same terms as in [FBRRS20, Lemma 4.1, (30)] with $a_{k,n}=1$. Then, proceeding
as in the proof of [FBRRS20, Lemma 4.1], we obtain

$\displaystyle
h^{d}\sum_{j\in\mathbb{Z}^{d}}u_{j}^{2}(1)\eta_{j}^{2}+\int_{0}^{1}\sum\limits_{k=1}^{d}\|h^{-1}(u(\cdot+he_{k},t)-u(\cdot,t))\eta\|^{2}_{\ell^{2}((h\mathbb{Z})^{d})}\,dt$
$\displaystyle\leq
h^{d}\sum_{j\in\mathbb{Z}^{d}}u_{j}^{2}(0)\eta_{j}^{2}+\frac{1}{4}\int_{0}^{1}\sum\limits_{k=1}^{d}\|h^{-1}(u(\cdot+he_{k},t)-u(\cdot,t))\eta\|^{2}_{\ell^{2}((h\mathbb{Z})^{d})}\,dt+C_{2}\int_{0}^{1}\|u(\cdot,t)\|^{2}_{\ell^{2}(A_{2})}\,dt,$

where $C_{2}$ depends on $\|V\|_{l^{\infty}(A_{2})}$. From here, we conclude

$C_{1}\int_{0}^{1}\sum\limits_{k=1}^{d}\|h^{-1}(u(\cdot+he_{k},t)-u(h\cdot,t))\|^{2}_{\ell^{2}(A_{1})}\,dt\\\
\leq
h^{d}\sum_{j\in\mathbb{Z}^{d}}u_{j}^{2}(0)\eta_{j}^{2}-h^{d}\sum_{j\in\mathbb{Z}^{d}}u_{j}^{2}(1)\eta_{j}^{2}+C_{2}\int_{0}^{1}\|u(\cdot,t)\|^{2}_{\ell^{2}(A_{2})}\,dt.$

as desired. ∎

##  3\. Semidiscrete heat equation: upper bounds

This section is dedicated to proving the upper bounds in each of the two
scenarios: when $Rh$ small and when $Rh$ large. In Subsection 3.3 we supply
the proof of Theorem 1. The main ingredient of the proof is a weighted log-
convexity property of the solutions to (1.5), which we will establish in
Subsection 3.2. Specifically, in Proposition 3.5 we demonstrate that the
quantity

(3.1)
$\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}\frac{K_{j_{k}}^{2}(\frac{\gamma}{h^{2}})}{K_{0}^{2}(\frac{\gamma}{h^{2}})}|u_{j}(t)|^{2}$

is logarithmically convex with respect to the time variable. Thus, we can
control this quantity by the corresponding weighted norms for the solution $u$
to (1.5) at times $0$ and $1$, as stated in (1.7). Our strategy is as follows:
firstly, we establish the well-defined nature of the quantity

$H_{\delta}(t):=\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}K_{j_{k}}^{2\delta}\big{(}\frac{\gamma}{h^{2}}\big{)}|u_{j}(t)|^{2},\quad\text{for
}t\in[0,1].$

This step relies on weighted energy estimates, detailed in Proposition 3.1 in
Subsection 3.1. Subsequently, we prove its log-convexity in Proposition 3.4.
Next, in Proposition 3.5 we conclude that the above energy (3.1) is finite by
using the upper bound for $H_{\delta}(t)$ and allowing $\delta\to 1$. Finally,
we employ an abstract convexity result from [EKPV08] (which we recall here in
Lemma 3.3) in order to derive the desired log-convexity estimate.

In Subsection 3.4, we will provide the proof of Theorem 2.

###  3.1. Weighted energy estimates

######  Proposition 3.1.

Let $u\in C^{1}([0,1]:\ell^{2}((h\mathbb{Z})^{d}))$ be a solution to the
initial value problem (1.5) with $V\in
L^{\infty}((h\mathbb{Z})^{d}\times\mathbb{R}_{+})$ and
$u(0)\in\ell^{2}((h\mathbb{Z})^{d})$. Let

$H(t)=\sum_{j\in\mathbb{Z}^{d}}|\omega_{j}u_{j}|^{2}\quad\text{ with
}\quad\omega_{j}=\omega(hj,t):=\prod_{k=1}^{d}K_{j_{k}}\big{(}\frac{\gamma}{h^{2}}+\frac{2t}{h^{2}}\big{)},$

for some $\gamma>0$. Then $H(t)\leq e^{2t\|V\|_{\infty}}H(0)$.

###### Proof.

We split the proof into two parts. In the first part we assume that all the
weighted energy terms are finite and we perform some formal computations in
order to derive the desired inequality. In the second part of the proof we
validate the assertions made in the first part by employing a truncation
argument.

Part I. Let $f_{j}:=\omega_{j}u_{j}$. Then
$\partial_{t}f_{j}=Sf_{j}+Af_{j}+V_{j}f_{j}$, where

(3.2)
$Sf_{j}=\frac{\partial_{t}\omega_{j}}{\omega_{j}}f_{j}+\frac{1}{2}\omega_{j}\Delta_{\operatorname{\operatorname{d}}}\Big{(}\frac{f_{j}}{\omega_{j}}\Big{)}+\frac{1}{2}\frac{1}{\omega_{j}}\,\Delta_{\operatorname{\operatorname{d}}}(f_{j}\omega_{j})$

and

(3.3)
$Af_{j}=\frac{1}{2}\omega_{j}\Delta_{\operatorname{\operatorname{d}}}\Big{(}\frac{f_{j}}{\omega_{j}}\Big{)}-\frac{1}{2}\frac{1}{\omega_{j}}\Delta_{\operatorname{\operatorname{d}}}(f_{j}\omega_{j}).$

Notice that $S$ is a symmetric operator, while $A$ is an antisymmetric
operator. Then

$H^{\prime}(t)=2\sum_{j\in\mathbb{Z}^{d}}f_{j}(f_{j})_{t}=2\operatorname{Re}\sum_{j\in\mathbb{Z}^{d}}f_{j}Sf_{j}+2\sum_{j\in\mathbb{Z}^{d}}V_{j}f_{j}^{2},$

since $\sum_{j\in\mathbb{Z}^{d}}f_{j}Af_{j}=0$. By using (3.2), we can deduce
that

$\displaystyle
H^{\prime}(t)=2\sum_{j\in\mathbb{Z}^{d}}\frac{\partial_{t}\omega_{j}}{\omega_{j}}f_{j}^{2}+\sum_{j\in\mathbb{Z}^{d}}\omega_{j}\Delta_{\operatorname{\operatorname{d}}}\Big{(}\frac{f_{j}}{\omega_{j}}\Big{)}f_{j}+\sum_{j\in\mathbb{Z}^{d}}\frac{1}{\omega_{j}}\Delta_{\operatorname{\operatorname{d}}}(f_{j}\omega_{j})f_{j}+2\sum_{j\in\mathbb{Z}^{d}}V_{j}f_{j}^{2}$
$\displaystyle=2\sum_{j\in\mathbb{Z}^{d}}\frac{\partial_{t}\omega_{j}}{\omega_{j}}f_{j}^{2}-\frac{4d}{h^{2}}\sum_{j\in\mathbb{Z}^{d}}f_{j}^{2}+\frac{2}{h^{2}}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\frac{\omega_{j}}{\omega_{j+e_{k}}}f_{j+e_{k}}f_{j}+\frac{2}{h^{2}}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\frac{\omega_{j+e_{k}}}{\omega_{j}}f_{j+e_{k}}f_{j}+2\sum_{j\in\mathbb{Z}^{d}}V_{j}f_{j}^{2}$
$\displaystyle\leq
2\sum_{j\in\mathbb{Z}^{d}}\frac{\partial_{t}\omega_{j}}{\omega_{j}}f_{j}^{2}-\frac{4d}{h^{2}}\sum_{j\in\mathbb{Z}^{d}}f_{j}^{2}+\frac{1}{h^{2}}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\frac{\omega_{j-e_{k}}}{\omega_{j}}f_{j}^{2}$
$\displaystyle\qquad+\frac{1}{h^{2}}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\frac{\omega_{j}}{\omega_{j+e_{k}}}f_{j}^{2}+\frac{1}{h^{2}}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\frac{\omega_{j}}{\omega_{j-e_{k}}}f_{j}^{2}+\frac{1}{h^{2}}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\frac{\omega_{j+e_{k}}}{\omega_{j}}f_{j}^{2}+2\sum_{j\in\mathbb{Z}^{d}}V_{j}f_{j}^{2},$

where we employed the inequality $2f_{j+e_{k}}f_{j}\leq
f_{j+e_{k}}^{2}+f_{j}^{2}$. Let
$\omega_{j}=\prod_{k=1}^{d}K_{j_{k}}\big{(}\frac{\gamma}{h^{2}}+\frac{2t}{h^{2}}\big{)}$
as in the hypothesis. Thus,

$\displaystyle\partial_{t}\omega_{j}$
$\displaystyle=\frac{2}{h^{2}}\sum_{k=1}^{d}K_{j_{k}}^{\prime}\big{(}\frac{\gamma}{h^{2}}+\frac{2t}{h^{2}}\big{)}\prod_{m\neq
k,m=1}^{d}K_{j_{m}}\big{(}\frac{\gamma}{h^{2}}+\frac{2t}{h^{2}}\big{)}$
$\displaystyle=\frac{2}{h^{2}}\sum_{k=1}^{d}\frac{K_{j_{k}}^{\prime}\big{(}\frac{\gamma}{h^{2}}+\frac{2t}{h^{2}}\big{)}}{K_{j_{k}}\big{(}\frac{\gamma}{h^{2}}+\frac{2t}{h^{2}}\big{)}}\prod_{m=1}^{d}K_{j_{m}}\big{(}\frac{\gamma}{h^{2}}+\frac{2t}{h^{2}}\big{)}=\frac{2}{h^{2}}\sum_{k=1}^{d}\frac{K_{j_{k}}^{\prime}\big{(}\frac{\gamma}{h^{2}}+\frac{2t}{h^{2}}\big{)}}{K_{j_{k}}\big{(}\frac{\gamma}{h^{2}}+\frac{2t}{h^{2}}\big{)}}\omega_{j},$

where $K_{j_{k}}^{\prime}(z)$ denotes $\partial_{z}K_{j_{k}}(z).$ Thus, we can
write

$H^{\prime}(t)\leq\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}E_{j,k}f_{j}^{2}-\frac{4d}{h^{2}}H(t)+2\sum_{j\in\mathbb{Z}^{d}}V_{j}f_{j}^{2},$

with

$E_{j,k}=\frac{4}{h^{2}}\frac{K_{j_{k}}^{\prime}}{K_{j_{k}}}+\frac{1}{h^{2}}\frac{K_{j_{k}-1}}{K_{j_{k}}}+\frac{1}{h^{2}}\frac{K_{j_{k}}}{K_{j_{k}+1}}+\frac{1}{h^{2}}\frac{K_{j_{k}}}{K_{j_{k}-1}}+\frac{1}{h^{2}}\frac{K_{j_{k}+1}}{K_{j_{k}}},$

where all the Bessel functions are evaluated at
$z=\frac{\gamma}{h^{2}}+\frac{2t}{h^{2}}$. We check now that $E_{j,k}\leq 0$.
Using the recurrence formula (B.4), we obtain

$\displaystyle E_{j,k}$
$\displaystyle=-\frac{2}{h^{2}}\frac{K_{j_{k}+1}}{K_{j_{k}}}-\frac{2}{h^{2}}\frac{K_{j_{k}-1}}{K_{j_{k}}}+\frac{1}{h^{2}}\frac{K_{j_{k}-1}}{K_{j_{k}}}+\frac{1}{h^{2}}\frac{K_{j_{k}}}{K_{j_{k}+1}}+\frac{1}{h^{2}}\frac{K_{j_{k}}}{K_{j_{k}-1}}+\frac{1}{h^{2}}\frac{K_{j_{k}+1}}{K_{j_{k}}}$
$\displaystyle=\frac{1}{h^{2}}\Big{(}-\frac{K_{j_{k}+1}}{K_{j_{k}}}+\frac{K_{j_{k}}}{K_{j_{k}-1}}\Big{)}+\frac{1}{h^{2}}\Big{(}-\frac{K_{j_{k}-1}}{K_{j_{k}}}+\frac{K_{j_{k}}}{K_{j_{k}+1}}\Big{)}.$

By Turán inequality (B.6), we can deduce that $E_{j,k}\leq 0$. Thus, we have

$H^{\prime}(t)\leq\Big{(}2\|V\|_{\infty}-\frac{4d}{h^{2}}\Big{)}H(t)\leq
2\|V\|_{\infty}H(t)\leq 2\|V\|_{\infty}H(t),$

and integrating the above inequality from $0$ to $t$ for $t>0$, we deduce that
the quantity $e^{-2t\|V\|_{\infty}}H(t)$ is monotone decreasing in time.
Therefore, the conclusion follows.

Part II. For $m\in\mathbb{Z}$, let us define $\psi^{R}_{m}$ by

$\psi^{R}_{m}:=\begin{cases}K_{m}\big{(}\frac{\gamma}{h^{2}}+\frac{2t}{h^{2}}\big{)}\mbox{
if }|m|\leq R,\\\\[5.69054pt]
K_{R}\big{(}\frac{\gamma}{h^{2}}+\frac{2t}{h^{2}}\big{)}\mbox{ if }|m|\geq
R,\end{cases}$

and let, for $j\in\mathbb{Z}^{d}$,
$\omega_{j}^{R}=\prod_{k=1}^{d}\psi^{R}_{j_{k}}$. Observe that all the
inequalities in Part I, up to the choice of the weight $w_{j}$, are valid in
this case and they are finite since we work with the truncated weight
$\omega_{j}^{R}$, for each $j\in\mathbb{Z}^{d}$. We now continue the argument
in Part I, but instead of choosing
$\omega_{j}=\prod_{k=1}^{d}K_{j_{k}}\big{(}\frac{\gamma}{h^{2}}+\frac{2t}{h^{2}}\big{)}$,
we carry out the computations with the weight $\omega_{j}^{R}$ just defined.
Let $H_{R}(t)=\sum_{j\in\mathbb{Z}^{d}}|\omega^{R}_{j}u_{j}|^{2}$. Arguing as
above we get

$\displaystyle H_{R}^{\prime}(t)$ $\displaystyle\leq
2\sum_{j\in\mathbb{Z}^{d}}\frac{\partial_{t}\omega^{R}_{j}}{\omega^{R}_{j}}f_{j}^{2}-\frac{4d}{h^{2}}\sum_{j\in\mathbb{Z}^{d}}f_{j}^{2}+\frac{1}{h^{2}}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\frac{\omega^{R}_{j-e_{k}}}{\omega^{R}_{j}}f_{j}^{2}+\frac{1}{h^{2}}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\frac{\omega^{R}_{j}}{\omega^{R}_{j+e_{k}}}f_{j}^{2}$
$\displaystyle\qquad+\frac{1}{h^{2}}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\frac{\omega^{R}_{j}}{\omega^{R}_{j-e_{k}}}f_{j}^{2}+\frac{1}{h^{2}}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\frac{\omega^{R}_{j+e_{k}}}{\omega^{R}_{j}}f_{j}^{2}+2\sum_{j\in\mathbb{Z}^{d}}V_{j}f_{j}^{2}$
$\displaystyle\leq-\frac{4d}{h^{2}}H_{R}(t)+2\|V\|_{\infty}H_{R}(t)+\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}E^{R}_{j,k}f_{j}^{2},$

where the quantities $E^{R}_{j,k}$ enclose the error terms. After tedious
computations, it is possible to check that for every $k\in\\{1,\dots,d\\}$ the
error terms are bounded as $E^{R}_{j,k}\leq\frac{2}{h^{2}}$, so the conclusion
of the proof is true for $H_{R}$. Finally, we let $R$ tend to $\infty$ to
complete the proof. ∎

###  3.2. Log-convexity estimates

First, we require a technical result.

######  Lemma 3.2.

Let $\delta\in[0,1]$. For any $x>0$ and any $j\in\mathbb{Z}$, the following
holds

(3.4)
$\Lambda_{\delta}:=\frac{K_{j+1}^{2\delta}(x)}{K_{j}^{2\delta}(x)}+\frac{K_{j-1}^{2\delta}(x)}{K_{j}^{2\delta}(x)}-\frac{K_{j}^{2\delta}(x)}{K_{j-1}^{2\delta}(x)}-\frac{K_{j}^{2\delta}(x)}{K_{j+1}^{2\delta}(x)}\\\
+\frac{K_{j+1}^{2\delta}(x)}{K_{j}^{\delta}(x)K_{j+2}^{\delta}(x)}-\frac{K_{j}^{\delta}(x)K_{j+2}^{\delta}(x)}{K_{j+1}^{2\delta}(x)}+\frac{K_{j-1}^{2\delta}(x)}{K_{j}^{\delta}(x)K_{j-2}^{\delta}(x)}-\frac{K_{j}^{\delta}(x)K_{j-2}^{\delta}(x)}{K_{j-1}^{2\delta}(x)}\geq-2\Big{(}1+\frac{1}{x}+\frac{1}{4x^{3}}\Big{)}.$

Moreover, when $\delta=1$, $\Lambda_{1}>0$.

