# Two-sided Lieb-Thirring bounds

Sven Bachmann1, Richard Froese1, Severin Schraven1

1 Department of Mathematics, The University of British Columbia,
Vancouver, BC V6T 1Z2, Canada

###### Abstract

We prove upper and lower bounds for the number of eigenvalues of semi-bounded
Schrödinger operators in all spatial dimensions. As a corollary, we obtain
two-sided estimates for the sum of the negative eigenvalues of atomic
Hamiltonians with Kato potentials. Instead of being in terms of the potential
itself, as in the usual Lieb-Thirring result, the bounds are in terms of the
landscape function, also known as the torsion function, which is a solution of
$(-\Delta+V+M)u_{M}=1$ in ${\mathbb{R}}^{d}$; here $M\in{\mathbb{R}}$ is
chosen so that the operator is positive. We further prove that the infimum of
$(u_{M}^{-1}-M)$ is a lower bound for the ground state energy $E_{0}$ and
derive a simple iteration scheme converging to $E_{0}$.

##  1 Introduction

The Lieb-Thirring (LT) inequalities

$\mathrm{tr}((-\Delta+V)_{-}^{\gamma})\leq
L_{\gamma,d}\int_{\mathbb{R}^{d}}V_{-}(x)^{\gamma+\frac{d}{2}}dx$

are bounds on the moments of the negative eigenvalues of atomic Hamiltonians
in terms of the negative part of the potential $V=V_{+}-V_{-}$. In the context
of the stability of matter for which it was originally developed [1], only an
upper bound was needed and much of the vast amount of subsequent work
extending and refining the estimates, see for example [2, 3], has similarly
focussed on upper bounds. Among the few results about lower bounds, we mention
[4, 5, 6] in special cases as well as [7, 8] for somewhat different estimates.

In this work, we give a simple proof of a two-sided estimate on the number of
eigenvalues and their moments. The upper and lower bounds are matching in the
sense that they differ only by a scaling and a multiplicative constant, but we
are not able to conjecture the sharp values. The main difference with the LT
inequality is that the bounds are not in terms of the potential $V$, but
rather in terms of what we shall refer to as the _effective potential_
$\frac{1}{u_{M}}$ where $u_{M}$ is a solution of

$(-\Delta+V+M\cdot\mathds{1})u_{M}=1.$ (1.1)

Unlike in [9] where the potential $V$ is assumed (among other hypotheses) to
be a nonnegative function, our assumption here is that the corresponding
Schrödinger operator is semi-bounded below,

$E_{0}=\inf\mathrm{spec}(-\Delta+V)>-\infty.$

The constant in (1.1) is then $M>-E_{0}$. Among the auxiliary results of this
paper, we will in particular show the existence of a weak solution $u_{M}$ in
the whole space, see also [10].

The equation (1.1) has been extensively studied without the potential term,
i.e. for $-\Delta u_{M}=1$, and its solution is known as the _torsion
function_. It has been used to establish optimal Sobolev constant, estimate
the ground state energy of the Laplacian and to obtain pointwise estimates for
the eigenfunctions (and more general solutions) of elliptic operators [11, 12,
13, 14, 15, 16, 17, 18]. Following this line of work, the more general case
with a potential term was considered only recently [19].

The connection of the torsion function with the phenomenon of localization for
random Schrödinger operators was discovered in [20] in the case $M=0$. In this
context, it is mostly called the _landscape function_ and it has been observed
to yield excellent estimates on the density of states throughout the spectrum
[21, 22, 23, 24] and to predict both the localization centres and exponential
decay of localized eigenfunctions [25, 26, 27], see also [28]. As pointed out
in [29] (or also implicitly in [30, page 2903]) the predicting power of the
landscape function for the localization of eigenfunctions relies solely on the
fact that the resolvent of the Schrödinger operator is positivity-preserving,
a key property that we will use here as well.

The present work and our previous [9] focus on spectral estimates using the
landscape function in the non-random case. In this context, the effective
potential plays the role of a smoothing of the original potential at the right
scale: It allows in particular for Cwikel-Lieb-Rozenblum (CLR)-type bounds
[31, 32, 33] in cases where a strict CLR bound in terms of $V$ fails.
Specifically, the eigenvalue counting function can be estimated in terms of
the volume of the sublevel sets of the effective potential. In this context,
we mention [10, 34] for a magnetic version of the landscape function as well
as the closely related approaches via Brownian motion [30], regularizing
kernel [35], respectively maximal functions [36, 37, 38, 39, 40, 41, 42].

We shall see here that the key property for the effective potential to be
useful in estimating the eigenvalues of the Schrödinger operator is that
$u_{M}$ satisfies a Harnack inequality, see (2.3) below. While we are not
aware of the most general assumptions under which this holds, the results of
[43, 44] yield that it does for atomic potentials, namely for negative $V$’s
that are also in the Kato class (4), which are our main motivation here.

We state our main results in details in Section 2. In Section 3 we prove an
abstract two-sided CLR-type bound conditional on the existence of a landscape
function in the whole space and on a Harnack inequality. We then prove the
validity of these assumptions in the case of Kato-class potentials in Section
4. In this case, the bounds can in fact be notably improved. In Section 5, we
concentrate on the relationship between the infimum of the effective potential
and the bottom of the spectrum: In particular, we propose an iterative scheme
which we converges to the ground state energy. Section 6 serves as both
illustration of the results and analysis of their sharpness: On the one hand,
in the case of the shallow square well in one dimension, we show how well the
bottom of the effective potential approximates the ground state energy, and on
the other hand we prove that our bounds match the asymptotic distribution of
the eigenvalues at the bottom of the essential spectrum, in the examples of
attractive radial potentials with a power law singularities. The proof of the
two-sided Lieb-Thirring inequalities in terms of $\frac{1}{u_{M}}-M$ is
provided in Section 7.