###### Proof.

Using inequality (B.6), we have that

$\frac{K_{j+1}^{2\delta}}{K_{j}^{2\delta}}\geq\frac{K_{j}^{2\delta}}{K_{j-1}^{2\delta}}\quad\text{and}\quad\frac{K_{j-1}^{2\delta}}{K_{j}^{2\delta}}\geq\frac{K_{j}^{2\delta}}{K_{j+1}^{2\delta}}.$

For the remaining terms, we combine the bounds (B.7) and (B.8) to conclude
that
$\displaystyle\frac{K_{j+1}^{2}(x)}{K_{j}(x)K_{j+2}(x)}\geq\frac{1}{1+\frac{1}{x}+\frac{1}{4x^{3}}}$
for all $j\in\mathbb{Z}.$ Thus, by estimating from below the negative terms in
the second line of (3.4), we obtain that

$-\frac{K_{j}^{\delta}(x)K_{j+2}^{\delta}(x)}{K_{j+1}^{2\delta}(x)}-\frac{K_{j}^{\delta}(x)K_{j-2}^{\delta}(x)}{K_{j-1}^{2\delta}(x)}\geq-2\Big{(}1+\frac{1}{x}+\frac{1}{4x^{3}}\Big{)}^{\delta}\geq-2\Big{(}1+\frac{1}{x}+\frac{1}{4x^{3}}\Big{)},$

for any $x>0$ and $j\in\mathbb{Z}$.

When $\delta=1$, it was proved in [FB18, p. 269] that $\Lambda_{1}>0$. ∎

It will be convenient to recall here part of [EKPV08, Lemma 2] which relates
the log-convexity of the $L^{2}$-norm of a function with a weak “pseudo-
positivity” condition on the commutator of symmetric and antisymmetric part of
an operator.

######  Lemma 3.3 ([EKPV08, Lemma 2]).

Let $S$ be a symmetric operator and let $A$ be an antisymmetric operator, both
allowed to depend on the time variable. Let $G$ be a positive function,
$f(x,t)$ be a reasonable function,

$H(t)=\langle f,f\rangle,\quad D(t)=\langle
Sf,f\rangle,\quad\partial_{t}S=S_{t}.$

Then, if

$|\partial_{t}f-Af-Sf|\leq M_{1}|f|+G\quad\text{ in
}\mathbb{R}^{n}\times[0,1],\quad S_{t}+[S,A]\geq-M_{0},$

and $M_{2}=\sup_{[0,1]}\|G(t)\|/\|f(t)\|$ is finite, then $\log H(t)$ is
“convex” in $[0,1]$, in the sense that there is a universal constant $N$ such
that

$H(t)\leq
e^{N(M_{0}+M_{1}+M_{2}+M_{1}^{2}+M_{2}^{2})}H(0)^{1-t}H(1)^{t},\text{ when
}0\leq t\leq 1.$

######  Proposition 3.4.

Let $\delta\in(0,1),\,h,\gamma>0$ and define

$H_{\delta}(t):=\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}K_{j_{k}}^{2\delta}\big{(}\frac{\gamma}{h^{2}}\big{)}|u_{j}(t)|^{2},\quad\text{for
}t\in[0,1].$

Then, $H_{\delta}$ is log-convex, in the sense there exists a constant
$N_{1}>0$ such that

$H_{\delta}(t)\leq
e^{N_{1}(\|V\|_{\infty}+\frac{4d}{h^{4}}(1+\frac{h^{2}}{\gamma}+\frac{h^{6}}{4\gamma^{3}}))}H_{\delta}(0)^{1-t}H_{\delta}(1)^{t},$

for every $t\in[0,1]$.

###### Proof.

First of all, we need to justify that $H_{\delta}(t)$ is well defined. This
follows using Proposition 3.1 and the following claim

(3.5)
$K_{j}^{\delta}\Big{(}\frac{\gamma}{h^{2}}\Big{)}<cK_{j}\Big{(}\frac{\gamma}{h^{2}}+\frac{2t}{h^{2}}\Big{)}\quad\text{for
all }t\in[0,1],\,\,j\in\mathbb{Z},$

for some positive $c$. Let us prove the claim. First, we argue for large $j$.
Let $t=1$. We utilize the asymptotic approximation for large order (B.10).
Therefore, there exists a $j_{0}\in\mathbb{Z}$ such that, for $j\geq j_{0}$,
the following estimate holds

$\frac{K_{j}(\frac{\gamma}{h^{2}}+\frac{2}{h^{2}})}{K_{j}^{\delta}(\frac{\gamma}{h^{2}})}\sim
j^{j(1-\delta)}j^{-\frac{1}{2}(1-\delta)}2^{j(1-\delta)}e^{-j(1-\delta)}\bigg{(}\frac{\big{(}\frac{\gamma}{h^{2}}\big{)}^{\delta}}{\frac{\gamma}{h^{2}}+\frac{2}{h^{2}}}\bigg{)}^{j}.$

First, we will prove that this quantity is larger than $1$ for large $j.$ The
dominant term is the first one, $j^{j(1-\delta)}$, and we will use it to
control all the small terms. Indeed, we have the lower bound

$\frac{K_{j}(\frac{\gamma}{h^{2}}+\frac{2}{h^{2}})}{K_{j}^{\delta}(\frac{\gamma}{h^{2}})}\geq
j^{\frac{1}{4}j(1-\delta)}j^{(\frac{1}{4}j-\frac{1}{2})(1-\delta)}\Big{(}2\frac{j^{\frac{1}{4}}}{e}\Big{)}^{j(1-\delta)}\Bigg{(}\frac{j^{\frac{1}{4}(1-\delta)}}{\frac{\frac{\gamma}{h^{2}}+\frac{2}{h^{2}}}{(\frac{\gamma}{h^{2}})^{\delta}}}\Bigg{)}^{j}.$

Let $j_{1}$ be such that
$j_{1}^{\frac{1}{4}(1-\delta)}\geq\frac{\frac{\gamma}{h^{2}}+\frac{2}{h^{2}}}{(\frac{\gamma}{h^{2}})^{\delta}}$.
For $j\geq\max\\{j_{0},2,e^{4},j_{1}\\}$ we have that
$\frac{K_{j}(\frac{\gamma}{h^{2}}+\frac{2}{h^{2}})}{K_{j}^{\delta}(\frac{\gamma}{h^{2}})}\geq
j^{\frac{1}{4}j(1-\delta)}>1$. Now, we argue for smaller values of $j$. For
$j\in(0,\max\\{j_{0},2,e^{4},j_{1}\\})\cap\mathbb{Z}$, by the positivity of
$K_{j}(x)$, there exists $b>0$ such that
$\frac{K_{j}(\frac{\gamma}{h^{2}}+\frac{2}{h^{2}})}{K_{j}^{\delta}(\frac{\gamma}{h^{2}})}\geq
b$. Thus, for all $j\in\mathbb{Z}$ we have that
$\frac{K_{j}(\frac{\gamma}{h^{2}}+\frac{2}{h^{2}})}{K_{j}^{\delta}(\frac{\gamma}{h^{2}})}\geq\min\\{1,b\\}$.
Since $K_{j}(\cdot)$ is monotone decreasing, we conclude that

$\frac{K_{j}(\frac{\gamma}{h^{2}}+\frac{2t}{h^{2}})}{K_{j}^{\delta}(\frac{\gamma}{h^{2}})}\geq\min\\{1,b\\},\quad\text{for
all }t\in[0,1],\,\,j\in\mathbb{Z}.$

Finally, by taking $\displaystyle c=\frac{1}{\min\\{1,b\\}}$, we conclude the
proof of inequality (3.5).

We continue the proof of Proposition 3.4. We will make use of [EKPV08, Lemma
2] (that we reproduced in Lemma 3.3 above). Let $f_{j}:=\omega_{j}u_{j}$, with

(3.6) $\omega_{j}=\prod_{k=1}^{d}\omega_{j_{k}}\quad\text{with
}\omega_{j_{k}}:=K_{j_{k}}^{\delta}\big{(}\frac{\gamma}{h^{2}}\big{)}.$

Then $\partial_{t}f_{j}=Sf_{j}+Af_{j}+V_{j}f_{j}$ where $S$ and $A$ are
defined in (3.2) and (3.3). Notice that the weight $\omega$ in this situation
does not depend on time.

Moreover, since the weight is given by a tensorial product, we can rewrite $S$
and $A$ in an equivalent form as $S=\sum_{k=1}^{d}S_{k}$ and
$A=\sum_{k=1}^{d}A_{k}$, with

$S_{k}f_{j}=\frac{1}{2}\omega_{j_{k}}\Delta_{\operatorname{\operatorname{d}},k}\Big{(}\frac{f_{j}}{\omega_{j_{k}}}\Big{)}+\frac{1}{2}\frac{1}{\omega_{j_{k}}}\,\Delta_{\operatorname{\operatorname{d}},k}(f_{j}\omega_{j_{k}}),\qquad
A_{k}f_{j}=\frac{1}{2}\omega_{j_{k}}\Delta_{\operatorname{\operatorname{d}},k}\Big{(}\frac{f_{j}}{\omega_{j_{k}}}\Big{)}-\frac{1}{2}\frac{1}{\omega_{j_{k}}}\Delta_{\operatorname{\operatorname{d}},k}(f_{j}\omega_{j_{k}}).$

After a tedious computation, we obtain that $[S_{k},A_{m}]f_{j}=0$ for all
$k\neq m$, and all $j\in\mathbb{Z}^{d}.$ Thus, the commutator’s expression
reduces to $[S,A]f_{j}=\sum_{k=1}^{d}[S_{k},A_{k}]f_{j}$. We verify that the
conditions of [EKPV08, Lemma 2] are satisfied, namely:
$|\partial_{t}f_{j}-Sf_{j}-Af_{j}|\leq M_{1}|f_{j}|$ and
$\langle[S,A]f_{j},f_{j}\rangle\geq-M_{0}\|f_{j}\|_{2}^{2}$ (note that the
weight $w_{j}$ does not depend on the variable $t$, so the action of $S_{t}$
is irrelevant here) for some constants $M_{0},M_{1}>0$. The first estimate can
be easily derived since the potential $V_{j}$ is bounded, namely
$|\partial_{t}f_{j}-Sf_{j}-Af_{j}|=|V_{j}f_{j}|\leq\|V_{j}\|_{\infty}|f_{j}|$.

For the second estimate, we notice that

$\displaystyle\sum_{j\in\mathbb{Z}^{d}}\langle[S,A]f_{j},f_{j}\rangle$
$\displaystyle=\sum_{k=1}^{d}\sum_{j\in\mathbb{Z}^{d}}\langle[S_{k},A_{k}]f_{j},f_{j}\rangle=2\operatorname{Re}\sum_{k=1}^{d}\sum_{j\in\mathbb{Z}^{d}}\langle
S_{k}f_{j},A_{k}f_{j}\rangle$
$\displaystyle=\frac{1}{2}\sum_{k=1}^{d}\Bigg{[}\sum_{j\in\mathbb{Z}^{d}}\Big{(}\omega_{j_{k}}\Delta_{\operatorname{\operatorname{d}},k}\Big{(}\frac{f_{j}}{\omega_{j_{k}}}\Big{)}\Big{)}^{2}-\sum_{j\in\mathbb{Z}^{d}}\Big{(}\frac{1}{\omega_{j_{k}}}\Delta_{\operatorname{\operatorname{d}},k}\big{(}f_{j}\omega_{j_{k}}\big{)}\Big{)}^{2}\Bigg{]}=:\frac{1}{2}\sum_{k=1}^{d}\Lambda_{k}.$

We prove that each $\Lambda_{k}$ is bounded from below by
$-c\|f_{j}\|_{2}^{2}$ for some $c>0$. Indeed, we have that

$\displaystyle\Lambda_{k}$
$\displaystyle=\frac{1}{h^{4}}\sum_{j\in\mathbb{Z}^{d}}\Big{(}\frac{\omega_{j_{k}-e_{k}}\omega_{j_{k}+e_{k}}}{\omega^{2}_{j_{k}}}-\frac{\omega^{2}_{j_{k}}}{\omega_{j_{k}-e_{k}}\omega_{{j_{k}}+e_{k}}}\Big{)}(f_{j+e_{k}}-f_{j_{-}e_{k}})^{2}+\frac{1}{h^{4}}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\Big{(}\frac{\omega^{2}_{j_{k}-e_{k}}}{\omega^{2}_{j_{k}}}+\frac{\omega^{2}_{j_{k}+e_{k}}}{\omega^{2}_{j_{k}}}$
$\displaystyle\qquad-\frac{\omega^{2}_{j_{k}}}{\omega^{2}_{j_{k}-e_{k}}}-\frac{\omega^{2}_{j_{k}}}{\omega^{2}_{j_{k}+e_{k}}}+\frac{\omega^{2}_{j_{k}-e_{k}}}{\omega_{j_{k}}\omega_{j_{k}-2e_{k}}}-\frac{\omega_{j_{k}}\omega_{j_{k}+2e_{k}}}{\omega^{2}_{j_{k}+e_{k}}}-\frac{\omega_{j_{k}}\omega_{j_{k}-2e_{k}}}{\omega^{2}_{j_{k}-e_{k}}}+\frac{\omega^{2}_{j_{k}+e_{k}}}{\omega_{j_{k}}\omega_{j_{k}+2e_{k}}}\Big{)}f^{2}_{j}.$

Now we take into account the definition of the weight $\omega_{j}$ in (3.6).
The first sum is positive since, by the Turán inequality (B.6), the following
coefficient is positive, namely
$\frac{\omega_{j_{k}-e_{k}}\omega_{j_{k}+e_{k}}}{\omega^{2}_{j_{k}}}-\frac{\omega^{2}_{j_{k}}}{\omega_{{j_{k}}-e_{k}}\omega_{{j_{k}}+e_{k}}}>0$.
The coefficient in the second sum can be rewritten in terms of the Macdonald’s
function $K_{{j_{k}}}$, and, by Lemma 3.2, this quantity is bounded from below
by $-2(1+\frac{h^{2}}{\gamma}+\frac{h^{6}}{4\gamma^{3}}).$ Thus

$\langle[S,A]f,f\rangle=\sum_{k=1}^{d}\sum_{j\in\mathbb{Z}^{d}}\langle[S_{k},A_{k}]f_{j},f_{j}\rangle\geq-\frac{4d}{h^{4}}\Big{(}1+\frac{h^{2}}{\gamma}+\frac{h^{6}}{4\gamma^{3}}\Big{)}\|f\|_{2}^{2}.$

Thus, by [EKPV08, Lemma 2], there is a universal constant $N_{1}$ such that

$H_{\delta}(t)\leq
e^{N_{1}(\|V\|_{\infty}+\frac{4d}{h^{4}}(1+\frac{h^{2}}{\gamma}+\frac{h^{6}}{4\gamma^{3}}))}H_{\delta}(0)^{1-t}H_{\delta}(1)^{t}.$

The proof is complete. ∎

The fact that the lower bound has a bad dependence on $h$ in Proposition 3.4
(it blows up as $h\to 0$) can be improved. Indeed, the latter proposition can
be used to justify the previous computations in the case $\delta=1$. In [FB18,
Theorem 2.1], the positivity of the commutator (which is the same for both
heat and Schrödinger evolutions) is proved, and therefore we have the
following

######  Proposition 3.5.

Let $u$ be a solution to (1.5). Assume that

(3.7)
$h^{d}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}\Big{(}\frac{K_{j_{k}}^{2}(\frac{\gamma}{h^{2}})}{K_{0}^{2}(\frac{\gamma}{h^{2}})}\Big{)}\big{(}u^{2}_{j}(0)+u_{j}^{2}(1)\big{)}<c$

for a positive constant independent of $h$. Then, there exists a constant
$N_{2}>0$ such that, for every $t\in(0,1)$, the following log-convexity
estimate holds

$h^{d}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}\Big{(}\frac{K_{j_{k}}^{2}(\frac{\gamma}{h^{2}})}{K_{0}^{2}(\frac{\gamma}{h^{2}})}\Big{)}u^{2}_{j}(t)\\\
\leq
e^{N_{2}\|V\|_{\infty}}h^{d}\Big{[}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}\Big{(}\frac{K_{j_{k}}^{2}(\frac{\gamma}{h^{2}})}{K_{0}^{2}(\frac{\gamma}{h^{2}})}\Big{)}u^{2}_{j}(0)\Big{]}^{1-t}\Big{[}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}\Big{(}\frac{K_{j_{k}}^{2}(\frac{\gamma}{h^{2}})}{K_{0}^{2}(\frac{\gamma}{h^{2}})}\Big{)}u_{j}^{2}(1)\Big{]}^{t}.$

###### Proof.

Let
$H(t):=\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}\frac{K_{j_{k}}^{2}(\frac{\gamma}{h^{2}})}{K_{0}^{2}(\frac{\gamma}{h^{2}})}u_{j}^{2}(t)$.
By Fatou’s Lemma, Proposition 3.4 and Dominated Convergence Theorem, we have
that

$\displaystyle\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}K_{j_{k}}^{2}\big{(}\frac{\gamma}{h^{2}}\big{)}u^{2}_{j}=\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}\lim_{\delta\to
1}K_{j_{k}}^{2\delta}\big{(}\frac{\gamma}{h^{2}}\big{)}u^{2}_{j}\leq\lim_{\delta\to
1}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}K_{j_{k}}^{2\delta}\big{(}\frac{\gamma}{h^{2}}\big{)}u^{2}_{j}$
$\displaystyle\quad\leq\lim_{\delta\to
1}e^{N(\|V\|_{\infty}+\frac{4d}{h^{4}}(1+\frac{h^{2}}{\gamma}+\frac{h^{6}}{4\gamma^{3}}))}\Big{[}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}K_{j_{k}}^{2\delta}\big{(}\frac{\gamma}{h^{2}}\big{)}u^{2}_{j}(0)\Big{]}^{1-t}\Big{[}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}K_{j_{k}}^{2\delta}\big{(}\frac{\gamma}{h^{2}}\big{)}u_{j}^{2}(1)\Big{]}^{t}$
$\displaystyle\quad=e^{N(\|V\|_{\infty}+\frac{4d}{h^{4}}(1+\frac{h^{2}}{\gamma}+\frac{h^{6}}{4\gamma^{3}}))}\Big{[}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}K_{j_{k}}^{2}\big{(}\frac{\gamma}{h^{2}}\big{)}u^{2}_{j}(0)\Big{]}^{1-t}\Big{[}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}K_{j_{k}}^{2}\big{(}\frac{\gamma}{h^{2}}\big{)}u_{j}^{2}(1)\Big{]}^{t}.$

Thus, by virtue of (3.7), we proved that
$\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}K_{j_{k}}^{2}(\frac{\gamma}{h^{2}})u^{2}_{j}$
is finite for all $0\leq t\leq 1$. This allows us to prove log-convexity for
the quantity $H(t)$. We follow the same approach as in the proof of
Proposition 3.4 and utilize the fact that $\Lambda_{1}>0$. Here, $\Lambda_{1}$
is defined in equation (3.4). Thus, in this case, we have
$\langle[S,A]f,f\rangle\geq 0$. Furthermore, by employing Lemma 3.3 once
again, we can establish the existence of a universal constant $N_{2}$ such
that the following inequality holds

$H(t)\leq e^{N_{2}\|V\|_{\infty}}H(0)^{1-t}H(1)^{t}\quad\text{for all
}t\in[0,1].$

With this, we conclude the proof. ∎

###  3.3. Upper bound: proof of Theorem 1 (case $Rh$ small).

Observe that Proposition 3.5 guarantees that for all $t\in[0,1]$, the quantity
$h^{d}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}\big{(}\frac{K_{j_{k}}^{2}(\frac{\gamma}{h^{2}})}{K_{0}^{2}(\frac{\gamma}{h^{2}})}\big{)}|u_{j}(t)|^{2}$
is finite. Consequently,

$\displaystyle h^{d}\int_{0}^{1}\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|jh|<R+1\end{subarray}}|u_{j}(t)|^{2}\,dt$
$\displaystyle=h^{d}\int_{0}^{1}\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|jh|<R+1\end{subarray}}\prod_{k=1}^{d}\Big{(}\frac{K_{0}^{2}(\frac{\gamma}{h^{2}})}{K_{j_{k}}^{2}(\frac{\gamma}{h^{2}})}\frac{K_{j_{k}}^{2}(\frac{\gamma}{h^{2}})}{K_{0}^{2}(\frac{\gamma}{h^{2}})}\Big{)}|u_{j}(t)|^{2}\,dt$
$\displaystyle\leq h^{d}\sup_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|jh|<R+1\end{subarray}}\prod_{k=1}^{d}\Big{(}\frac{K_{0}^{2}(\frac{\gamma}{h^{2}})}{K_{j_{k}}^{2}(\frac{\gamma}{h^{2}})}\Big{)}\,\sup_{t\in[0,1]}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}\Big{(}\frac{K_{j_{k}}^{2}(\frac{\gamma}{h^{2}})}{K_{0}^{2}(\frac{\gamma}{h^{2}})}\Big{)}|u_{j}(t)|^{2},$

and, by virtue of Proposition 3.5, the quantity with the $\sup_{t\in[0,1]}$
can be bounded by a finite constant $C$ uniformly in $h$. Hence, our goal is
to provide an upper bound estimation for the following quantity

(3.8) $\sup_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|jh|<R+1\end{subarray}}\prod_{k=1}^{d}\Big{(}\frac{K_{0}^{2}(\frac{\gamma}{h^{2}})}{K_{j_{k}}^{2}(\frac{\gamma}{h^{2}})}\Big{)}.$

To facilitate the analysis, let us first consider the one-dimensional case. It
suffices to examine nonnegative values of $j\in\mathbb{Z}$, as per the
observation in (B.3). We will consider $jh\simeq R$ and
$K_{j}^{2}(\frac{\gamma}{h^{2}})=K_{j}^{2}(j\frac{\gamma}{jh^{2}})$ (which
implies that $\gamma/(jh^{2})\simeq\gamma/(Rh)$ is fixed). Then, we will use
the asymptotic relations (B.11) for large argument and (B.12) for large index
(we bring them here for the sake of the reading)

$K_{0}(z)\leq C_{M}\frac{1}{\sqrt{z}}e^{-z},\quad\text{ for all }z\geq M,$

and

$K_{j}(jz)\geq
c_{M}\frac{1}{\sqrt{j}(1+z^{2})^{1/4}}e^{-j\big{(}\sqrt{1+z^{2}}+\log\frac{z}{1+\sqrt{1+z^{2}}}\big{)}},\quad\text{
for all }|j|\geq M,$

where $M=100$ has been fixed in the hypotheses. Thus, for $h$ as in the
hypothesis, which ensures that $\frac{\gamma}{h^{2}}\geq M$ and
$j\simeq\frac{R}{h}\geq M$, we have

$\displaystyle\frac{K_{0}^{2}(\frac{\gamma}{h^{2}})}{K_{j}^{2}(\frac{\gamma}{h^{2}})}$
$\displaystyle\lesssim\frac{jh^{2}\sqrt{1+\frac{\gamma^{2}}{j^{2}h^{4}}}}{\gamma}\exp\Bigg{[}-2\Bigg{(}\frac{\gamma}{h^{2}}-j\sqrt{1+\frac{\gamma^{2}}{j^{2}h^{4}}}-j\log\Big{(}\frac{\frac{\gamma}{jh^{2}}}{1+\sqrt{1+\frac{\gamma^{2}}{j^{2}h^{4}}}}\Big{)}\Bigg{)}\Bigg{]}$
(3.9)
$\displaystyle\simeq\frac{Rh}{\gamma}\sqrt{1+\frac{\gamma^{2}}{R^{2}h^{2}}}\exp\Bigg{[}-2\Bigg{(}\frac{\gamma}{h^{2}}-\frac{R}{h}\sqrt{1+\frac{\gamma^{2}}{R^{2}h^{2}}}-\frac{R}{h}\log\Big{(}\frac{\frac{\gamma}{Rh}}{1+\sqrt{1+\frac{\gamma^{2}}{R^{2}h^{2}}}}\Big{)}\Bigg{)}\Bigg{]}.$

We rewrite the estimate (3.3) in the form

$\frac{K_{0}^{2}(\frac{\gamma}{h^{2}})}{K_{j}^{2}(\frac{\gamma}{h^{2}})}\lesssim\bigg{(}\sqrt{\frac{R^{2}h^{2}}{\gamma^{2}}+1}\bigg{)}\exp\Bigg{[}-2\Bigg{(}\frac{\gamma}{h^{2}}-\frac{\gamma}{h^{2}}\bigg{(}\sqrt{\frac{R^{2}h^{2}}{\gamma^{2}}+1}\bigg{)}+\frac{R}{h}\log\Big{(}\frac{Rh+\gamma\sqrt{\frac{R^{2}h^{2}}{\gamma^{2}}+1}}{\gamma}-1+1\Big{)}\Bigg{)}\Bigg{]}.$

We will use the Taylor expansion $\sqrt{1+z^{2}}\sim 1+z^{2}/2$ and
$\log(1+z)\sim z$ as $|z|<1$. In order to do this, we recall that, by
hypothesis $\frac{Rh}{\gamma}<\frac{1}{2}$, thus
$\sqrt{\frac{R^{2}h^{2}}{\gamma^{2}}+1}\sim\frac{R^{2}h^{2}}{2\gamma^{2}}+1$
and

$\frac{Rh+\gamma\sqrt{\frac{R^{2}h^{2}}{\gamma^{2}}+1}}{\gamma}-1\sim\frac{Rh+\gamma(\frac{R^{2}h^{2}}{2\gamma^{2}}+1)}{\gamma}-1=\frac{Rh}{\gamma}+\frac{R^{2}h^{2}}{2\gamma^{2}}$

which has modulus less than $1$, using again the fact that
$\frac{Rh}{\gamma}<\frac{1}{2}$. Then,

$\displaystyle\frac{K_{0}^{2}(\frac{\gamma}{h^{2}})}{K_{j}^{2}(\frac{\gamma}{h^{2}})}$
$\displaystyle\lesssim\bigg{(}1+\frac{R^{2}h^{2}}{2\gamma^{2}}\bigg{)}\exp\Big{[}-2\frac{\gamma}{h^{2}}+2\frac{\gamma}{h^{2}}\bigg{(}\frac{R^{2}h^{2}}{2\gamma^{2}}+1\bigg{)}-2\frac{R}{h}\cdot\frac{1}{\gamma}\left(Rh+\frac{R^{2}h^{2}}{2\gamma}\right)\Big{]}$
$\displaystyle\lesssim\Big{(}1+\frac{1}{2\gamma^{2}}\Big{)}\exp\Big{[}-\frac{R^{2}}{\gamma}\Big{(}1+\frac{Rh}{\gamma}\Big{)}\Big{]}\leq
C_{\gamma}e^{-R^{2}/\gamma}.$

We extend this estimate to the multidimensional case as follows. Let us
consider the tensorial product (3.8) for some fixed $j\in\mathbb{Z}^{d}$ such
that $R-2<|jh|<R+1$. Then, there exists at least one component $j_{k}$ such
that $|j_{k}h|\sim R$, let us assume this is $j_{1}$. We split the product in

$\displaystyle\frac{K_{0}^{2}(\frac{\gamma}{h^{2}})}{K_{j_{1}}^{2}(\frac{\gamma}{h^{2}})}\prod_{k=2}^{d}\Big{(}\frac{K_{0}^{2}(\frac{\gamma}{h^{2}})}{K_{j_{k}}^{2}(\frac{\gamma}{h^{2}})}\Big{)}\lesssim
C_{\gamma}e^{-R^{2}/\gamma}\cdot 1,$

where the quotients in the tensorial product are simply bounded by $1$, since
the modified Bessel functions are monotone increasing with respect to order
(see (B.5)). The proof is complete now.

######  Remark 3.6.

Let $d\geq 1$ and $u\in C^{1}([0,1]:\ell^{2}((h\mathbb{Z})^{d}))$ be a
solution to (1.5). Assume that for some $\gamma>0$ there exists a finite
positive constant $c(\gamma)$ independent of $h$ such that

$h^{d}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}\Big{(}\frac{K_{j_{k}}^{2}(\frac{\gamma}{h^{2}})}{K_{0}^{2}(\frac{\gamma}{h^{2}})}\Big{)}\big{(}|u_{j}(0)|^{2}+|u_{j}(1)|^{2}\big{)}<c(\gamma)$

and that there exists $h_{0}>0$ with $\frac{\gamma}{h_{0}^{2}}\geq M=100$. If
now we choose $R>1$ such that $\frac{Rh}{\gamma}>1$ and $\frac{R}{h}\geq M$,
for $0<h<h_{0}$, then arguing as in the previous proof, the quantity in (3.3)
can be estimated by

$\displaystyle\frac{Rh}{\gamma}\Big{(}1+\frac{\gamma^{2}}{2R^{2}h^{2}}\Big{)}\exp\Big{[}-\frac{2\gamma}{h^{2}}+\frac{2R}{h}\Big{(}1+\frac{\gamma^{2}}{R^{2}h^{2}}\Big{)}-\frac{2R}{h}\log(Rh)-\frac{2R}{h}\log\Big{(}\frac{2}{\gamma}\Big{(}1+\frac{\gamma^{2}}{4R^{2}h^{2}}\Big{)}\Big{)}\Big{]}$
$\displaystyle\qquad\qquad\lesssim\frac{1}{\gamma}\exp\Big{[}c\Big{(}-\frac{2R}{h}\log(Rh)-\frac{2R}{h}\Big{(}\log\Big{(}\frac{2}{\gamma}\Big{)}-1\Big{)}+\log(Rh)\Big{)}\Big{]},$

and observe that the leading term on the right hand side is
$e^{-c\frac{R}{h}\log(Rh)}$, $c$ being independent of all relevant parameters.
Notice that the term $Rh$ in front of the exponential on the left hand side
has been absorbed by the exponential, with the cost of certain constant. On
the other hand, the term $1/\gamma$ in front of the exponential is not playing
a relevant role. Eventually, this estimate does not produce a contradiction if
we combine it with the lower bound in Theorem 3. This motivates to consider a
different condition for the regime $Rh$ large.

###  3.4. Upper bound: proof of Theorem 2 (case $Rh$ large)

Analogously to the approach in [FB18, Lemma 3.1 and Remark 3.2] (see also
[FBV17, Lemma 2.2]), we can prove that

$h^{d}\sum_{j\in\mathbb{Z}^{d}}e^{\beta\cdot j}|u_{j}(t)|^{2}\leq
e^{C\|V\|_{\infty}}h^{d}\sum_{j\in\mathbb{Z}^{d}}e^{\beta\cdot
j}\big{(}|u_{j}(0)|^{2}+|u_{j}(1)|^{2}\big{)}.$

In fact, it can be verified that the commutator computation in the proof of
Proposition 3.4 is identical to that in [FB18, proof of Theorem 2.2]. This
commutator computation, with the specific weight $e^{\beta\cdot j}$ delivers
the result in [FB18, Lemma 3.1 and Remark 3.2], scaled in $h$.

If we multiply by $\prod_{k=1}^{d}e^{-\frac{2}{eh^{2}}\cosh{(\beta_{k}/\mu)}}$
and integrate in $\beta\in\mathbb{R}^{d}$ we have, in virtue of (B.14),

$\displaystyle\int_{\mathbb{R}^{d}}e^{\beta\cdot
j}\prod_{k=1}^{d}e^{-\frac{2}{eh^{2}}\cosh{(\beta_{k}/\mu)}}\,d\beta=2^{d}\int_{0}^{\infty}\cdots\int_{0}^{\infty}\prod_{k=1}^{d}\frac{e^{\beta_{k}j_{k}}+e^{-\beta_{k}j_{k}}}{2}e^{-\frac{2}{eh^{2}}\cosh{(\beta_{k}/\mu)}}\,d\beta_{1}\cdots\beta_{d}$
$\displaystyle\quad=2^{d}\int_{0}^{\infty}\cdots\int_{0}^{\infty}\prod_{k=1}^{d}\cosh(\beta_{k}j_{k})e^{-\frac{2}{eh^{2}}\cosh{(\beta_{k}/\mu)}}\,d\beta_{1}\cdots\beta_{d}=(2\mu)^{d}\prod_{k=1}^{d}K_{j_{k}\mu}\Big{(}\frac{2}{eh^{2}}\Big{)}.$

By Tonelli, we deduce that

$h^{d}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}K_{j_{k}\mu}\Big{(}\frac{2}{eh^{2}}\Big{)}|u_{j}(t)|^{2}\leq
e^{C\|V\|_{\infty}}h^{d}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}K_{j_{k}\mu}\Big{(}\frac{2}{eh^{2}}\Big{)}\big{(}|u_{j}(0)|^{2}+|u_{j}(1)|^{2}\big{)}$

and from here trivially

(3.10)
$h^{d}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}\frac{K_{j_{k}\mu}(\frac{2}{eh^{2}})}{K_{0}(\frac{2}{eh^{2}})}|u_{j}(t)|^{2}\leq
e^{C\|V\|_{\infty}}h^{d}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}\frac{K_{j_{k}\mu}(\frac{2}{eh^{2}})}{K_{0}(\frac{2}{eh^{2}})})\big{(}|u_{j}(0)|^{2}+|u_{j}(1)|^{2}\big{)},$

which is bounded by a constant $C$ uniformly in $h$, by hypothesis.