##  2 Results

One of the central objects of interest in this paper is the eigenvalue
counting function for the Schrödinger operator $-\Delta+V$ on
$L^{2}({\mathbb{R}}^{d})$, which is the rank of the spectral projection
associated with $(-\infty,\mu]$, namely

$\mathcal{N}^{V}(\mu)=\mathrm{dim}\,\mathrm{Ran}\left(\mathbbm{1}_{(-\infty,\mu]}(-\Delta+V)\right).$
(2.1)

While most of the existing results in the literature are bounds on
$\mathcal{N}^{V}$ in terms of certain integrals of $V$, we aim here for
estimates in terms of the effective potential, $\frac{1}{u_{M}}$ where $u_{M}$
is, formally, the solution of (1.1). To start with, we shall assume its
existence, namely that there is a $u_{M}\in
H_{\mathrm{loc}}^{1}(\mathbb{R}^{d})$ such that

$\int_{\mathbb{R}^{d}}\big{(}\nabla
u_{M}(y)\cdot\nabla\varphi(y)+(V(y)+M)u_{M}(y)\varphi(y)\big{)}dy=\int_{\mathbb{R}^{d}}\varphi(y)dy$
(2.2)

for all $\varphi\in C_{c}^{\infty}(\mathbb{R}^{d})$. Our first bounds will be
in terms of the volume $\mathcal{V}_{M}(\mu)$ of the sublevel sets of the
effective potential

$\mathcal{V}_{M}(\mu)=\int_{\\{x\in\mathbb{R}^{d}\ :\
\frac{1}{u_{M}}\leq\mu\\}}dx.$

For the bound to hold, we assume what we shall refer to as a Harnack-Moser
inequality, namely that there is $C_{HM}>0$ such that

$\sup_{Q}u_{M}\leq C_{HM}\left(\inf_{Q}u_{M}+\ell(Q)^{2}\right)$ (2.3)

for all cubes $Q\subseteq\mathbb{R}^{d}$ with sidelength $\ell(Q)$ (for exact
definitions, see Section 3).

######  Theorem 2.1.

Let $V\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{d})$ be such that

$E_{0}=\inf\mathrm{spec}(-\Delta+V)>-\infty.$

Let $M>-E_{0}$ and let the solution $u_{M}$ of (2.2) be positive, such that
$u_{M}\in H_{\mathrm{loc}}^{1}(\mathbb{R}^{d})\cap
L^{\infty}_{\mathrm{loc}}(\mathbb{R}^{d})$ and $\frac{1}{u_{M}}\in
L^{\infty}_{\mathrm{loc}}(\mathbb{R}^{d})$. If (2.3) holds, then there exist
constants $c_{0,d},C_{0,d}>0$ depending only on $d,C_{HM}$ such that

$(c_{0,d}\mu)^{d/2}\mathcal{V}_{M}(c_{0,d}\mu)\leq\mathcal{N}^{V+M}(\mu)\leq(C_{0,d}\mu)^{d/2}\mathcal{V}_{M}(C_{0,d}\mu),$
(2.4)

for all $\mu\in\mathbb{R}$.

Note that the constants $c_{0,d},C_{0,d}$ depend on $M$ only via $C_{HM}$.

This bound in terms of the volume of the sublevel sets implies the following
Lieb-Thirring bounds (see [45] for a study on how shifting affects the Lieb-
Thirring inequalities in a similar setting).

######  Corollary 2.2.

Under the assumptions of Theorem 2.1 and if $E_{0}<0$, then

$c_{\gamma,d}\int_{\mathbb{R}^{d}}\left(|E_{0}|+\delta-\frac{1}{c_{0,d}u_{M}(x)}\right)_{+}^{\gamma+\frac{d}{2}}dx\leq\mathrm{tr}((-\Delta+V)_{-}^{\gamma})$
(2.5)

for all $\gamma>0$, and if $\gamma\geq 1$,

$\mathrm{tr}((-\Delta+V)_{-}^{\gamma})\leq
C_{\gamma,d}\int_{\mathbb{R}^{d}}\left(|E_{0}|+2\delta-\frac{1}{C_{0,d}u_{M}(x)}\right)_{+}^{\gamma+\frac{d}{2}}dx.$
(2.6)

Here, $\delta=M-|E_{0}|$ and

$c_{\gamma,d}=\frac{c_{0,d}^{d/2}\gamma}{\frac{d}{2}+\gamma}\min\left\\{1,\frac{\delta}{|E_{0}|}\right\\}^{\frac{d}{2}},\qquad
C_{\gamma,d}=\frac{C_{0,d}^{\frac{d}{2}}\gamma}{\frac{d}{2}+\gamma}\left(1+\frac{|E_{0}|}{\delta}\right)^{\frac{d}{2}},$

where $c_{0,d},C_{0,d}$ are the constants of Theorem 2.1 for the potential
$V+M$.

The Harnack-Moser inequality (2.3) controls the oscillations of $u_{M}$ at all
scales and is the natural choice in the present context. It originates in the
PDE literature in slightly different settings. On finite domains with $V\equiv
0$ this inequality holds always true for nonnegative solutions of (2.2), see
[46, Theorem 3.3]). On the other hand, for Kato-class potentials (see [44,
Theorem 2.5]) it is known that nonnegative solutions of the homogeneous
equation $(-\Delta+V)u=0$ satisfy a Harnack inequality for balls with small
radius, namely $\sup_{B(x,r)}u\leq C_{H}\inf_{B(x,r)}u$. The key difficulty to
overcome here is the lack of compactness as we are working in $\mathbb{R}^{d}$
and need the inequality to hold for arbitrary large cubes.

The assumptions of the above theorem, and in particular (2.3) can be verified
for the physically relevant potentials in the Kato-class ${\mathcal{K}}_{d}$
(see Section 4 for a precise definition). This includes all bounded potentials
as well as power laws singularities $|x|^{-\rho}$ for
$\rho\in[0,\min\\{d,2\\})$. A ‘crystal’ of Coulomb singularities is allowed
too.

######  Proposition 2.3.

Let $V$ be such that
$V_{+}\in\mathcal{K}_{d}^{\mathrm{loc}},V_{-}\in\mathcal{K}_{d}$. Then
$E_{0}=\inf\mathrm{spec}(-\Delta+V)>-\infty$. For all $M>-E_{0}$ there exists
a positive solution $u_{M}$ of (2.2) such that $u_{M}\in
L^{\infty}(\mathbb{R}^{d})\cap C^{0}(\mathbb{R}^{d})\cap
H_{\mathrm{loc}}^{1}(\mathbb{R}^{d})$ and $\frac{1}{u_{M}}\in
L^{\infty}_{\mathrm{loc}}(\mathbb{R}^{d})$. If, additionally,
$V_{+}\in\mathcal{K}_{d}$, then $u_{M}^{-1}\in L^{\infty}(\mathbb{R}^{d})$ and
$u_{M}$ satisfies (2.3). In fact,

$0<A_{M}=\frac{\sup_{\mathbb{R}^{d}}u_{M}}{\inf_{\mathbb{R}^{d}}u_{M}}<\infty.$

The fact that, with a Kato condition, the operator $-\Delta+V$ is bounded from
below is well-known, see [47, Theorem A.2.7.]). Note that the very last claim
of the theorem is a global Harnack inequality which implies (2.3), although
for a potentially suboptimal constant since $C_{HM}\leq A_{M}$.

The bound (2.4), while very general indeed, is not optimal as the spectral
parameter is shifted by $M$. In other words, a sharp estimate would use
$\frac{1}{u_{M}}-M$ instead of $\frac{1}{u_{M}}$ as the effective potential.
The following theorem realizes this improvement for Kato-class potentials.
Specifically, we consider the unshifted counting function
$\mathcal{N}^{V}(\mu)$ and show (i) that a lower bound in terms of coarsed-
grained volume of the sublevel sets of $\frac{1}{u_{M}}-M$ always holds, (ii)
that the coarse-graining can be removed if $\frac{1}{u_{M}}-M$ satisfies a
scale-invariant Harnack inequality and (iii) that a CLR-type upper bound, with
$V$ replaced by $\frac{1}{u_{M}}-M$, holds in dimensions $3$ and higher.

######  Theorem 2.4.

Let $V\in\mathcal{K}_{d}$ and $M>-E_{0}$. Let $u_{M}$ be the function given by
Proposition 2.3.

  1. i)

Let $\mu<0$. For any $c>1$, let

$n_{c}(\mu)=\left|\left\\{Q\in\mathcal{Q}_{(C_{c}|\mu|)^{-1/2}}\ :\
\sup_{Q}\left(\frac{1}{u_{M}}-M\right)\leq c\mu\right\\}\right|,\quad
C_{c}=\frac{c-1}{2^{d}(5A_{M})^{2}},$ (2.7)

where ${\mathcal{Q}}_{\ell}$ is a partition of $\mathbb{R}^{d}$ into cubes of
side length $\ell$, see Section 3. Then

$\mathcal{N}^{V}(\mu)\geq n_{c}(\mu).$ (2.8)

  2. ii)

Let $c>1$. Assume there exists $\widetilde{C}_{H}>0$ such that for all
$\ell>0$ and all $Q\in\mathcal{Q}_{\ell}$ with
$\sup_{Q}(\frac{1}{u_{M}}-M)\leq-\frac{c}{C_{c}\ell^{2}}$,

$\sup_{Q}\left(\frac{1}{u_{M}}-M\right)\leq\widetilde{C}_{H}\inf_{Q}\left(\frac{1}{u_{M}}-M\right).$
(2.9)

Then

$\mathcal{N}^{V}(\mu)\geq(C_{c}|\mu|)^{d/2}\left|\left\\{x\in\mathbb{R}^{d}\
:\ \frac{1}{u_{M}(x)}-M\leq\frac{c\mu}{\widetilde{C}_{H}}\right\\}\right|$
(2.10)

for all $\mu<0$.

  3. iii)

Let $d\geq 3$. There exist a constant $C_{\mathrm{CLR}}>0$ depending only on
$d$ such that

$\mathcal{N}^{V}(\mu)\leq
C_{\mathrm{CLR}}A_{M}^{d}\int_{\mathbb{R}^{d}}\left((\mu+\varepsilon)-\left(\frac{1}{u_{M}(x)}-M\right)\right)_{+}^{d/2}dx,$
(2.11)

for all $\mu\in\mathbb{R}$ and all $\varepsilon>0$.

Here again, these bounds yield upper and lower Lieb-Thirring inequalities. The
difference with the bounds of Corollary 2.2 cannot be understated: Here, the
effective potential appears as it really should, without an additional scaling
and shift.