On the one hand, taking $h$ small enough, we can use the asymptotics (B.11) so
that

(3.11) $K_{0}\Big{(}\frac{2}{eh^{2}}\Big{)}\sim
h\exp\big{[}-\frac{c}{h^{2}}\big{]}.$

On the other hand, taking into account that $|jh|\simeq R$ let us assume,
without loss of generality, that $|j_{1}h|\simeq R$. By using the asymptotics
in (B.10) we obtain, for $Rh>\frac{2}{e\mu}$,

(3.12) $2\mu
K_{j_{1}\mu}\Big{(}\frac{2}{eh^{2}}\Big{)}\sim\sqrt{\frac{h}{R}}\exp{\Big{[}\frac{\mu
R}{h}\log(Rh)+\frac{\mu R}{h}\log\mu\Big{]}}$

since the exponential absorbs the terms multiplying by $\sqrt{\mu}$ in front
of the exponential. Now we have

$\displaystyle h^{d}\int_{0}^{1}\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|jh|<R+1\end{subarray}}|u_{j}(t)|^{2}\,dt=h^{d}\int_{0}^{1}\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|jh|<R+1\end{subarray}}\prod_{k=1}^{d}\Big{(}\frac{K_{0}(\frac{2}{eh^{2}})}{K_{j_{k}\mu}(\frac{2}{eh^{2}})}\frac{K_{j_{k}\mu}(\frac{2}{eh^{2}})}{K_{0}(\frac{2}{eh^{2}})}\Big{)}|u_{j}(t)|^{2}\,dt$
$\displaystyle\qquad\leq h^{d}\sup_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|jh|<R+1\end{subarray}}\prod_{k=1}^{d}\frac{K_{0}(\frac{2}{eh^{2}})}{K_{j_{k}\mu}(\frac{2}{eh^{2}})}\,\sup_{t\in[0,1]}\sum_{j\in\mathbb{Z}^{d}}\prod_{k=1}^{d}\frac{K_{j_{k}\mu}(\frac{2}{eh^{2}})}{K_{0}(\frac{2}{eh^{2}})}|u_{j}(t)|^{2}.$

By (3.10), the quantity with $\sup_{t\in[0,1]}$ is bounded uniformly in $h$.
On the other hand, considering the tensorial product
$\prod_{k=1}^{d}\frac{K_{0}(\frac{2}{eh^{2}})}{K_{j_{k}\mu}(\frac{2}{eh^{2}})}$
for a fixed $j\in\mathbb{Z}^{d}$ such that $|jh|\simeq R$ we have, by (3.12),
(3.11), and (B.5)

$\displaystyle\frac{K_{0}(\frac{2}{eh^{2}})}{K_{j_{1}\mu}(\frac{2}{eh^{2}})}\prod_{k=2}^{d}\frac{K_{0}(\frac{2}{eh^{2}})}{K_{j_{k}\mu}(\frac{2}{eh^{2}})}$
$\displaystyle\lesssim
h\sqrt{\frac{R}{h}}\exp[-\frac{c}{h^{2}}]\exp{\Big{[}-\frac{\mu
R}{h}\log(Rh)-\frac{\mu R}{h}\log\mu\Big{]}}$
$\displaystyle\lesssim\exp{\Big{[}-c_{0}\Big{(}\frac{\mu
R}{h}\log(Rh)+\frac{\mu R}{h}\log\mu\Big{)}\Big{]}}$

which leads to the desired estimate.

##  4\. Semidiscrete heat equation: lower bound

The lower bound in Theorem 3 will be obtained through a Carleman estimate,
which is stated and proven in the next subsection.

###  4.1. A Carleman inequality for the heat operator

######  Theorem 5 (Carleman inequality).

Let $R\geq 1$ and $f:(h\mathbb{Z})^{d}\times[0,1]\to\mathbb{R}$ be such that

$\operatorname{supp}(f)\subset\Big{\\{}1\leq\big{|}\frac{hj}{R}+\varphi(t)e_{1}\big{|}\leq
4\Big{\\}}\times(0,1).$

Let $\phi:(h\mathbb{Z})^{d}\times[0,1]\to\mathbb{R}$,
$\phi_{j}(t)=\alpha\big{|}\frac{hj}{R}+\varphi(t)e_{1}\big{|}^{2},$ where
$\varphi:[0,1]\to[0,\infty)$ is a smooth function such that
$\operatorname{supp}(\varphi)\subset(0,1)$. Then there exists $h_{0}>0$ and
$C>1$ such that

(4.1) $\displaystyle\frac{1}{h^{2}}\sqrt{\sinh\frac{2\alpha
h^{2}}{R^{2}}}\sinh\Big{(}\frac{2\alpha
h}{R\sqrt{d}}\Big{)}\|f\|_{L^{2}([0,1]:\ell^{2}(h\mathbb{Z}^{d}))}$
$\displaystyle\quad+\frac{2}{h^{2}}\sqrt{\sinh\frac{2\alpha
h^{2}}{R^{2}}}\Big{(}h^{d}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\Big{|}\frac{f_{j+e_{k}}-f_{j-e_{k}}}{2}\Big{|}^{2}dt\Big{)}^{1/2}\leq
C\|e^{\phi}(\partial_{t}-\Delta_{\operatorname{\operatorname{d}}})(e^{-\phi}f)\|_{L^{2}([0,1]:\ell^{2}(h\mathbb{Z}^{d}))}$

where $\alpha$, $0<h<h_{0}$, and $R$ satisfy the relations

(4.2) $\alpha\leq c_{\varphi}\frac{1}{h^{4}}\sinh\Big{(}\frac{2\alpha
h^{2}}{R^{2}}\Big{)}\sinh^{2}\Big{(}\frac{2\alpha h}{R\sqrt{d}}\Big{)}$

for a constant
$c_{\varphi}=c_{\varphi}(d,\|\varphi^{\prime}\|_{\infty},\|\varphi^{\prime\prime}\|_{\infty})$
and

(4.3) $\alpha\geq cR^{2}\qquad\text{ if }\quad\frac{\alpha h}{R}\leq 1/10$

or

(4.4) $1\lesssim\frac{1}{Rh}e^{(2-\varepsilon)\frac{\alpha
h}{R\sqrt{d}}}\qquad\text{ if }\quad\frac{\alpha h}{R}\geq\sqrt{d}/2$

for a small, universal $\varepsilon$ such that $h/R<\varepsilon/\sqrt{d}$.

###### Proof.

We argue in two steps. First we derive the commutator contributions and in the
second step we explain how to absorb the unsigned contributions.

_Step 1: Commutator._ Let
$\phi_{j}(t)=\alpha\big{|}\frac{hj}{R}+\varphi(t)e_{1}\big{|}^{2},$ where
$\varphi:[0,1]\to[0,\infty)$ is a smooth function supported in
$[\frac{1}{4},\frac{3}{4}]$. As explained, we will often omit the time
variable to simplify notation, that is, we will frequently write
$f_{j}:=f_{j}(t)$. In view of [FBV17], we can decompose
$-e^{\phi}\Delta_{d}(e^{-\phi}f)=Sf+Af$, where

$S\,f_{j}=\frac{1}{h^{2}}\Big{\\{}-2df_{j}+\sum_{k=1}^{d}\cosh\big{(}\frac{2\alpha}{R}\big{(}\frac{j_{k}+1/2}{R}+\varphi(t)\delta_{1k}\big{)}\big{)}f_{j+e_{k}}+\sum_{k=1}^{d}\cosh\big{(}\frac{2\alpha}{R}\big{(}\frac{j_{k}-1/2}{R}+\varphi(t)\delta_{1k}\big{)}\big{)}f_{j-e_{k}}\Big{\\}}$

and

$A\,f_{j}=\frac{1}{h^{2}}\Big{\\{}-\sum_{k=1}^{d}\sinh\big{(}\frac{2\alpha}{R}\big{(}\frac{j_{k}+1/2}{R}+\varphi(t)\delta_{1k}\big{)}\big{)}f_{j+e_{k}}+\sum_{k=1}^{d}\sinh\big{(}\frac{2\alpha}{R}\big{(}\frac{j_{k}-1/2}{R}+\varphi(t)\delta_{1k}\big{)}\big{)}f_{j-e_{k}}\Big{\\}}.$

Thus, we have $e^{\phi}(\partial_{t}-\Delta_{d})g=-\partial_{t}\phi
f+\partial_{t}f+Sf+Af=\widetilde{S}f+\widetilde{A}f$, with
$\widetilde{S}=S-\partial_{t}\phi$, $\widetilde{A}=A+\partial_{t}$.

We compute the expression
$\langle[\widetilde{S},\widetilde{A}]f,f\rangle_{L^{2}([0,1]:\ell^{2}(h\mathbb{Z}^{d}))}=(I)+(II)+(III)+(IV)$,
where the terms $(I),\,(II),\,(III)$ and $(IV)$ are explained and discussed in
the sequel. More precisely, we have

$(I):=\langle[-\partial_{t}\phi,\partial_{t}]f,f\rangle,\quad(II):=\langle[-\partial_{t}\phi,A]f,f\rangle,\quad(III):=\langle[S,\partial_{t}]f,f\rangle,\quad(IV):=\langle[S,A]f,f\rangle.$

We next study these contributions individually. By the observations from
[FBV17], it is known that $(IV)$ gives rise to positive contributions, it will
thus be our main aim to either deduce positivity also for $(I)-(III)$ or to
absorb the possibly non-negative contributions into $(IV)$.

We begin by computing $(I)$

$\displaystyle(I)=h^{d}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\partial_{t}^{2}\phi_{j}(t)|f_{j}|^{2}dt=2\alpha
h^{d}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\Big{(}(\varphi^{\prime}(t))^{2}+\Big{(}\frac{hj_{1}}{R}+\varphi(t)\Big{)}\varphi^{\prime\prime}(t)\Big{)}|f_{j}|^{2}dt.$

Next, for $(II)$ we obtain

$\displaystyle(II)$ $\displaystyle=2\alpha
h^{d-2}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\sinh\Big{(}\frac{2\alpha
h}{R}\Big{(}\frac{h(j_{k}+1/2)}{R}+\varphi(t)\delta_{1k}\Big{)}\Big{)}\Big{(}\frac{hj_{1}}{R}+\varphi(t)\Big{)}\varphi^{\prime}(t)\,f_{j+e_{k}}f_{j}\,dt$
$\displaystyle\quad-2\alpha
h^{d-2}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\sinh\Big{(}\frac{2\alpha
h}{R}\Big{(}\frac{h(j_{k}-1/2)}{R}+\varphi(t)\delta_{1k}\Big{)}\Big{)}\Big{(}\frac{hj_{1}}{R}+\varphi(t)\Big{)}\varphi^{\prime}(t)\,f_{j-e_{k}}f_{j}\,dt$
$\displaystyle=-\frac{2\alpha
h}{R}h^{d-2}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sinh\Big{(}\frac{2\alpha
h}{R}\Big{(}\frac{h(j_{1}+1/2)}{R}+\varphi(t)\Big{)}\Big{)}\varphi^{\prime}(t)f_{j}f_{j+e_{1}}\,dt.$

For $(III)$ we infer

$\displaystyle(III)$ $\displaystyle=2\langle
Sf,\partial_{t}f\rangle=-4d\,h^{d-2}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\,f_{j}\,\partial_{t}f_{j}\,dt$
$\displaystyle\quad+2h^{d-2}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\cosh\Big{(}\frac{2\alpha
h}{R}\Big{(}\frac{h(j_{k}+1/2)}{R}+\varphi(t)\delta_{1k}\Big{)}\Big{)}f_{j+e_{k}}\partial_{t}f_{j}\,dt$
$\displaystyle\quad+2h^{d-2}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\cosh\Big{(}\frac{2\alpha
h}{R}\Big{(}\frac{h(j_{k}-1/2)}{R}+\varphi(t)\delta_{1k}\Big{)}\Big{)}f_{j-e_{k}}\partial_{t}f_{j}\,dt$
$\displaystyle=:(III_{1})+(III_{2})+(III_{3}).$

In $(III)$ we observe the first term $(III_{1})=0$ (since after integration by
parts in time we get $(III_{1})=-(III_{1})$). For the second sum in $(III)$ we
also integrate by parts in time and obtain that

$\displaystyle(III_{2})$
$\displaystyle=-2h^{d-2}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\Big{[}\partial_{t}\cosh\Big{(}\frac{2\alpha
h}{R}\Big{(}\frac{h(j_{k}+1/2)}{R}+\varphi(t)\delta_{1k}\Big{)}\Big{)}\Big{]}\,f_{j+e_{k}}f_{j}\,dt$
$\displaystyle\quad-2h^{d-2}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\cosh\Big{(}\frac{2\alpha
h}{R}\Big{(}\frac{h(j_{k}+1/2)}{R}+\varphi(t)\delta_{1k}\Big{)}\Big{)}(\partial_{t}f_{j+e_{k}})\,f_{j}\,dt$
$\displaystyle=-2h^{d-2}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\frac{2\alpha
h}{R}\varphi^{\prime}(t)\delta_{1k}\sinh\Big{(}\frac{2\alpha
h}{R}\Big{(}\frac{h(j_{k}+1/2)}{R}+\varphi(t)\delta_{1k}\Big{)}\Big{)}f_{j+e_{k}}f_{j}\,dt$
$\displaystyle\quad-2h^{d-2}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\cosh\Big{(}\frac{2\alpha
h}{R}\Big{(}\frac{h(j_{k}-1/2)}{R}+\varphi(t)\delta_{1k}\Big{)}\Big{)}(\partial_{t}f_{j})f_{j-e_{k}}\,dt.$

Now, observe that the last term above is just $-(III_{3})$ and thus

$(III)=(III_{2})+(III_{3})=-\frac{4\alpha
h}{R}h^{d-2}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sinh\Big{(}\frac{2\alpha
h}{R}\Big{(}\frac{h(j_{1}+1/2)}{R}+\varphi(t)\Big{)}\Big{)}\varphi^{\prime}(t)f_{j+e_{1}}f_{j}\,dt,$

so that $(III)=2(II)$.

Finally, for $(IV)$, we use observations from [FBV17] adapted to the rescaled
setting. This yields

$\displaystyle(IV)$ $\displaystyle=4h^{d-4}\sinh\frac{2\alpha
h^{2}}{R^{2}}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\sinh^{2}\Big{(}\frac{2\alpha
h}{R}\Big{(}\frac{hj_{k}}{R}+\varphi(t)\delta_{1k}\Big{)}\Big{)}|f_{j}|^{2}dt$
$\displaystyle\quad+4h^{d-4}\sinh\frac{2\alpha
h^{2}}{R^{2}}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\Big{|}\frac{f_{j+e_{k}}-f_{j-e_{k}}}{2}\Big{|}^{2}dt.$

Combining the computations for $(I)-(IV)$, we obtain, for the full parabolic
commutator,

$\displaystyle\langle[\widetilde{S},\widetilde{A}]f,f\rangle=2\alpha
h^{d}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\Big{(}(\varphi^{\prime}(t))^{2}+\Big{(}\frac{hj_{1}}{R}+\varphi(t)\Big{)}\varphi^{\prime\prime}(t)\Big{)}(f_{j})^{2}dt$
$\displaystyle\quad-6\alpha
h^{d-2}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sinh\Big{(}\frac{2\alpha
h}{R}\Big{(}\frac{h(j_{1}+1/2)}{R}+\varphi(t)\Big{)}\Big{)}\frac{h}{R}\varphi^{\prime}(t)\,f_{j+e_{1}}f_{j}dt$
$\displaystyle\quad+4h^{d-4}\sinh\frac{2\alpha
h^{2}}{R^{2}}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\sinh^{2}\Big{(}\frac{2\alpha
h}{R}\Big{(}\frac{hj_{k}}{R}+\varphi(t)\delta_{1k}\Big{)}\Big{)}|f_{j}|^{2}dt$
$\displaystyle\quad+4h^{d-4}\sinh\frac{2\alpha
h^{2}}{R^{2}}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\Big{|}\frac{f_{j+e_{k}}-f_{j-e_{k}}}{2}\Big{|}^{2}dt.$

As already indicated above, we view $(IV)$ as the dominant term and seek to
control all possibly non-signed commutator contributions by this term. Thus,
we search for sufficient conditions on the parameters such that all not
correctly signed contributions from $(I)-(III)$ can be absorbed into
$\frac{1}{2}(IV)$. Since all terms carry a factor $h^{d}$, in the sequel, we
will simply drop this in comparing the different contributions. As above, we
will deduce sufficient conditions allowing to absorb the possibly non-signed
terms by studying the non-signed contributions individually next and compare
their coefficients to the ones from $(IV)$.

_Step 2: Absorption arguments._ For the first contribution $(I)$ we only seek
to absorb the second term, since the first one is positive. Thus, comparing
the coefficients of the second term in $(I)$ with the first term in $(IV)$, we
are led to impose the following sufficient condition

$2\alpha\|\varphi^{\prime\prime}\|_{\infty}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\Big{|}\frac{hj_{1}}{R}+\varphi(t)\Big{|}|f_{j}|^{2}\\\
\leq\frac{2}{3}\frac{1}{h^{4}}\sinh\frac{2\alpha
h^{2}}{R^{2}}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\sinh^{2}\Big{(}\frac{2\alpha
h}{R}\Big{(}\frac{hj_{k}}{R}+\varphi(t)\delta_{1k}\Big{)}\Big{)}|f_{j}|^{2}.$

Due to the support of $f$, it turns out that for any point in the support we
have $\big{|}\frac{hj_{k}}{R}+\varphi(t)\delta_{1k}\big{|}\leq 4$, $\forall
k\in\\{1,\dots,d\\}$, while there exists $l\in\\{1,\dots,d\\}$ such that
$\big{|}\frac{hj_{l}}{R}+\varphi(t)\delta_{1l}\big{|}\geq\frac{1}{\sqrt{d}}$.
This implies that the first contribution can be absorbed as long as

(4.5)
$8\alpha\|\varphi^{\prime\prime}\|_{\infty}\leq\frac{2}{3h^{4}}\sinh\Big{(}\frac{2\alpha
h^{2}}{R^{2}}\Big{)}\sinh^{2}\Big{(}\frac{2\alpha h}{R\sqrt{d}}\Big{)}.$

For $(II)$ we first apply Young’s inequality
$2f_{j+e_{1}}f_{j}\leq(f_{j+e_{1}})^{2}+(f_{j})^{2}$ and, after translating in
$j$, we need the condition

(4.6)
$\frac{\alpha}{hR}\|\varphi^{\prime}\|_{\infty}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\Big{|}\sinh\Big{(}\frac{2\alpha
h}{R}\Big{(}\frac{h(j_{1}\pm
1/2)}{R}+\varphi(t)\Big{)}\Big{)}\Big{|}|f_{j}|^{2}\\\
\leq\frac{1}{3h^{4}}\sinh\frac{2\alpha
h^{2}}{R^{2}}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\sinh^{2}\Big{(}\frac{2\alpha
h}{R}\Big{(}\frac{hj_{k}}{R}+\varphi(t)\delta_{1k}\Big{)}\Big{)}|f_{j}|^{2}.$

We are reduced to analyze under which assumptions on $\alpha,h$ and $R$,
inequalities (4.5) and (4.6) hold. Let us consider two cases. Without loss of
generality, let $\|f_{j}\|_{L^{2}([0,1]:\ell^{2}(h\mathbb{Z}^{d}))}=1$ below.