######  Corollary 2.5.

Under the assumptions of Theorem 2.4(ii), respectively (iii),

$k_{\gamma,d}\int_{\mathbb{R}^{d}}\left(\frac{1}{u_{M}(x)}-M\right)_{-}^{\gamma+\frac{d}{2}}dx\leq\mathrm{tr}((-\Delta+V)_{-}^{\gamma})$
(2.12)

for all $\gamma>0$, and if $\gamma\geq 1$,

$\mathrm{tr}((-\Delta+V)_{-}^{\gamma})\leq
L_{\gamma,d}A_{M}^{2\gamma+d}\int_{\mathbb{R}^{d}}\left(\frac{1}{u_{M}(x)}-M\right)_{-}^{\gamma+\frac{d}{2}}dx,$
(2.13)

where $L_{\gamma,d}$ are the standard Lieb-Thirring constants and
$k_{\gamma,d}=C_{c}^{\frac{d}{2}}\frac{\gamma}{\gamma+\frac{d}{2}}\left(\frac{\widetilde{C}_{H}}{c}\right)^{\gamma+\frac{d}{2}}$.

While the above results are valid throughout the spectrum, their sharpness
comes into focus at two critical points: the bottom of the essential spectrum
and the bottom of the spectrum. The first proposition below shows that the
infimum of $\frac{1}{u_{M}}-M$ is a lower bound for the ground state energy
$E_{0}$, see also [28] for nonnegative potentials as well as the previous work
[13]. This is turn allows us to set up an iterative procedure converging to
$E_{0}$. Finally, we show that the CLR bound using $\frac{1}{u_{M}}-M$ yields
the exact asymptotics of the negative eigenvalues accumulating at $0^{-}$ for
‘atomic’ potentials $V(x)=-|x|^{-\rho}$.

######  Proposition 2.6.

Let $V\in\mathcal{K}_{d}$ and $M>-E_{0}$. Let $u_{M}$ be the function given by
Proposition 2.3. We have

$E_{0}\geq\inf_{\mathbb{R}^{d}}\left(\frac{1}{u_{M}}-M\right).$ (2.14)

It may seem that the proposition is relatively useless without a priori
information on the ground state energy. This is erroneous as the result can be
bootstrapped to provide a convergent sequence converging to $E_{0}$. As we
shall see in Section 6, the approximation may already be very good even after
one iteration, even if $M-E_{0}$ is very large.

######  Corollary 2.7.

Let $V\in\mathcal{K}_{d}$ and let

$\displaystyle F:(-E_{0},\infty)\rightarrow(-E_{0},\infty),\quad M\mapsto
M-\inf_{\mathbb{R}^{d}}\frac{1}{u_{M}},$

Let $M_{0}\in(-E_{0},\infty)$ and define recursively $M_{n+1}=F(M_{n})$ for
$n\in\mathbb{N}$. Then $M_{n+1}\leq M_{n}$ and $M_{n}\to-E_{0}$ as
$n\rightarrow\infty$. Moreover,

$\lim_{M\rightarrow(-E_{0})^{+}}\|u_{M}\|_{L^{\infty}(\mathbb{R}^{d})}=\infty.$
(2.15)

We remark that (2.15) is in stark contrast to the random case. A shifted
landscape function was considered in [48] in the case of a random potential.
There, it is shown that the limit $M\to-E_{0}^{+}$ can be taken after
averaging over the random potential. In the present case where no averaging is
available, we will see explicitly in Section 6 in the case of a radial
potential well that $u_{M}$ blows up at every single point in the same limit.

Finally, we turn to the asymptotics of the eigenvalue counting function as
$\mu\to 0^{-}$.

######  Proposition 2.8.

Let $\rho_{d}=\min\\{d,2\\}$. For $\rho\in(0,\rho_{d})$ the potential
$V(x)=-|x|^{-\rho}$ is in the Kato class ${\mathcal{K}}_{d}$. Let $M>-E_{0}$
and let $u_{M}$ be the function given by Proposition 2.3. Then as
$\mu\rightarrow 0^{-}$

$\displaystyle{\mathcal{N}}^{V}(\mu)$
$\displaystyle=(1+o(1))\left[(2\sqrt{\pi})^{d}\Gamma\left(\frac{d}{2}+1\right)\right]^{-1}\int_{\mathbb{R}^{d}}\left(\mu-\left(\frac{1}{u_{M}(x)}-M\right)\right)_{+}^{d/2}dx$
(2.16) $\displaystyle=(1+o(1)){\mathcal{N}}^{\frac{1}{u_{M}}-M}(\mu).$ (2.17)

The first equality above shows the exactness of the CLR asymptotics when using
the landscape function $\frac{1}{u_{M}}-M$, with the same constant as with the
original potential $V$. In the second equality, we point out that
${\mathcal{N}}^{\frac{1}{u_{M}}-M}$ is the counting function for the operator
$-\Delta+(\frac{1}{u_{M}}-M)$: This is non-trivial since $-\Delta+V$ is
unitarily equivalent to $-\frac{1}{u_{M}^{2}}\nabla\cdot
u_{M}^{2}\nabla+(\frac{1}{u_{M}}-M)$, namely for a modified kinetic energy.
This proves a conjecture of [22, Equation $(1.5)$] in this particular case.

To conclude this section, we point out that our proofs allow us to treat
magnetic Schrödinger operators with suitably well-behaved magnetic fields.

##  3 Variational argument: Proof for Theorem 2.1

We follow the strategy of [9] and first estimate $\mathcal{N}^{V}$ in terms of
some coarse-grained volume of the sublevel sets of the effective potential. In
a second step we will relate this to the measure of the sublevel set of
$1/u_{M}$.

We first introduce some notation. A box of sidelength $\ell$ is a set of the
form $\times_{i=1}^{d}[a_{i},b_{i}]$ where $b_{i}-a_{i}=\ell$ and
$a_{i},b_{i}\in\ell\mathbb{Z}$. For any $\ell>0$, we consider the collection
${\mathcal{Q}}_{\ell}$ of boxes of sidelength $\ell$ such that
$\bigcup_{Q\in{\mathcal{Q}}_{\ell}}Q={\mathbb{R}}^{d}$ and
$\mathring{Q}\cap\mathring{Q}^{\prime}=\emptyset$ whenever $Q\neq Q^{\prime}$.
We define for any $\mu>0$

$\displaystyle N(\mu)=\left|\left\\{Q\in{\mathcal{Q}}_{\mu^{-1/2}}\ :\
\inf_{Q}\frac{1}{u_{M}}\leq\mu\right\\}\right|$

and

$\displaystyle n(\mu)=\left|\left\\{Q\in{\mathcal{Q}}_{\mu^{-1/2}}\ :\
\sup_{Q}\,\frac{1}{u_{M}}\leq\mu\right\\}\right|.$

The reason we are considering only cubes with corner on $\ell\mathbb{Z}^{d}$
is to make it possible to compare ${\mathcal{Q}}_{\ell}$ with
${\mathcal{Q}}_{n\ell}$ for $n\in\mathbb{N}$, the former being a refinement of
the later.

For the class of potentials considered here, namely those satisfying the
Harnack condition (2.3), both coarse-grained volumes are directly related to
the measure ${\mathcal{V}}(\mu)$ of the sublevel set. The following lemma is
essentially the same as [9, Lemma 4.1].

######  Lemma 3.1.

Let $V$ satisfy the conditions of Theorem 2.1. Then

$\displaystyle n(\mu)\leq\mu^{d/2}{\mathcal{V}}_{M}(\mu)\leq N(\mu)\leq
n\Big{(}\left\lceil\sqrt{2C_{HM}}\right\rceil^{2}\mu\Big{)}$

for all $\mu\in\mathbb{R}$. Here $C_{HM}$ is the Harnack-Moser constant of
(2.3).

###### Proof.

The first two inequalities are immediate as, up to null sets,
$n(\mu)/\mu^{d/2}$ is the measure of all boxes that are strictly contained in
the sublevel set $\\{1/u_{M}\leq\mu\\}$ and $N(\mu)/\mu^{d/2}$ is the measure
of all the boxes that intersect the sublevel set.

Let $K=\lceil\sqrt{2C_{HM}}\rceil^{2}$, then the cubes in
${\mathcal{Q}}_{(K\mu)^{-1/2}}$ are subcubes of exactly one cube in
${\mathcal{Q}}_{\mu^{-1/2}}$ since $\sqrt{K}$ is an integer. To prove the last
inequality, it is sufficient to show that every cube in
${\mathcal{Q}}_{\mu^{-1/2}}$ which contributes to $N(\mu)$ admits a subcube in
${\mathcal{Q}}_{(K\mu)^{-1/2}}$ which contributes to $n(K\mu)$. If
$Q\in{\mathcal{Q}}_{\mu^{-1/2}}$ satisfies $\sup_{Q}\frac{1}{u_{M}}\leq\mu$
then all its subcubes contribute to $n(K\mu)$. The only other option for $Q$
to contribute to $N(\mu)$ is if
$\inf_{Q}\frac{1}{u_{M}}\leq\mu<\sup_{Q}\frac{1}{u_{M}}$. For those, we have
in particular $\sup_{Q}u_{M}\geq\frac{1}{\mu}$. Now pick a subcube
$\widetilde{Q}\in{\mathcal{Q}}_{(K\mu)^{-1/2}}$ of $Q$ such that
$\sup_{\widetilde{Q}}u_{M}\geq\frac{1}{\mu}$. Then by (2.3) we get, as
$K=\lceil\sqrt{2C_{HM}}\rceil^{2}\geq 2C_{HM}$,

$\displaystyle\inf_{\widetilde{Q}}u_{M}\geq\frac{1}{C_{HM}}\sup_{\widetilde{Q}}u_{M}-\frac{1}{K\mu}\geq\frac{1}{C_{HM}\mu}-\frac{1}{2C_{HM}\mu}=\frac{1}{2C_{HM}\mu}.$

Thus, $\sup_{\widetilde{Q}}\frac{1}{u_{M}}\leq 2C_{HM}\mu\leq K\mu$, and so
$\widetilde{Q}$ contributes to $n(K\mu)$. ∎

We now structure the proof of Theorem 2.1 in two lemmas for the upper,
respectively the lower bound.

######  Lemma 3.2.

Let $V$ satisfy the conditions of Theorem 2.1. Then

$\displaystyle\mathcal{N}^{V+M}(\mu)\leq N(C\mu)$

for all $C>\frac{4dC_{H}^{2}}{\pi^{2}}$ and all $\mu\in\mathbb{R}$.