Case 1: $\frac{\alpha h}{R}\leq\frac{1}{10}$. Thanks to the support condition,
the left hand side of (4.6) is bounded from above by

(4.7) $\|\varphi^{\prime}\|_{\infty}h^{-2}\frac{\alpha
h}{R}\sinh\Big{(}\frac{9\alpha h}{R}\Big{)}$

and the right hand side of (4.6) is bounded from below by

(4.8) $\frac{2}{3}h^{-4}\sinh\Big{(}\frac{2\alpha
h^{2}}{R^{2}}\Big{)}\sinh^{2}\Big{(}\frac{2\alpha h}{R\sqrt{d}}\Big{)}.$

Since $\sinh(x)\sim x$ for $x$ small, we have that (4.7) is bounded by (4.8)
whenever $\alpha\geq cR^{2}$ (in particular when $\alpha=CR^{2}$ for a
suitable constant $C=C(\|\varphi^{\prime}\|_{\infty})>0$).

Case 2: $\frac{\alpha h}{R}\geq\frac{\sqrt{d}}{2}$. Since $h/R$ is small, we
can assume that there exists a small, universal $\varepsilon$ such that
$h/R<\varepsilon/\sqrt{d}$222Indeed, we can assume that
$\varepsilon/\sqrt{d}=1/M$, where $M$ is the constant in Theorem 1.. For each
$j=(j_{1},\ldots,j_{d})$, we distinguish two situations.

Situation 2.1: the component $j_{1}$ satisfies
$\big{|}\frac{hj_{1}}{R}+\varphi(t)\big{|}\geq\frac{1}{\sqrt{d}}$. Let us look
at the argument of $\sinh$ in the left hand side of (4.6), namely
$\frac{2\alpha h}{R}\Big{(}\frac{h(j_{1}\pm
1/2)}{R}+\varphi(t)\Big{)}=\frac{2\alpha
h}{R}\Big{(}\frac{hj_{1}}{R}+\varphi(t)\Big{)}\pm\frac{\alpha h^{2}}{R^{2}}$.
Hence,

$\frac{\alpha h^{2}}{R^{2}}\leq\frac{\alpha
h}{R}\frac{\varepsilon}{\sqrt{d}}\lesssim\varepsilon\frac{\alpha
h}{R}\Big{|}\frac{hj_{1}}{R}+\varphi(t)\Big{|}$

and so the term on left hand side of (4.6) is bounded from above by a constant
times

$h^{-2}\frac{\alpha h}{R}\sinh\Big{(}(2+\varepsilon)\frac{\alpha
h}{R}\Big{|}\frac{hj_{1}}{R}+\varphi(t)\Big{|}\Big{)}.$

Now we would like to use that $\sinh(x)\sim e^{x}/2$ for $x$ large (say $x\geq
1$), this being possible since

$(2+\varepsilon)\frac{\alpha
h}{R}\Big{|}\frac{hj_{1}}{R}+\varphi(t)\Big{|}>\frac{\alpha
h}{R}\frac{2}{\sqrt{d}}.$

Thus

(4.9) $h^{-2}\frac{\alpha h}{R}\sinh\Big{(}(2+\varepsilon)\frac{\alpha
h}{R}\Big{|}\frac{hj_{1}}{R}+\varphi(t)\Big{|}\Big{)}\leq h^{-2}\frac{\alpha
h}{R}e^{(2+\varepsilon)\frac{\alpha
h}{R}\big{|}\frac{hj_{1}}{R}+\varphi(t)\big{|}}.$

On the other hand, the term corresponding with the component $j_{1}$ on the
right hand side of (4.6) is bounded from below by a constant times

(4.10) $h^{-4}\sinh\Big{(}\frac{2\alpha
h^{2}}{R^{2}}\Big{)}\sinh^{1+\varepsilon/2}\Big{(}\frac{2\alpha
h}{R}\Big{|}\frac{hj_{1}}{R}+\varphi(t)\Big{|}\Big{)}\sinh^{1-\varepsilon/2}\Big{(}\frac{2\alpha
h}{R}\Big{|}\frac{hj_{1}}{R}+\varphi(t)\Big{|}\Big{)}\\\ \geq
h^{-4}\sinh\Big{(}\frac{2\alpha
h^{2}}{R^{2}}\Big{)}e^{(2+\varepsilon)\frac{\alpha
h}{R}\big{|}\frac{hj_{1}}{R}+\varphi(t)\big{|}}\sinh^{1-\varepsilon/2}\Big{(}\frac{2\alpha
h}{R}\frac{1}{\sqrt{d}}\Big{)}.$

Thus (4.9) is bounded by (4.10) whenever

(4.11) $h^{-2}\frac{\alpha h}{R}\lesssim h^{-4}\sinh\Big{(}\frac{2\alpha
h^{2}}{R^{2}}\Big{)}\sinh^{1-\varepsilon/2}\Big{(}\frac{2\alpha
h}{R}\frac{1}{\sqrt{d}}\Big{)}.$

Nevertheless, since $x\leq\sinh x$ for all positive $x$, it can be easily
checked that the condition
$1\lesssim\frac{1}{Rh}e^{(2-\varepsilon)\frac{\alpha h}{R\sqrt{d}}}$ implies
(4.11).

Situation 2.2: the component $j_{1}$ satisfies
$\big{|}\frac{hj_{1}}{R}+\varphi(t)\big{|}\leq\frac{1}{\sqrt{d}}$. Recall that
there exists $l\neq 1$ such that
$\big{|}\frac{hj_{l}}{R}+\varphi(t)\delta_{1l}\big{|}\geq\frac{1}{\sqrt{d}}$.
Then the left hand side of (4.6) is bounded from above by a constant times
$h^{-2}\frac{\alpha h}{R}\sinh\big{(}\frac{2\alpha
h}{R}\frac{1}{\sqrt{d}}+\frac{\alpha
h}{R}\frac{\varepsilon}{\sqrt{d}}\big{)}\leq h^{-2}\frac{\alpha
h}{R}\sinh\big{(}(2+\varepsilon)\frac{\alpha h}{R\sqrt{d}}\big{)}$. On the
other hand, the right hand side of (4.6) is bounded from below by a constant
times $h^{-4}\sinh\big{(}\frac{2\alpha
h^{2}}{R^{2}}\big{)}\sinh^{2}\big{(}\frac{2\alpha h}{R\sqrt{d}}\big{)}$.
Therefore, we require that

(4.12) $h^{-2}\frac{\alpha h}{R}e^{(2+\varepsilon)\frac{\alpha
h}{R\sqrt{d}}}\lesssim h^{-4}\frac{2\alpha h^{2}}{R^{2}}e^{\frac{4\alpha
h}{R\sqrt{d}}}$

and, as before, the condition
$1\lesssim\frac{1}{Rh}e^{(2-\varepsilon)\frac{\alpha h}{R\sqrt{d}}}$ is
sufficient to ensure (4.12). The study of $(III)$ is identical to that of
$(II)$ and we arrive at the same condition, up to a dimensional constant.

If these conditions (which are the conditions given at (4.2)) are satisfied,
then the unsigned contributions of the commutator are absorbed in a constant
multiple of $(IV)$, which is bounded by the left-hand side of (4.1). This
finishes the proof since

$\|e^{\phi}\big{(}\partial_{t}-\Delta_{d}\big{)}g\|^{2}_{L^{2}([0,1]:\ell^{2}(h\mathbb{Z}^{d}))}=\|\widetilde{S}f+\widetilde{A}f\|^{2}_{L^{2}([0,1]:\ell^{2}(h\mathbb{Z}^{d}))}\geq\langle[\widetilde{S},\widetilde{A}]f,f\rangle_{L^{2}([0,1]:\ell^{2}(h\mathbb{Z}^{d}))},$

as desired. ∎

###  4.2. Lower bound: proof of Theorem 3

The quantitative lower bound in Theorem 3 will be derived from the Carleman
estimate in Theorem 5. We first state and prove a technical lemma.

######  Lemma 4.1.

Let $u\in C^{1}([0,1]:\ell^{2}(h\mathbb{Z}^{d}))$ be a non-trivial solution to
(1.5). Assume that $R\geq 1$. There exists $h_{0}>0$ such that if (4.2),(4.3),
(4.4) holds for $h\in(0,h_{0})$, and for some $C>1$,

(4.13)
$C\big{(}u(0)+\|V\|_{\infty}^{2}\big{)}\leq\frac{1}{h^{4}}\sinh\frac{2\alpha
h^{2}}{R^{2}}\sinh^{2}\Big{(}\frac{2\alpha h}{R\sqrt{d}}\Big{)},$

are satisfied, then

(4.14) $\frac{1}{h^{4}}\sinh\frac{2\alpha
h^{2}}{R^{2}}\sinh^{2}\Big{(}\frac{2\alpha
h}{R\sqrt{d}}\Big{)}e^{-14\alpha}\leq
Ch^{d}\int_{0}^{1}\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|hj|<R+1\end{subarray}}(|u_{j}(0)|^{2}+|u_{j}(t)|^{2})\,dt.$

###### Proof.

Let $\theta_{R}:\mathbb{R}^{d}\to\mathbb{R}$ be a smooth function such that
$\theta_{R}\in C_{0}^{\infty}(\mathbb{R}^{d})$, $\theta_{R}(x)=0$ for $|x|\geq
R$ and $\theta_{R}(x)=1$ for $|x|\leq R-1$. Let
$\phi:\mathbb{R}^{d}\times[0,\infty)\to\mathbb{R}$ be defined by

$\phi(x,t)=\alpha\Big{|}\frac{x}{R}+\varphi(t)e_{1}\Big{|}^{2},$

where $\varphi:[0,1]\to[0,\infty)$ is a smooth function such that

$0\leq\varphi\leq 3,\quad\varphi(t)=0\text{ for
}t\in\Big{[}0,\frac{1}{4}\Big{]}\cup\Big{[}\frac{3}{4},1\Big{]},\quad\varphi(t)=3\text{
for }t\in\Big{[}\frac{3}{8},\frac{5}{8}\Big{]}.$

Let $\eta:\mathbb{R}^{d}\to\mathbb{R}$ be a smooth function such that
$\eta(x)=0$ if $|x|\leq 1$ and $\eta(x)=1$ if $|x|\geq 2$. Thus

$g_{j}(t):=u_{j}(t)\,\theta_{R}(hj)\,\eta\Big{(}\frac{hj}{R}+\varphi(t)e_{1}\Big{)}$

is a compactly supported function for which the Carleman inequality (4.1)
holds true.

We will use the decomposition

$\Delta_{\operatorname{\operatorname{d}},k}(fg)_{j}=g_{j}\Delta_{\operatorname{\operatorname{d}},k}f_{j}+2D^{s}_{k}f_{j}D_{+,k}g_{j}+f_{j-e_{k}}\Delta_{\operatorname{\operatorname{d}},k}g_{j}.$

This will give

$\displaystyle(\partial_{t}-\Delta_{\operatorname{\operatorname{d}}})[g_{j}(t)]=(\partial_{t}-\Delta_{\operatorname{\operatorname{d}}})[u_{j}(t)]\,\theta_{R}(hj)\,\eta\Big{(}\frac{hj}{R}+\varphi(t)e_{1}\Big{)}$
$\displaystyle\quad-2\sum_{k=1}^{d}D^{s}_{k}u_{j}(t)D_{+,k}\Big{(}\theta_{R}(hj)\eta\Big{(}\frac{hj}{R}+\varphi(t)e_{1}\Big{)}\Big{)}-\sum_{k=1}^{d}u_{j-e_{k}}(t)\Delta_{\operatorname{\operatorname{d}},k}\Big{(}\theta_{R}(hj)\,\eta\Big{(}\frac{hj}{R}+\varphi(t)e_{1}\Big{)}\Big{)}$
$\displaystyle\quad+u_{j}(t)\partial_{t}\Big{(}\theta_{R}(hj)\eta\Big{(}\frac{hj}{R}+\varphi(t)e_{1}\Big{)}\Big{)}.$

Thus, the Carleman inequality (4.1) gives (notice that the $h^{d}$ term can be
simplified from all terms, so we refrain from writing it and will include it
after these computations)

$\displaystyle\frac{1}{Ch^{4}}$ $\displaystyle\sinh\frac{2\alpha
h^{2}}{R^{2}}\sinh^{2}\Big{(}\frac{2\alpha
h}{R\sqrt{d}}\Big{)}\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}(e^{\phi_{j}(t)}g_{j})^{2}dt$
(4.15)
$\displaystyle\leq\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\Big{(}e^{\phi_{j}(t)}V_{j}(t)u_{j}(t)\,\theta_{R}(hj)\,\eta\Big{(}\frac{hj}{R}+\varphi(t)e_{1}\Big{)}\Big{)}^{2}$
$\displaystyle\quad+2\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}e^{2\phi_{j}(t)}(D^{s}_{k}u_{j}(t))^{2}\Big{(}D_{+,k}\Big{(}\theta_{R}(hj)\eta\Big{(}\frac{hj}{R}+\varphi(t)e_{1}\Big{)}\Big{)}\Big{)}^{2}$
$\displaystyle\quad+\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}e^{2\phi_{j}(t)}(u_{j-e_{k}}(t))^{2}\Big{(}\Delta_{\operatorname{\operatorname{d}},k}\Big{(}\theta_{R}(hj)\eta\Big{(}\frac{hj}{R}+\varphi(t)e_{1}\Big{)}\Big{)}\Big{)}^{2}$
$\displaystyle\quad+\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}e^{2\phi_{j}(t)}(u_{j}(t))^{2}\Big{(}\partial_{t}\Big{(}\theta_{R}(hj)\eta\Big{(}\frac{hj}{R}+\varphi(t)e_{1}\Big{)}\Big{)}\Big{)}^{2}.$

First we aim to absorb the term with $V$ into the left hand side. Thus, we
need to ensure

(4.16) $C\|V\|_{\infty}^{2}\leq\frac{1}{10}\frac{1}{h^{4}}\sinh\frac{2\alpha
h^{2}}{R^{2}}\sinh^{2}\Big{(}\frac{2\alpha h}{R\sqrt{d}}\Big{)}.$

We analyze each of the other terms individually. We have

$\displaystyle
D_{+,k}\Big{(}\theta_{R}(hj)\eta\Big{(}\frac{hj}{R}+\varphi(t)e_{1}\Big{)}\Big{)}=D_{+,k}\theta_{R}(hj)\,\eta\Big{(}\frac{hj}{R}+\varphi(t)e_{1}\Big{)}+\theta_{R}((j+e_{k})h)D_{+,k}\eta\Big{(}\frac{hj}{R}+\varphi(t)e_{1}\Big{)}.$

Notice that $D_{+,k}\theta_{R}$ is supported inside $\\{R-1{-h}<|hj|<R{+h}\\}$
and $|D_{+,k}\theta_{R}(hj)|\leq\|\nabla\theta_{R}\|_{\infty}\leq C$. Thus in
this region $|\frac{hj}{R}+\varphi e_{1}|\leq 4+h$ and, by inequality (2.3),
we obtain, for $h_{0}$ small enough,

$\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}e^{2\phi_{j}(t)}(D^{s}_{k}u_{j}(t))^{2}\Big{(}D_{+,k}\theta_{R}(hj)\,\eta\Big{(}\frac{hj}{R}+\varphi(t)e_{1}\Big{)}\Big{)}^{2}\\\
\leq
Ce^{33\alpha}\int_{0}^{1}\sum_{R-2<|hj|<R+1}(|u_{j}(0)|^{2}+|u_{j}(t)|^{2})\,dt.$

We observe that the support of
$D_{+,k}\eta\big{(}\frac{hj}{R}+\varphi(t)e_{1}\big{)}$ is contained in
$\\{1-\frac{h}{R}\leq|\frac{hj}{R}+\varphi e_{1}|\leq 2+\frac{h}{R}\\}$ and
$\big{|}D_{+,k}\eta\big{(}\big{|}\frac{hj}{R}+\varphi(t)e_{1}\big{|}\big{)}\big{|}\leq
C\frac{1}{R}$. Hence, for any $k=1,\dots,d$,

$\displaystyle\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}e^{2\phi_{j}(t)}(D^{s}_{k}u_{j}(t))^{2}\Big{(}\theta_{R}((j+e_{k})h)D_{+,k}\eta\Big{(}\frac{hj}{R}+\varphi(t)e_{1}\Big{)}\Big{)}^{2}dt$
$\displaystyle\quad\leq
C\frac{e^{2\alpha(2+\frac{h}{R})^{2}}}{R^{2}}\int_{0}^{1}\sum_{\\{|hj|<R+h\\}\cap\\{1-\frac{h}{R}\leq|\frac{hj}{R}+\varphi
e_{1}|\leq 2+\frac{h}{R}\\}}(D^{s}_{k}u_{j}(t))^{2}\,dt\leq
C\frac{e^{2\alpha(2+\frac{h}{R})^{2}}}{R^{2}}\sum_{j\in\mathbb{Z}^{d}}|u_{j}(0)|^{2},$

where we used the energy estimate (Lemma 2.1) in the last inequality and $C$
depends on $\|V\|_{\infty}$.