###### Proof.

For $\mu\leq 0$ we have $\mathcal{N}^{V+M}(\mu)=0=N(C\mu)$. Thus, we will
assume for the rest of the proof that $\mu>0$. In order to have that
$\mathcal{N}^{V}(\mu)\leq N$ it suffices, by the Min-Max Principle (see [49,
Theorem XIII.2]), to exhibit a subspace
$\mathcal{H}_{N}\subseteq\mathrm{dom}(H^{1/2})$ with codimension at most $N$
such that

$\displaystyle\int_{\mathbb{R}^{d}}\left(|\nabla
v|^{2}+(V+M)|v|^{2}\right)>\mu\int_{\mathbb{R}^{d}}|v|^{2}$

for all $v\in\mathcal{H}_{N}$. Let $\mathcal{F}$ be the collection of boxes
such that

$\displaystyle\mathcal{F}=\left\\{Q\in{\mathcal{Q}}_{(C\mu)^{-1/2}}\ :\
\inf_{Q}\,\frac{1}{u_{M}}\leq\mu\right\\},$

where $C>0$ will be chosen later, and let

$\displaystyle\mathcal{H}_{N}=\left\\{v\in\mathrm{dom}(H^{1/2})\ :\
\int_{Q}\frac{v}{u_{M}}=0\quad\forall Q\in\mathcal{F}\right\\}.$

The codimension of $\mathcal{H}_{N}$ is equal to $|\mathcal{F}|=N(C\mu)$.

We first claim that

$\int_{\mathbb{R}^{d}}\left(|\nabla\varphi|^{2}+(V+M)|\varphi|^{2}\right)=\int_{\mathbb{R}^{d}}u_{M}^{2}\bigg{(}\left|\nabla\left(\frac{\varphi}{u_{M}}\right)\right|^{2}+\frac{1}{u_{M}}\left|\frac{\varphi}{u_{M}}\right|^{2}\bigg{)}$
(3.1)

for all $\varphi\in H^{1}(\mathbb{R}^{d})\cap L^{\infty}(\mathbb{R}^{d})$ with
compact support. As $u_{M}\in H^{1}_{\mathrm{loc}}(\mathbb{R}^{d})$ and
$1/u_{M}\in L^{\infty}_{\mathrm{loc}}(\mathbb{R}^{d})$ by assumption, we get
that $\varphi^{2}/u_{M}\in H^{1}(\mathbb{R}^{d})\cap
L^{\infty}(\mathbb{R}^{d})$ with compact support and thus, after a simple
approximation argument, we have

$\displaystyle\int_{\mathbb{R}^{d}}\left(\nabla
u_{M}\cdot\nabla\left(\frac{|\varphi|^{2}}{u_{M}}\right)+(V+M)u_{M}\frac{|\varphi|^{2}}{u_{M}}\right)=\int_{\mathbb{R}^{d}}\frac{|\varphi|^{2}}{u_{M}},$

since $u_{M}$ solves (2.2). Furthermore, using the product rule for Sobolev
functions [50, Lemma 7.4] yields

$\displaystyle\nabla
u_{M}\cdot\nabla\left(\frac{|\varphi|^{2}}{u_{M}}\right)=|\nabla\varphi|^{2}-u_{M}^{2}\left|\nabla\left(\frac{\varphi}{u_{M}}\right)\right|^{2},$

which readily implies (3.1). As $C_{c}^{\infty}(\mathbb{R}^{d})$ is dense in
the form domain of $-\Delta+V+M$ (see [47, Theorem A.2.8.]) the statement of
the lemma follows if we can show that there exists $\varepsilon>0$ such that
for all non-zero $\varphi\in
C_{c}^{\infty}(\mathbb{R}^{d})\cap{\mathcal{H}}_{N}$,

$\displaystyle\int_{\mathbb{R}^{d}}\bigg{(}u_{M}^{2}\left|\nabla\left(\frac{\varphi}{u_{M}}\right)\right|^{2}+\frac{|\varphi|^{2}}{u_{M}}\bigg{)}>(1+\varepsilon)\mu\int_{\mathbb{R}^{d}}|\varphi|^{2}.$

The additional $\varepsilon$ is needed to ensure that the strict inequality
(without $\varepsilon$) holds true on all of ${\mathcal{H}}_{N}$. We check
this inequality using the partition into boxes. In any box
$Q\notin\mathcal{F}$, the bound $\inf_{Q}1/u_{M}>\mu$ and the nonnegativity of
the first term yield the claim. If $Q\in\mathcal{F}$, we use that
$\varphi\in{\mathcal{H}}_{N}$, namely that the integral of $\varphi/u_{M}$
vanishes, together with Poincaré inequality on $Q$ with optimal constant
$\frac{\pi^{2}}{d}(C\mu)$, see [51], to conclude that

$\displaystyle\int_{Q}u_{M}^{2}|\nabla\left(\frac{\varphi}{u_{M}}\right)|^{2}\geq\left(\frac{\inf_{Q}u_{M}}{\sup_{Q}u_{M}}\right)^{2}\frac{\pi^{2}C\mu}{d}\int_{Q}|\varphi|^{2}.$

However, using the Harnack-Moser inequality (2.3) and the definition of
$\mathcal{F}$ we obtain

$\displaystyle
C_{HM}\inf_{Q}u_{M}\geq\sup_{Q}u_{M}-\frac{C_{HM}}{C\mu}\geq\left(1-\frac{C_{HM}}{C}\right)\frac{1}{\mu}.$

Thus, $\inf_{Q}u_{M}\geq\frac{1}{2C_{HM}\mu}$ for any $C\geq 2C_{HM}$ and
hence, again by (2.3)

$\displaystyle\sup_{Q}u_{M}\leq
C_{HM}\left(\inf_{Q}u_{M}+(C\mu)^{-1}\right)\leq 2C_{HM}\inf_{Q}u_{M}.$

This yields

$\displaystyle\int_{Q}u_{M}^{2}|\nabla\left(\frac{\varphi}{u_{M}}\right)|^{2}\geq\frac{C\pi^{2}}{4dC_{HM}^{2}}\mu\int_{Q}|\varphi|^{2}$

and so any choice of $C>\frac{4dC_{HM}^{2}}{\pi^{2}}$ will give the desired
estimate. ∎

######  Lemma 3.3.

Let $V$ satisfy the conditions of Theorem 2.1. Then

$\displaystyle n(\mu)\leq\mathcal{N}^{V+M}\left(C\mu\right)$

where $C=1+2^{d+2}C_{HM}^{2}$ and for all $\mu\in{\mathbb{R}}$.