For next terms we proceed similarly, delivering that for any $k=1,\dots,d$,

$\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}e^{2\phi_{j}(t)}|u_{j-e_{k}}(t)|^{2}\Big{(}\Delta_{\operatorname{\operatorname{d}},k}\Big{(}\theta_{R}(hj)\,\eta\Big{(}\frac{hj}{R}+\varphi(t)e_{1}\Big{)}\Big{)}\Big{)}^{2}\\\
\quad\leq
Ce^{33\alpha}\int_{0}^{1}\sum_{R-1{-h}<|hj|<R{+h}}|u_{j-e_{k}}(t)|^{2}\,dt+C\frac{e^{2\alpha(2+\frac{h}{R})^{2}}}{R^{2}}\int_{0}^{1}\sum_{\\{1-\frac{h}{R}\leq|\frac{hj}{R}+\varphi
e_{1}|\leq 2+\frac{h}{R}\\}}|u_{j-e_{k}}(t)|^{2}\,dt.$

Finally, for the last term notice that
$\partial_{t}\big{[}\eta\big{(}\frac{hj}{R}+\varphi(t)e_{1}\big{)}\big{]}=\partial_{x_{1}}\eta\big{(}\frac{hj}{R}+\varphi(t)e_{1}\big{)}\varphi^{\prime}(t)$
so that

$\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}e^{2\phi_{j}(t)}(u_{j}(t)\,\theta_{R}(hj))^{2}\Big{(}\partial_{x_{1}}\eta\Big{(}\frac{hj}{R}+\varphi(t)e_{1}\Big{)}\varphi^{\prime}(t)\Big{)}^{2}\\\
\leq
C\|\varphi^{\prime}\|_{\infty}\|\nabla\eta\|_{\infty}\,e^{8\alpha}\int_{0}^{1}\sum_{|jh|<R}|u_{j}(t)|^{2}\,dt.$

For the left-hand of side (4.2) we notice that $g\equiv u$ in the set
$\\{|hj|\leq R-1\\}\times[\frac{1}{2}-\frac{1}{8},\frac{1}{2}+\frac{1}{8}].$
In this set $\varphi\equiv 3$ and thus $|\frac{hj}{R}+\varphi(t)e_{1}|\geq
3-\frac{R-1}{R}=2+\frac{1}{R}.$ Thus,

$\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}(e^{\phi_{j}(t)}g_{j})^{2}dt\geq\int_{\frac{1}{2}-\frac{1}{8}}^{\frac{1}{2}+\frac{1}{8}}\sum_{|hj|<R-1}(e^{\phi_{j}(t)}u_{j}(t))^{2}\,dt\geq
e^{2(2+\frac{1}{R})^{2}\alpha}\int_{\frac{1}{2}-\frac{1}{8}}^{\frac{1}{2}+\frac{1}{8}}\sum_{|hj|<R-1}|u_{j}(t)|^{2}\,dt.$

Since $u$ is nontrivial, after a suitable scaling one can assume that

$h^{d}\int_{\frac{1}{2}-\frac{1}{8}}^{\frac{1}{2}+\frac{1}{8}}\sum_{|hj|<R-1}|u_{j}(t)|^{2}\,dt\geq
1.$

Indeed, if the latter integral is zero for every $R$, automatically $u\equiv
0$. So there must exist some $R_{0}$ and some positive constant $c$ such that
for $R\geq R_{0}$ the integral is greater than $c$. Since a constant times the
solution solves the same equation, one can assume without loss of generality
that $c\geq 1$.

Altogether we obtain, from (4.2), after including the factor $h^{d}$ that we
removed,

$\frac{1}{h^{4}}\sinh\frac{2\alpha h^{2}}{R^{2}}\sinh^{2}\Big{(}\frac{2\alpha
h}{R\sqrt{d}}\Big{)}e^{2(2+\frac{1}{R})^{2}\alpha}\\\ \leq
Ch^{d}\Big{(}\frac{e^{2\alpha(2+\frac{h}{R})^{2}}}{R^{2}}\sum_{j\in\mathbb{Z}^{d}}|u_{j}(0)|^{2}+e^{33\alpha}\int_{0}^{1}\sum_{R-2<|hj|<R+1}(|u_{j}(0)|^{2}+|u_{j}(t)|^{2})\,dt\Big{)},$

where we also used the energy estimate in Lemma 2.1, namely
$\int_{0}^{1}\sum_{j\in\mathbb{Z}^{d}}|u_{j}(t)|^{2}\,dt\leq
C\sum_{j\in\mathbb{Z}^{d}}|u_{j}(0)|^{2}$. Thus, if

(4.17) $\frac{1}{2}\frac{1}{h^{4}}\sinh\frac{2\alpha
h^{2}}{R^{2}}\sinh^{2}\Big{(}\frac{2\alpha
h}{R\sqrt{d}}\Big{)}\geq\frac{\|u(\cdot,0)\|_{\ell^{2}}^{2}}{R^{2}}$

then for $R\geq 1$

$\frac{1}{2}\frac{1}{h^{4}}\sinh\frac{2\alpha
h^{2}}{R^{2}}\sinh^{2}\Big{(}\frac{2\alpha
h}{R\sqrt{d}}\Big{)}e^{-14\alpha}\leq
h^{d}\int_{0}^{1}\sum_{R-2<|hj|<R+1}(|u_{j}(0)|^{2}+|u_{j}(t)|^{2})\,dt.$

Now, to summarize, we just need to ensure that conditions (4.16) and (4.17)
hold, or stated altogether, that

$C\big{(}u(0),\|V\|_{\infty}\big{)}\leq\frac{1}{h^{4}}\sinh\frac{2\alpha
h^{2}}{R^{2}}\sinh^{2}\Big{(}\frac{2\alpha h}{R\sqrt{d}}\Big{)}$

holds. ∎

Now we are ready to prove the lower bound of Theorem 3.

###### Proof of Theorem 3.

We have to optimize the lower bound from the previous Lemma 4.1 in $\alpha$
and $R$. This amounts to studying the relation between $\alpha,h,R$ such that
(4.2) and either (4.3) or (4.4), hold. In words, besides (4.13) we need the
condition (4.2)

$c_{\varphi}h^{-4}\sinh^{2}(2\alpha h/R\sqrt{d})\sinh(2\alpha
h^{2}/R^{2})\geq\alpha$

and

$\alpha\geq cR^{2}\,\,\,\text{ if }\,\,\,\frac{\alpha h}{R}\leq
1/10\qquad\text{ or }\qquad
1\lesssim\frac{1}{Rh}e^{(2-\varepsilon)\frac{\alpha h}{R\sqrt{d}}}\,\,\,\text{
if }\,\,\,\frac{\alpha h}{R}\geq\sqrt{d}/2$

for a small, universal $\varepsilon$ such that $h/R<\varepsilon/\sqrt{d}$. We
have to take into account that $h$ is small, $R$ large, and we would like to
have $\alpha$ large, but going to infinity in the slowest possible way. Let us
first link $R$ and $h$ by taking $R=h^{-\beta}$ for $\beta>0$ and do some
casuistic to determine $\alpha$ at every scale $|hj|\sim R$. Observe that at
each scale, (4.2) reads

(4.18)
$c_{\varphi}R^{4/\beta}\sinh^{2}(2\alpha/R^{1+1/\beta}\sqrt{d})\sinh(2\alpha/R^{2+2/\beta})\geq\alpha.$

We recall that $\sinh(x)\sim x$ and $\sinh(x)\sim e^{|x|}/2$ for $x$ small and
large, respectively.
Case 1: Choice of $\alpha$ such that $\alpha\leq R^{1+1/\beta}/10$. In this
case, since $\alpha h/R\leq 1/10$, we require $\alpha\geq cR^{2}$. Besides,
all hyperbolic functions are evaluated at small argument. Therefore in this
regime we can use the relation $2x\geq\sinh x\geq x$, for $x$ positive, and
condition (4.18) is satisfied if

$c_{\varphi}R^{4/\beta}\frac{4\alpha^{2}}{R^{2+2/\beta}d}\frac{2\alpha}{R^{2+2/\beta}}\geq\alpha\Longleftrightarrow
8c_{\varphi}\alpha^{2}\geq dR^{4}.$

Hence, a choice to get a constant size and answer positively to our question
is

$\alpha=cR^{2},$

with $c$ large enough independent of $R$. This is the same as in the continuum
regime but this choice forces the requirement $cR^{2}\leq R^{1+1/\beta}/10$
or, in other words, $\beta<1$. If $\beta=1$ we require some control on the
constant $c$ (which depends on the dimension and the function $\varphi$).
Case 2: Choice of $\alpha$ such that $\alpha\geq\sqrt{d}R^{1+1/\beta}/2$ and
$\alpha\leq R^{2+2/\beta}/10$. This is the range for $\alpha$ to satisfy
(4.4), so at the scale $R=h^{-\beta}$ we require

(4.19)
$R^{-1+1/\beta}e^{(2-\varepsilon)\frac{\alpha}{R^{1+1/\beta}\sqrt{d}}}\geq
C_{1}.$

We recall that $\varepsilon$ is a universal small number such that
$\varepsilon>\sqrt{d}h/R$. Now, when the hyperbolic sine is evaluated at small
argument, we can use as before $2x\geq\sinh x\geq x$, while at large arguments
one has $e^{x}/2\geq\sinh x\geq e^{x}/4$. Therefore, to ensure (4.18) we
require

$c_{\varphi}R^{4/\beta}\frac{e^{\frac{4\alpha}{R^{1+1/\beta}\sqrt{d}}}}{4}\frac{2\alpha}{R^{2+2/\beta}}\geq\alpha\Longleftrightarrow\frac{c_{\varphi}e^{\frac{4\alpha}{R^{1+1/\beta}\sqrt{d}}}}{2R^{2-2/\beta}}\geq
1,$

which is a weaker relation than (4.19). Hence, we have to analyze the best
relation in (4.19). If $\beta<1$ we take $\alpha=\sqrt{d}R^{1+1/\beta}/2$, if
$\beta=1$ we take $\alpha=cR^{2}$ with
$c=\max\\{\frac{\sqrt{d}}{2},\frac{\sqrt{d}}{4}\log\left(\frac{2}{c_{\varphi}}\right)\\}$,
and if $\beta>1$ we take $\alpha=\tilde{c}R^{1+1/\beta}\log(R^{1-1/\beta})$,
for which we need

$R^{((2-\epsilon)\tilde{c}/\sqrt{d}-1)(1-1/\beta)}\geq C_{1}$

which is satisfied if $\tilde{c}>\frac{\sqrt{d}}{2-\epsilon}$.
Case 3: Choice of $\alpha$ such that $\alpha\geq R^{2+2/\beta}$. In this range
we require (4.19) and, since both hyperbolic functions are evaluated at large
argument,

$c_{\varphi}R^{4/\beta}e^{\frac{4\alpha}{R^{1+1/\beta}\sqrt{d}}+\frac{2\alpha}{R^{2+2\beta}}}\geq
16\alpha.$

The best choice in this case is then $\alpha=R^{2+2/\beta}$.

Combining the three cases, if $\beta\leq 1$,
$\alpha=\min\\{cR^{2},CR^{1+1/\beta},R^{2+2/\beta}\\}=cR^{2}$. In other words,
if $Rh\leq 1$, the choice of $\alpha$ is the same as in the continuum case. If
$\beta>1$, then

$\alpha=\min\\{\tilde{c}R^{1+1/\beta}\log(R^{1-1/\beta}),R^{2+2/\beta}\\}=\tilde{c}R^{1+1/\beta}\log(R^{1-1/\beta})$

with $\tilde{c}$ a universal constant independent of $\beta$. Observe that at
these scales $R=h^{-\beta}$, we can take $Rh\geq 1$ and
$\alpha=\tilde{c}R^{1+1/\beta}\log(R^{1-1/\beta})=\tilde{c}\frac{R}{h}\log(Rh)$.
We summarize this study as

(4.20) $\alpha=\begin{cases}cR^{2}\mbox{ if }Rh\leq 1,\\\
\tilde{c}\frac{R}{h}\log(Rh)\mbox{ if }Rh>1.\end{cases}$

Inserting these choices into (4.14), taking (4.2) into account, we obtain

(4.21) $Ch^{d}\int_{0}^{1}\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|hj|<R+1\end{subarray}}(|u_{j}(0)|^{2}+|u_{j}(t)|^{2})\,dt\geq
e^{-14\alpha}\alpha\geq\begin{cases}e^{-cR^{2}}\mbox{ if }Rh\leq 1,\\\
e^{-\tilde{c}\frac{R}{h}\log(Rh)}\mbox{ if }Rh>1,\end{cases}$

as desired.

If for fixed $h$ we want to find the best choice of $\alpha$, we notice that
only (4.4) can occur, and in order to satisfy this condition, we take
$\alpha=c\frac{R}{h}\log(Rh)$. For $c$ large enough, only depending on $d$,
(4.2) is sastisfied. This agrees with the previous lower bound. ∎

######  Remark 4.2.

In the case $R=h^{-\beta}$, $Rh>1$ we can write the lower bound as

$Ch^{d}\int_{0}^{1}\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|hj|<R+1\end{subarray}}(|u_{j}(0)|^{2}+|u_{j}(t)|^{2})\,dt\geq
e^{-14CR^{1+\frac{1}{\beta}}\log(R^{1-1/\beta})}\quad\mbox{ if }Rh\geq 1.$

It is possible to adapt the previous argument to have
$e^{-14CR^{1+\frac{1}{\beta}}\log(1+R^{1-1/\beta})}$ on the right hand side.
Then, the value $\beta=1$ leads us to the close-to-continuum region, and gives
us the $R^{2}$ factor.

##  5\. Landis-type results for the semidiscrete heat equation

###  5.1. The close-to-continuum regime: proof of Theorem A (1)

We present a proof of the Landis-type estimate for the discrete heat equation
in Theorem A $(1)$. Assume that $u$ is nontrivial. We focus on the regime $Rh$
small, so in this case, from the lower bound in Theorem 3,

$h^{d}\int_{0}^{1}\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|hj|<R+1\end{subarray}}(|u_{j}(0)|^{2}+|u_{j}(t)|^{2})\,dt\gtrsim
e^{-cR^{2}}.$

From the upper bound in Theorem 1,

$h^{d}\int_{0}^{1}\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|hj|<R+1\end{subarray}}(|u_{j}(0)|^{2}+|u_{j}(t)|^{2})\,dt\leq
C_{\gamma,d}e^{-dR^{2}/\gamma}$

and hence

$e^{-cR^{2}}\leq C_{\gamma,d}e^{-dR^{2}/\gamma}.$

Thus, for $R$ greater than some specific $\bar{R}(C,\gamma,d)$ we arrive at a
contradiction if $\gamma<dc^{-1}$. Notice this simultaneously gives the
existence of some $h_{0}=h_{0}(\gamma,c)$ such that for $h\in(0,h_{0})$ the
regime condition $Rh\leq\min\\{\gamma/2,1/10\\}$ is satisfied.

###  5.2. The purely discrete regime: proof of Theorem A (2)

The proof of Theorem A (2) follows the idea of the proof of Theorem A (1) and
makes use of the upper and lower bounds in Theorems 2 and 3, respectively.
This time, if we assume $\mu$ to be large enough, only depending on the
dimension, by letting $Rh\ggg 1$, we reach a contradiction, hence $u\equiv 0$,
see also [FBV17, p. 4864].

##  6\. Optimality

We can see with an example how the quantitative estimates we have obtained in
Theorems 1, 2, and 3 are optimal. For simplicity, we will restrict ourselves
to the one dimensional case in this section. Hence, we are going to provide a
nontrivial solution $u_{j}(t)$ to the discrete free heat equation with fast
decay and study the quantity

$h\int_{0}^{1}\sum_{\begin{subarray}{c}j\in\mathbb{Z}\\\
R-2<|hj|<R+1\end{subarray}}(|u_{j}(0)|^{2}+|u_{j}(t)|^{2})\,dt.$

This quantity behaves as the lower and upper bounds of the results above or,
in other words, these bounds are tight. This example does not contradict
Theorem A $(1)$ because the decay rate $\gamma$ is too large and the
hypothesis $\gamma<C$ of this result is not satisfied.

Indeed, let us consider the function

(6.1)
$u_{j}(t)=\sum_{j\in\mathbb{Z}}e^{-2t/h^{2}}\frac{I_{j}(2t/h^{2}+1/h^{2})}{I_{0}(1/h^{2})}.$

First of all, one can check that this function is a solution to equation (1.5)
such that fulfills the condition (1.7). By taking for instance $\gamma=4$ in
(1.7), it suffices to see that the quantities

$h\sum_{j\in\mathbb{Z}}\frac{K_{j}^{2}(\frac{4}{h^{2}})I_{j}^{2}(\frac{1}{h^{2}})}{K_{0}^{2}(\frac{4}{h^{2}})I_{0}^{2}(\frac{1}{h^{2}})}\quad\text{
and }\quad
e^{-\frac{4}{h^{2}}}h\sum_{j\in\mathbb{Z}}\frac{K_{j}^{2}(\frac{4}{h^{2}})I_{j}^{2}(\frac{3}{h^{2}})}{K_{0}^{2}(\frac{4}{h^{2}})I_{0}^{2}(\frac{3}{h^{2}})}$

are finite, with estimates uniform in $h$. Intuitively, by arguments similar
to those appearing in [FB17, Theorem 2.1] or [CJK10, Proposition 4.7] (see
also Appendix A), we can see that $u_{j}(t)$ is an approximation of the
solution to the free heat equation with Gaussian initial data, and that the
function $K_{j}(4/h^{2})/K_{0}(4/h^{2})$ approximates the inverse of another
Gaussian function. Therefore, both conditions above approximate an integral of
a certain Gaussian function. In any case, one can check this by means of
computations in the spirit of the ones carried out in Subsection 3.3. Now the
analysis of the sums has to distinguish the regimes in which $j^{2}h^{4}\leq
c_{1}$ for some $c_{1}$ positive and small and $j^{2}h^{4}\geq c_{2}$ for some
$c_{2}$ positive and large, and then make use of the uniform bounds (B.12) and
(B.13). For the sum
$\sum_{c_{1}<j^{2}h^{4}<c_{2}}\frac{K_{j}^{2}(\frac{4}{h^{2}})I_{j}^{2}(\frac{1}{h^{2}})}{K_{0}^{2}(\frac{4}{h^{2}})I_{0}^{2}(\frac{1}{h^{2}})}$
(respectively, the one concerning the condition on $u_{j}(1)$), it suffices to
notice that the $j\mapsto K_{j}(b)I_{j}(a)$, for $a<b$, is decreasing; this
follows from (B.9). An analogous computation can be carried out to see that
the function $u_{j}(t)$ in (6.1) satisfies that

$h\sum_{j\in\mathbb{Z}}\frac{K_{j\mu}(\frac{2}{eh^{2}})I_{j}^{2}(\frac{1}{h^{2}})}{K_{0}(\frac{2}{eh^{2}})I_{0}^{2}(\frac{1}{h^{2}})}\quad\text{
and }\quad
e^{-\frac{4}{h^{2}}}h\sum_{j\in\mathbb{Z}}\frac{K_{j\mu}(\frac{2}{eh^{2}})I_{j}^{2}(\frac{3}{h^{2}})}{K_{0}(\frac{2}{eh^{2}}))I_{0}^{2}(\frac{3}{h^{2}})}$

are finite (and actually uniformly in $h$, for $\mu=1/e$, for instance). This
is the condition in Theorem A $(2)$.

Now, let us consider a point in the lattice such that $jh\simeq R$. Using once
again the asymptotic expressions of the Bessel functions one can check that

$u_{j}(t)\sim
e^{-\frac{1+2t}{h^{2}}+\frac{R}{h}\left[\log\left(\frac{1+2t}{hR+\sqrt{h^{2}R^{2}+(1+2t)^{2}}}+\sqrt{1+\frac{(1+2t)^{2}}{h^{2}R^{2}}}\right)\right]}\frac{1}{(h^{2}R^{2}+(1+2t)^{2})^{1/4}}.$

If $Rh$ is small, we proceed as in the first part of Theorem 3; notice that
$R$ is a quantity assumed to be large, by using Taylor approximations we
deduce that the leading term of $u_{j}(t)$ behaves as

$u_{j}(t)\sim\frac{1}{\sqrt{1+2t}}e^{-\frac{R^{2}}{2(1+2t)}},$

which can be bounded uniformly with respect to $t\in(0,1)$. Thanks to the fact
that the number of lattice points $j$ such that $R-2<|hj|<R+1$ is comparable
to $h^{-1}$ we conclude

$h\int_{0}^{1}\sum_{\begin{subarray}{c}j\in\mathbb{Z}\\\
R-2<|hj|<R+1\end{subarray}}(|u_{j}(0)|^{2}+|u_{j}(t)|^{2})\,dt\sim
e^{-cR^{2}}.$

If $Rh$ is large, we proceed as in the second part of Theorem 3; similarly we
get $u_{j}(t)\sim e^{-c\frac{R}{h}\log(Rh)}$, and therefore

$h\int_{0}^{1}\sum_{\begin{subarray}{c}j\in\mathbb{Z}\\\
R-2<|hj|<R+1\end{subarray}}(|u_{j}(0)|^{2}+|u_{j}(t)|^{2})\,dt\sim
e^{-c\frac{R}{h}\log(Rh)}.$

As in Subsection 3.3, the argument can be adapted to the multidimensional
case.

##  7\. Landis-type result for the stationary discrete Schrödinger equation

The strategy carried out to deal with the parabolic setting can be adapted to
discuss the discrete elliptic framework of Landis conjecture. Now we are
concerned with the problem

(7.1) $\Delta_{\operatorname{\operatorname{d}}}u_{j}+V_{j}u_{j}=0\quad\text{in
}(h\mathbb{Z})^{d},$

where $V:(h\mathbb{Z})^{d}\to\mathbb{R}$ is a bounded potential and
$u:(h\mathbb{Z})^{d}\to\mathbb{R}$.

As before, we will prove a qualitative Landis-type result assuming decay of
the solution to (7.1), which will be based on lower quantitative bounds for
the solutions.

###  7.1. Lower bound: proof of Theorem 4

The lower bound relies on the Carleman inequality in Theorem 6. The proof of
this inequality follows the same steps as the one for the heat equation in
Theorem 5 but it is simpler, since there are no conditions between the
parameters thanks to the positivity of the commutator. We will omit such a
proof.

######  Theorem 6 (Carleman inequality).

Let $R>1$ and $f:(h\mathbb{Z})^{d}\to\mathbb{R}$ be such that

$\operatorname{supp}(f)\subset\Big{\\{}1\leq\big{|}\frac{hj}{R}+3e_{1}\big{|}\leq
4\Big{\\}}\times(0,1).$

Let $\phi:(h\mathbb{Z})^{d}\to\mathbb{R}$,
$\phi_{j}=\alpha\big{|}\frac{hj}{R}+3e_{1}\big{|}^{2}$. Then there exist
$R_{0}=R_{0}(d)>0$, $C>1$ and $h_{0}>0$ such that

(7.2) $\frac{1}{h^{2}}\sqrt{\sinh\frac{2\alpha
h^{2}}{R^{2}}}\sinh\Big{(}\frac{2\alpha
h}{R\sqrt{d}}\Big{)}\|f\|_{\ell^{2}(h\mathbb{Z}^{d})}+2\frac{1}{h^{2}}\sqrt{\sinh\frac{2\alpha
h^{2}}{R^{2}}}\Big{(}h^{d}\sum_{j\in\mathbb{Z}^{d}}\sum_{k=1}^{d}\Big{|}\frac{f_{j+e_{k}}-f_{j-e_{k}}}{2}\Big{|}^{2}\Big{)}^{1/2}\\\
\qquad\leq
C\|e^{\phi}\Delta_{\operatorname{\operatorname{d}}}(e^{-\phi}f)\|_{\ell^{2}(h\mathbb{Z}^{d})}$

for all $R\geq R_{0}$, $0<h<h_{0}$.

The Carleman inequality will lead to the quantitative lower bound stated in
Theorem 4.

###### Proof of Theorem 4.

The proof is similar to the proof of the lower bound in the parabolic setting
and therefore we will omit part of the details. In a first step, we apply the
previous Carleman estimate and conclude, analogously as in Lemma 4.1, that

$\frac{1}{h^{4}}\sinh\frac{2\alpha h^{2}}{R^{2}}\sinh^{2}\Big{(}\frac{2\alpha
h}{R\sqrt{d}}\Big{)}e^{-14\alpha}\leq
h^{d}\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|hj|<R+1\end{subarray}}|u_{j}|^{2}$

as long as

(7.3) $C\left(1+\|V\|_{\infty}^{2}\right)\leq\frac{1}{h^{4}}\sinh\frac{2\alpha
h^{2}}{R^{2}}\sinh^{2}\Big{(}\frac{2\alpha h}{R\sqrt{d}}\Big{)}.$

Analogously as in the parabolic case, let us first link $R$ and $h$, write
$R=h^{-\beta}$ for $\beta>0$, and do some casuistic to determine $\alpha$.
Since this choice is related to the amount of mass of the nontrivial solution
we capture at the scale $|hj|\sim R$, our goal is to find, at each scale, the
smallest possible value of $\alpha$, depending on $R$, such that (7.3) holds.
At the scale $R=h^{-\beta}$ the condition reads

(7.4)
$R^{4/\beta}\sinh\big{(}2\frac{\alpha}{R^{2+2/\beta}}\big{)}\sinh^{2}\big{(}2\frac{\alpha}{R^{1+1/\beta}\sqrt{d}}\big{)}\geq
C_{0}.$

Case 1: $\alpha\leq R^{1+1/\beta}$. In this first case both hyperbolic
functions are small and we can use the chain of inequalities
$2x\geq\sinh(x)\geq x$ as we did in the parabolic setting. This implies

$R^{4/\beta}\sinh\big{(}2\frac{\alpha}{R^{2+2/\beta}}\big{)}\sinh^{2}\big{(}2\frac{\alpha}{R^{1+1/\beta}\sqrt{d}}\big{)}\sim
c\frac{\alpha^{3}}{R^{4}}.$

Thus, a choice to get a constant size and answer positively to our question is

$\alpha=cR^{4/3},$

with an appropriate $c$, independent on $h$ and $R$. This choice is possible
only if $4/3\leq 1+1/\beta\Leftrightarrow\beta\leq 3$. Notice that $\beta=3$
requires $c\leq 1$, which may not be satisfied. If $\beta>3$, this case will
not lead to a choice of $\alpha$ such that (7.3) holds.
Case 2: $R^{2+2/\beta}\geq\alpha\geq\sqrt{d}R^{1+1/\beta}/2$. In this case, by
using the asymptotics of the $\sinh$ function

$R^{4/\beta}\sinh\big{(}2\frac{\alpha}{R^{2+2/\beta}}\big{)}\sinh^{2}\big{(}2\frac{\alpha}{R^{1+1/\beta}\sqrt{d}}\big{)}\sim\frac{\alpha}{R^{2-2/\beta}}e^{4\frac{\alpha}{R^{1+1/\beta}\sqrt{d}}}.$

This motivates the choice

$\alpha=cR^{1+\frac{1}{\beta}}\log(R^{1-1/\beta})$

with $c$ a positive constant. Hence,

$\frac{\alpha}{R^{2-2/\beta}}e^{\frac{4\alpha}{R^{1+1/\beta}\sqrt{d}}}=cR^{\frac{4c(1-1/\beta)}{\sqrt{d}}+\frac{3}{\beta}-1}\log(R^{1-1/\beta})$

and by taking $c=\sqrt{d}/4$ condition (7.4) is satisfied. If $\beta\leq 3$ we
can remove the logarithm from the choice of $\alpha$ and take
$\alpha=cR^{2+2/\beta}$ with $c$ a sufficiently large constant, independent of
$R,h$ and $\beta$.
Case 3: $\alpha\geq R^{2+2/\beta}$. Again, by the asymptotics of $\sinh$,

$R^{4/\beta}\sinh\big{(}2\frac{\alpha}{R^{2+2/\beta}}\big{)}\sinh^{2}\big{(}2\frac{\alpha}{R^{1+1/\beta}\sqrt{d}}\big{)}\sim
R^{4/\beta}\frac{\alpha}{R^{2-2/\beta}}e^{2\frac{\alpha}{R^{2+2/\beta}}+4\frac{\alpha}{R^{1+1/\beta}\sqrt{d}}}.$

And the appropriate choice of $\alpha$ is $R^{2+2/\beta}$. Gathering the
information of these three cases, if $\beta\leq 3$ we take
$\alpha=\min\\{cR^{4/3},cR^{1+1/\beta},R^{2+2/\beta}\\}=cR^{4/3}$, while, if
$\beta>3$ we take
$\alpha=\min\\{cR^{1+1/\beta}\log(R^{1-1/\beta}),R^{2+2\beta}\\}=cR^{1+1/\beta}\log(R^{1-1/\beta})$,
being all constants independent of $R$. After the choice of $\alpha$ is done,
we get the lower bound as in the parabolic setting.

If we look at condition (7.3) for fixed $h$, then by taking
$\alpha=c\frac{R}{h}\log(Rh)$ the condition is satisfied, by following the
previous strategy. ∎

###  7.2. Landis-type result for the solution to the discrete elliptic
equation

We conclude this section with the proof of the Landis-type results for the
elliptic equation.

###### Proof of Theorem B.

The proof follows from Theorem 4 and the assumptions on the solution. Assume
$u$ is nontrivial and let

$\Lambda_{R}:=h^{d}\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{d}\\\
R-2<|hj|<R+1\end{subarray}}|u_{j}|^{2}.$

If the decay assumption in $(1)$ is satisfied, it is easy to check that
$\Lambda_{R}\lesssim e^{-\mu_{0}R^{4/3}}$ for all $R$. On the other hand by
taking for instance $R=h^{-1/2}$, by the first part of Theorem 4 we get
$\Lambda_{R}\gtrsim e^{-CR^{4/3}}$. We arrive at a contradiction by letting
$R\to\infty$ (and therefore $h\to 0$), choosing $\mu_{0}>C$.

If the decay assumption in (2) is satisfied, $\Lambda_{R}\lesssim
e^{-\mu_{0}R^{1+1/\beta}\log(R^{1-1/\beta})}$ for all $R$. Next, we get from
the second part of Theorem 4 that $\Lambda_{R}\gtrsim
e^{-CR^{1+1/\beta}\log(R^{1-1/\beta})}$ by looking at the scale
$R=h^{-\beta}$, and we arrive at a contradiction by letting $R\to\infty$ (and
therefore $h\to 0$) if $\mu_{0}$ is large enough.

To finish with the proof, under decay assumption (3), we are dealing with
fixed $h>0$ and we get $\Lambda_{R}\lesssim e^{-\frac{\mu_{0}}{h}R\log(Rh)}$,
while, from Theorem 4 , $\Lambda_{R}\gtrsim e^{-C\frac{R}{h}\log(Rh)}$,
arriving to a contradiction by letting $R\to\infty$ if $\mu_{0}$ is large
enough. ∎

######  Remark 7.1.

It should be pointed out that the difference between the three parts of
Theorem B is that under the first decay condition we arrive at a contradiction
by letting $h\to 0$ looking at the values of the nontrivial solution at points
of the lattice that shrinks to the continuum, and therefore it has to be
understood as a close-to-continuum type result; under the second decay
condition we are looking at a spurious region of the lattice where we connect
the close-to-continuum region with the purely discrete situation, being this
the case where we do not require $h\to 0$, i. e., the third decay assumption.

###  7.3. Examples

We illustrate the Landis-type result in Theorem B with some examples in the
spirit of [LM18]. Recall that we are working with the problem

(7.5) $\Delta_{\operatorname{\operatorname{d}}}u(x)+V(x)u(x)=0,\quad
x\in(h\mathbb{Z})^{d},$

where $V:(h\mathbb{Z})^{d}\to\mathbb{R}$. Let
$|x|_{\infty}:=\max\\{|x_{1}|,\ldots,|x_{d}|\\}$, $x\in(h\mathbb{Z})^{d}$.

######  Lemma 7.2.

Let $h>0$ and $R>0$ be of the form $R=Nh$ with $N\in\mathbb{N}^{*}$. Let
$V:(h\mathbb{Z})^{d}\to\mathbb{R}$ and $u$ be a solution to (7.5). Denote
$q_{N}:=h^{2}\max_{|n|_{\infty}=N}|V_{n}|$, $n\in\mathbb{N}^{d}$. If $u(x)$
satisfies the following decay estimate

$\max_{|x|_{\infty}\in\\{R-h,R\\}}|u(x)|<\frac{\max_{|x|_{\infty}\in\\{0,h\\}}|u(x)|}{(4d-1+q_{N})(4d-1+q_{N-1})\cdots(4d-1+q_{1})},$

then $u\equiv 0$.

###### Proof.

The equation can be rewritten as
$\sum_{k=1}^{d}u(x-he_{k})=-\sum_{k=1}^{d}u(x+he_{k})+(2d-\widetilde{V}(x))u(x)$,
with $\widetilde{V}(x)=h^{2}V(x)$.