###### Proof.

For a lower bound $N\leq\mathcal{N}^{V}(C\mu)$ it suffices, again by the Min-
Max Principle, to find a subspace
$\mathcal{H}_{N}\subseteq\mathrm{dom}(H^{1/2})$ of dimension at least $N$ such
that

$\displaystyle\int_{\mathbb{R}^{d}}\left(|\nabla
v|^{2}+(V+M)|v|^{2}\right)\leq C\mu\int_{\mathbb{R}^{d}}|v|^{2}.$

We define

$\displaystyle\mathcal{F}=\left\\{Q\in\mathcal{Q}_{\mu^{-1/2}}\ :\
\sup_{Q}\,\frac{1}{u_{M}}\leq\mu\right\\}.$

Furthermore, for a box $Q$ we pick $\chi_{Q}\in C_{c}^{\infty}(Q)$ with
$0\leq\chi_{Q}\leq
1,\|\nabla\chi_{Q}\|_{L^{\infty}(\mathbb{R}^{d})}\leq\widetilde{C}\mu^{1/2}$
for some $\widetilde{C}>4$, $\chi_{Q}\equiv 1$ on $Q/2$. Since the functions
$\chi_{Q}u_{M}$ are non-zero and orthogonal to each other, the space

$\displaystyle\mathcal{H}_{N}=\mathrm{span}\\{\chi_{Q}u_{M}\ :\
Q\in\mathcal{F}\\}$

is of dimension $|\mathcal{F}|=n(\mu)$.

By assumption we have $u_{M}\in H_{\mathrm{loc}}^{1}(\mathbb{R}^{d})\cap
L_{\mathrm{loc}}^{\infty}(\mathbb{R}^{d})$ and thus $\chi_{Q}u_{M}$ is in
$H^{1}(\mathbb{R}^{d})\cap L^{\infty}(\mathbb{R}^{d})$ with compact support.
By (3.1) and an approximation argument for $\chi_{Q}u_{M}$, we get

$\int_{\mathbb{R}^{d}}\left(|\nabla(\chi_{Q}u_{M})|^{2}+(V+M)\chi_{Q}^{2}u_{M}^{2}\right)=\int_{\mathbb{R}^{d}}\left(\chi_{Q}^{2}u_{M}+|\nabla\chi_{Q}|^{2}u_{M}^{2}\right),$
(3.2)

and hence, since $Q\in{\mathcal{F}}$,

$\displaystyle\int_{\mathbb{R}^{d}}\left(|\nabla(\chi_{Q}u_{M})|^{2}+(V+M)\chi_{Q}^{2}u_{M}^{2}\right)$
$\displaystyle\leq\left(\sup_{Q}\,\frac{1}{u_{M}}\right)\int_{Q}\chi_{Q}^{2}u_{M}^{2}+\widetilde{C}^{2}\mu\int_{Q}u_{M}^{2}$
$\displaystyle\leq\mu\left(\int_{Q}\chi_{Q}^{2}u_{M}^{2}+\widetilde{C}^{2}\int_{Q}u_{M}^{2}\right).$
(3.3)

We are left to replace the integrand of the second term by
$(\chi_{Q}u_{M})^{2}$. As $Q\in\mathcal{F}$ we have $\inf_{Q}u_{M}\geq 1/\mu$
and hence, by the Harnack-Moser inequality (2.3) we get

$\displaystyle\sup_{Q}u_{M}\leq
C_{HM}\left(\inf_{Q}u_{M}+\frac{1}{\mu}\right)\leq 2C_{HM}\inf_{Q}u_{M}.$

We conclude that

$\int_{Q}u_{M}^{2}\leq|Q|\,\sup_{Q}\,u_{M}^{2}\leq|Q|\,(2C_{HM})^{2}\,\inf_{Q/2}\,u_{M}^{2}\leq
2^{d+2}C_{HM}^{2}\int_{Q/2}u_{M}^{2}\leq
2^{d+2}C_{HM}^{2}\int_{\mathbb{R}^{d}}\chi_{Q}^{2}u_{M}^{2},$

where the last inequality follows from the properties of $\chi_{Q}$. This
yields

$\displaystyle
n(\mu)\leq{\mathcal{N}}^{V+M}((1+\widetilde{C}2^{d+2}C_{HM}^{2})\mu).$

As $\mu\mapsto{\mathcal{N}}^{V+M}(\mu)$ is right-continuous and any
$\widetilde{C}>4$ is admissible, we obtain the claim. ∎

##  4 Kato-class potentials

We now turn to the case of potentials $V$ that are in the Kato-class
$\mathcal{K}_{d}$. We say that a measurable function
$V:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is in the Kato-class $\mathcal{K}_{d}$
if

$\displaystyle\lim_{\varepsilon\rightarrow
0^{+}}\sup_{x\in\mathbb{R}^{d}}\int_{B(x,\varepsilon)}|x-y|^{-(d-2)}|V(y)|dy$
$\displaystyle=0,$ $\displaystyle d\geq 3,$
$\displaystyle\lim_{\varepsilon\rightarrow
0^{+}}\sup_{x\in\mathbb{R}^{d}}\int_{B(x,\varepsilon)}\ln(|x-y|^{-1})|V(y)|dy$
$\displaystyle=0,$ $\displaystyle d=2,$ (4.1)
$\displaystyle\sup_{x\in\mathbb{R}^{d}}\int_{B(x,1)}|V(y)|dy$
$\displaystyle<\infty,$ $\displaystyle d=1.$