$-R$$(j,R+h)$$(j,R)$$R$$j$ Figure 1. Illustration in two dimensions; for a
point in the ball of center $(0,0)$ and radius $R$ and the metric
$|\cdot|_{\infty}$, say $(j,R)$ with $j,R\in h\mathbb{Z}$, $j<R$, and for a
solution of the equation, then
$u(j,R-h)=4u(j,R)-u(j-h,R)-u(j+h,R)-u(j,R+h)+Vu(j,R)$ holds. Observe that on
the right hand side of this equation there are $2\cdot 2+(2\cdot 2-2)$
evaluations on the ball of radius $R$ ($2\cdot 2$ are coming from the values
of $u(j,R)$ and $(2\cdot 2-2)$ from the values of $u(j-h,R),u(j+h,R)$), and
one evaluation of $u$ on the ball of radius $R+h$. In higher dimensions, this
corresponds to $2d+(2d-2)$ evaluations on the ball of radius $R$ and one on
the ball of radius $R+h$.

Let $y\in(h\mathbb{Z})^{d}$ such that $|y|_{\infty}=R-h$. For simplicity we
assume the maximum is attained in the first index, $|y|_{\infty}=|y_{1}|$.

Case $y_{1}>0$. Then $y_{1}=R-h$. Let $x=y+he_{1}=(R,y_{2},\dots,y_{d})$. Then
$|x|_{\infty}=R$ and $|x+he_{1}|_{\infty}=R+h$, $|x\pm he_{k}|_{\infty}=R$ if
$k\neq 1$. Hence

$u(y)=u(x-he_{1})=-\sum_{k=2}^{d}u(x-he_{k})-\sum_{k=2}^{d}u(x+he_{k})+u(x+he_{1})+(2d-\widetilde{V}(x))u(x)$

and thus (see Figure 1)

$\displaystyle|u(y)|$
$\displaystyle\leq\sum_{k=2}^{d}|u(x-he_{k})|+\sum_{k=2}^{d}|u(x+he_{k})|+|u(x+he_{1})|+|2d-\widetilde{V}(x))|u(x)|$
$\displaystyle\leq(2d+2d-2+\widetilde{V}(x))\max_{|x|=R}|u(x)|+\max_{|x|_{\infty}=R+h}|u(x)|.$

Case $y_{1}<0$. Then $y_{1}=-R+h$. Let
$x=y-he_{e_{1}}=(-R,y_{2},\dots,y_{d})$. Then $|x|_{\infty}=R$ and
$|x-he_{1}|=R+h$, $|x\pm he_{k}|_{\infty}=R+h$ for $k\geq 2$. We argue as
above writing the expression for $u(x+he_{1})$ using the equation

$u(y)=u(x+he_{1})=-\sum_{k=2}^{d}u(x+he_{k})-\sum_{k=2}^{d}u(x-he_{k})-u(x-he_{1})+(2d-\widetilde{V}(x))u(x).$

Since $y$ was chosen arbitrary with $|y|_{\infty}=R-h$, we conclude that
(after renaming variables)

$\max_{|x|=R-h}|u(x)|\leq(2d+2d-2+\widetilde{V}(x))\max_{|x|=R}|u(x)|+\max_{|x|_{\infty}=R+h}|u(x)|.$

Since $u(x)$ with $|x|=R$ also satisfies the previous inequality, then we can
conclude that

$\max_{|x|_{\infty}\in\\{R-h,R\\}}|u(x)|\
\leq(\max_{|x|_{\infty}=R}|\widetilde{V}(x)|+4d-1)\max_{|x|_{\infty}\in\\{R,R+h\\}}|u(x)|.$

Let $x=nh$, where $n\in\mathbb{Z}^{d}$. The latter is then rewritten as (here
$|n|_{\infty}$ has the same meaning as above, but with the maximum over
$\\{n_{1},\ldots,n_{d}\\}$, $n\in\mathbb{N}^{d}$)

$\max_{|n|_{\infty}\in\\{N,N-1\\}}|u_{n}|\leq(4d-1+\max_{|n|_{\infty}=N}|\widetilde{V}_{n}|)\max_{|n|_{\infty}\in\\{N,N+1\\}}|u_{n}|.$

If we write $M_{N}:=\max_{|n|_{\infty}\in\\{N,N-1\\}}|u_{n}|$ and
$q_{N}:=\max_{|n|_{\infty}=N}|\widetilde{V}_{n}|$, the recurrence inequality
$M_{N}\leq(4d-1+q_{N})M_{N+1}$ holds. Thus, repeating the same procedure

$M_{N+1}\geq\frac{M_{N-1}}{(4d-1+q_{N})(2d+1+q_{N-1})}\geq\cdots\geq\frac{M_{1}}{(2d+1+q_{N})(4d-1+q_{N-1})\cdots(2d+1+q_{1})}.$

This gives that if

$\max_{|n|_{\infty}\in\\{N+1,N\\}}|u_{n}|=M_{N+1}<\frac{M_{1}}{(4d-1+q_{N})(4d-1+q_{N-1})\cdots(4d-1+q_{1})},$

then $u\equiv 0$. By rescaling in $h$, we obtain the result. ∎

######  Corollary 7.3.

Let $h>0$ and $R>0$ be of the form $R=Nh$ with $N\in\mathbb{N}^{*}$. Let
$V:(h\mathbb{Z})^{d}\to\mathbb{R}$ be a bounded potential. Let $u$ be a
solution to (7.5) which satisfies the decay estimate

$\max_{|x|_{\infty}\in\\{R+h,R\\}}|u(x)|<e^{-\frac{R}{h}\log(4d-1+h^{2}\|V\|_{\infty})}\cdot
e^{\max_{|x|_{\infty}\in\\{h,0\\}}|u(x)|},$

then $u\equiv 0$.

###### Proof.

Since $V$ is bounded, we easily obtain that
$q_{n}\leq\|\widetilde{V}\|_{\infty}=h^{2}\|V\|_{\infty}$. Thus if

$\max_{|x|_{\infty}\in\\{R+h,R\\}}|u(x)|<\frac{\max_{|x|_{\infty}\in\\{h,0\\}}|u(x)|}{(\|\widetilde{V}\|_{\infty}+4d-1)^{N}},$

then $u$ satisfies the hypothesis of Lemma 7.2, and, therefore, $u\equiv 0.$
Notice that the right hand side of the above bound can be written as
$e^{-\frac{R}{h}\log(4d-1+\|\widetilde{V}\|_{\infty})}\max_{|x|_{\infty}\in\\{h,0\\}}|u(x)|$,
and we are done. ∎

######  Remark 7.4.

Observe that Corollary 7.3 is a scaled version of [LM18, Proposition 4.3].

As application of Lemma 7.2, we establish a unique continuation result with a
specific potential.

######  Example 7.5.

Let $h>0$ and $R>0$ be of the form $R=Nh$ with $N\in\mathbb{N}^{*}$. Let $u$
be a solution to

$\Delta_{d}u(x)+xu(x)=0,\quad x\in(h\mathbb{Z})^{d}.$

Then, if $u$ satisfies the decay condition

$|u(R)|<e^{-\frac{R}{h}\log(4d-1+h^{2}R)}\max_{|x|_{\infty}\in\\{h,0\\}}|u(x)|.$

then $u\equiv 0$.

###### Proof.

In this situation we obtain $\widetilde{V}(x)=h^{2}x$. Recalling the notation
from Lemma 7.2, in this case we have $q_{N}=h^{3}N$ and $q_{N-1}\cdots
q_{1}=(h^{3})^{N-1}(N-1)!$. Thus, by applying Lemma 7.2, we obtain that, if
$u$ satisfies the decay estimate

(7.6)
$|u(R)|<e^{-\log((4d-1+h^{3}N)(4d-1+h^{3}(N-1))\dots(4d-1+h^{3}))}\max_{|x|_{\infty}\in\\{h,0\\}}|u(x)|$

then $u\equiv 0.$ Notice that, a stronger decay condition on $u$ is

$|u(R)|<e^{-\log((4d-1+h^{3}N)^{N}}\max_{|x|_{\infty}\in\\{h,0\\}}|u(x)|=e^{-N\log(4d-1+h^{3}N)}\max_{|x|_{\infty}\in\\{h,0\\}}|u(x)|$

which can be rewritten as
$|u(R)|<e^{-\frac{R}{h}\log(4d-1+h^{2}R)}\max_{|x|_{\infty}\in\\{h,0\\}}|u(x)|$,
so the result holds. ∎

######  Remark 7.6.

Notice that the condition (7.6) implies:

$\displaystyle|u(R)|<e^{-\log((h^{3})^{N}N!)}\max_{|x|_{\infty}\in\\{h,0\\}}|u(x)|$
$\displaystyle\leq e^{-(3N)\log
h-\log(N!)}\max_{|x|_{\infty}\in\\{h,0\\}}|u(x)|$ $\displaystyle=e^{-(3N)\log
h-N\log N+N}\max_{|x|_{\infty}\in\\{h,0\\}}|u(x)|.$

For large $N$, assuming $h=1$ for simplicity, the aforementioned weaker decay
simplifies to $e^{-N\log N+N}$, which, in accordance with (B.15), is
essentially the behavior of the Bessel function $|J_{N}(t)|$ of order $N$ and
fixed argument $t$, as $N\to\infty$. Notably, the potential associated with
$J_{n}(t)$ is a solution to (7.5) is $2\big{(}\frac{n}{t}-1\big{)}$ and thus,
in this situation, $|J_{n}(t)|$ serves as an upper bound for solutions to
$\Delta_{d}u(n)+2\big{(}\frac{n}{t}-1\big{)}u(n)=0$. Hence, it would be
interesting to relax the condition (7.6) which, as we have just seen, it is
related to the behaviour of the Bessel function $J_{N}(t),$ to conclude that
$u\equiv 0$.

##  Appendix A Convergence to continuum Gaussian function

To support the fact that the weight involved in the decay conditions in
Theorem A and Theorem 1 can be seen as a discrete version of the inverse of
the Gaussian function, we rephrase here the convergence result in [CJK10,
Proposition 4.7] as follows.

######  Proposition A.1.

Let $h>0$ and $x\in\mathbb{R}$. For any $t>0$ we have

$\lim_{h\to
0}\frac{1}{h}e^{-2t/(xh)^{2}}I_{1/h}(2t/(xh)^{2})=\frac{x}{\sqrt{4\pi
t}}e^{-\frac{x^{2}}{4t}}.$

###### Proof.

The proof is the similar to[CJK10, Proposition 4.7] with the proper
modifications. We are going to prove that, for $j\in\mathbb{Z}$ and $h>0$ such
that $jh=1$, we have

$\lim_{\begin{subarray}{c}j\to\infty,h\to 0\\\
jh=1\end{subarray}}je^{-2tj^{2}/x^{2}}I_{j}(2tj^{2}/x^{2})=\frac{x}{\sqrt{4\pi
t}}e^{-\frac{x^{2}}{4t}}.$

We will use the identity
$I_{n}(t)=\frac{1}{\pi}\int_{0}^{\pi}e^{t\cos\theta}\cos\theta n\,d\theta$,
for $n\in\mathbb{Z}$ (see [PBM1, p. 456]). Then

$\displaystyle je^{-2t(j/x)^{2}}I_{j}\big{(}2t(j/x)^{2}\big{)}$
$\displaystyle=j\frac{e^{-2t(j/x)^{2}}}{\pi}\int_{0}^{\pi}e^{2t(j/x)^{2}\cos\theta}\cos(\theta
j)\,d\theta$ $\displaystyle=\frac{1}{\pi}\int_{0}^{\pi
j}e^{-2t(j/x)^{2}(1-\cos(y/j))}\cos y\,dy,$

where we made the change of variable $y=\theta j$ in the last equality. By
Taylor, for all $v\in[0,\pi]$

$0<c:=\frac{1}{2}-\frac{\pi^{2}}{24}\leq\frac{1}{2}-\frac{v^{2}}{24}\leq\frac{1-\cos
v}{v^{2}}.$

From here we obtain the uniform bound $\frac{1-\cos(y/j)}{(x/j)^{2}}\geq
c\big{(}\frac{y}{x}\big{)}^{2}$. Moreover
$\frac{1-\cos(y/j)}{(x/j)^{2}}\xrightarrow{j\to\infty}\frac{y^{2}}{2x^{2}}$.
Thus

$\int_{0}^{\pi j}|e^{-2t(j/x)^{2}(1-\cos(y/j))}\cos
y|\,dy\leq\int_{0}^{\infty}e^{-2t\frac{cy^{2}}{x^{2}}}\,dy=\frac{\sqrt{\pi}x}{2\sqrt{2c}t^{1/2}},$

so that by dominated convergence we have

$je^{-2t(j/x)^{2}}I_{j}\big{(}2t(j/x)^{2}\big{)}\xrightarrow{j\to\infty}\frac{1}{\pi}\int_{0}^{\infty}e^{-\frac{ty^{2}}{x^{2}}}\cos
y\,dy=\frac{x}{\sqrt{4\pi t}}e^{-\frac{x^{2}}{4t}},$

where the last identity can be found e.g. in [PBM1, p. 451]. ∎

######  Remark A.2.

The argument in the proof of Proposition A.1 can be extended to the
multidimensional case without effort. Indeed, we consider the multidimensional
discrete weight

$\frac{1}{h}e^{-\frac{2t}{h^{2}}\big{(}\frac{1}{x_{1}^{2}}+\cdots\frac{1}{x_{d}^{2}}\big{)}}\prod_{i=1}^{d}I_{1/h}(2t/(x_{i}h)^{2}),\quad\mathbf{x}=(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}.$

It holds, for $\mathbf{x}=(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}$,

$je^{-2tj^{2}\big{(}\frac{1}{x_{1}^{2}}+\cdots\frac{1}{x_{d}^{2}}\big{)}}\prod_{i=1}^{d}I_{j}(2t/(x_{i}h)^{2})=j\frac{1}{\pi}\prod_{i=1}^{d}\int_{0}^{\pi}e^{-2tj^{2}\frac{1}{x_{i}^{2}}}e^{2t\cos\theta_{i}}\cos(\theta_{i}j)\,d\theta_{i},$

and each integral can be treated separately as in the one dimensional case.
Observe also that we could also consider a multidimensional mesh with
different scale in every dimension, say $\mathbf{h}=(h_{1},\ldots,h_{d})$, and
then study the limits $h_{i}\to 0$ simultaneously.

##  Appendix B Some facts on Bessel functions

Let $I_{\nu}(z)$ be the modified Bessel function of first kind given by the
formula (see [Leb, Chapter 5, Section 5.7])

(B.1)
$I_{\nu}(z)=\sum_{k=0}^{\infty}\frac{(z/2)^{\nu+2k}}{\Gamma(k+1)\Gamma(k+\nu+1)},\qquad|z|<\infty,\quad|\operatorname{arg}z|<\pi$

and let $K_{\nu}$ be the Macdonald’s function of order $\nu$ (see also [Leb,
Chapter 5, Section 5.7])

(B.2)
$K_{\nu}(z)=\frac{\pi}{2}\frac{I_{-\nu}(z)-I_{\nu}(z)}{\sin\nu\pi},\qquad|\operatorname{arg}z|<\pi,\quad\nu\neq
0,\pm 1,\pm 2,\ldots.$

For integer $\nu=n$, we have that $K_{n}(z)=\lim_{\nu\to n}K_{\nu}(z)$,
$n=0,\pm 1,\pm 2,\ldots$. From (B.2),

(B.3) $K_{-\nu}(z)=K_{\nu}(z).$

Macdonald’s functions $K_{\nu}(z)$ satisfy

(B.4) $\frac{\partial}{\partial
z}K_{\nu}(z)=-\frac{1}{2}(K_{\nu+1}(z)+K_{\nu-1}(z)).$

They are increasing with respect to order (see [OlMax, (10.37)]), namely for
fixed argument,

(B.5) $K_{\nu}(z)<K_{\mu}(z),\quad\text{ for all }0\leq\nu<\mu,\quad\text{ for
all }z\in\mathbb{R}.$

They also fulfill the so-called Turán-type inequality (see for instance [S,
Theorem 8]),

(B.6) $K_{\nu}^{2}(z)\leq K_{\nu+1}(z)K_{\nu-1}(z),\quad\text{ for all
}\nu\in\mathbb{R},\,\,\,z>0$

and (see [Ba, Theorem 2])

(B.7)
$\frac{(K_{\nu}(z))^{2}}{K_{\nu+1}(z)K_{\nu-1}(z)}\geq\frac{z}{1+z},\quad|\nu|\geq\frac{1}{2},\,\,\,z>0.$

From [Ba, Theorem 2] by taking $\nu=0$ we deduce

(B.8)
$\frac{1}{1+\frac{1}{z}+\frac{1}{4z^{3}}}<\frac{K_{0}^{2}(z)}{K_{-1}(z)K_{1}(z)}<\frac{z}{z+1},\,\,\,z>0.$

The following monotonicity property of the product of Macdonald’s and modified
Bessel functions is proven in [S, Theorem 1]

(B.9)
$\frac{I_{\nu+1/2}(x)}{I_{\nu-1/2}(x)}<\frac{x}{\nu+\sqrt{\nu^{2}+x^{2}}}\leq\frac{K_{\nu-1/2}(x)}{K_{\nu+1/2}(x)}.$

We have the following asymptotics:

  * •

for large order and fixed $z\neq 0$, see [OlMax, (10.41.2)]

(B.10)
$K_{\nu}(z)\sim\sqrt{\frac{\pi}{2\nu}}\Big{(}\frac{ez}{2\nu}\Big{)}^{-\nu}\quad\nu\to\infty.$

  * •