We equip the sets ${\mathcal{K}}_{d}$ with the following norms:

$\displaystyle\|V\|_{\mathcal{K}_{d}}=\begin{cases}\displaystyle\sup_{x\in\mathbb{R}^{d}}\int_{B(x,1)}|x-y|^{-(d-2)}|V(y)|dy,&\qquad
d\geq 3,\\\
\displaystyle\sup_{x\in\mathbb{R}^{d}}\int_{B(x,1/2)}\ln(|x-y|^{-1})|V(y)|dy,&\qquad
d=2,\\\ \displaystyle\sup_{x\in\mathbb{R}^{d}}\int_{B(x,1)}|V(y)|dy,&\qquad
d=1.\end{cases}$

Furthermore, we say $V\in{\mathcal{K}}_{d}^{\mathrm{loc}}$ if
$\mathbbm{1}_{B(0,R)}V\in{\mathcal{K}}_{d}$ for all $R>0$.

We start by recalling some properties about the Green’s function of
$-\Delta+V$ in this particular case.

######  Theorem 4.1.

[47, Theorem B.7.2.] Suppose
$V_{+}\in\mathcal{K}_{d}^{\mathrm{loc}},V_{-}\in\mathcal{K}_{d}$. Let
$0<\alpha<d/2$ and $\mathrm{Re}(z)<\inf\mathrm{spec}(-\Delta+V)$ or
$z\notin\mathrm{spec}(-\Delta+V)$, $\alpha$ integral. Then
$(-\Delta+V-z)^{-\alpha}$ is an integral operator with integral kernel
$G^{(\alpha)}(x,y;z)$ obeying

  1. 1.

$G^{(\alpha)}(\cdot,\cdot,z)$ is continuous away from $x=y$ and bounded
uniformly in each region $\\{(x,y)\in\mathbb{R}^{d}\times\mathbb{R}^{d}\ :\
|x-y|\geq a\\}$ for $a>0$.

  2. 2.

We have

$|G^{(\alpha)}(x,y;z)|\leq C|x-y|^{-d+2\alpha}.$ (4.2)

  3. 3.

For $|x-y|$ sufficiently small and $|x|<R$

$|G^{(\alpha)}(x,y;z)|\geq\widetilde{C}_{R}|x-y|^{-d+2\alpha}.$ (4.3)

If $V\in\mathcal{K}_{d}$, $\widetilde{C}_{R}$ can be chosen independent of
$R$.

  4. 4.

For $|x-y|\geq 1$

$|G^{(\alpha)}(x,y;z)|\leq C_{\delta,\alpha,z}\exp(-\delta|x-y|)$ (4.4)

for some $\delta>0$ and if $\mathrm{Re}(z)<\inf\mathrm{spec}(-\Delta+V)$, for
all $\delta$ with
$\frac{1}{2}\delta^{2}<\inf\mathrm{spec}(-\Delta+V)-\mathrm{Re}(z)$.

This allows us to prove next that the Green’s function is nonnegative.

######  Lemma 4.2.

Under the assumptions of Proposition 2.3 we have for $d\geq 3$ that
$G^{(1)}(x,y;-M)\geq 0$ and for all $f\in L^{2}(\mathbb{R}^{d})$ we have that
$f\geq 0$ implies $(-\Delta+V+M)^{-1}f\geq 0$.

###### Proof.

By Theorem 4.1 we know that $G^{(1)}(\cdot,\cdot;-M)$ is continuous away from
the diagonal. Hence, it is enough to prove that $(-\Delta+V+M)^{-1}$ maps
nonnegative $L^{2}$ functions to nonnegative $L^{2}$ functions (otherwise
consider a positive bump function localize where the Green’s function is
negative to get a contradiction).

We note that for all $t>0$ we have that $(-\Delta+t)^{-1}$ maps nonnegative
$L^{2}$ functions to nonnegative $L^{2}$ functions, which can easily be seen
from the fact that the corresponding explicit integral kernel is positive. As
$V_{-}$ is form bounded by the Laplacian with relative bound zero (see [47,
Proposition A.2.3]) and
$\mathcal{D}((-\Delta)^{1/2})\cap\mathcal{D}(V_{+}^{1/2})$ is dense in
$L^{2}(\mathbb{R}^{d})$ (as $C_{c}^{\infty}(\mathbb{R}^{d})$ is contained in
this intersection), we get by [52, Lemma 2.1, Lemma 2.2] applied to $V_{+}$
and $V_{-}$ that $(-\Delta+V+M)^{-1}$ maps nonnegative $L^{2}$ functions to
nonnegative $L^{2}$ functions. ∎

###### Proof of Proposition 2.3.

We start with the case $d\geq 3$. In the end we will use Hadamard’s method of
descent to obtain all the properties for $d\in\\{1,2\\}$ from the case $d=3$.

By assumption $\inf\mathrm{spec}(-\Delta+V+M)>0$ and thus, the Green’s
function $G_{M}(x,y):=G^{(1)}(x,y;-M)$ satisfies the properties of Theorem
4.1. We define

$\displaystyle u_{M}(x)=\int_{\mathbb{R}^{d}}G_{M}(x,y)dy$

and will check that this function satisfies indeed all the properties of
Proposition 2.3.

The boundedness of $u_{M}$ follows immediately from (4.2) and (4.4). In order
to check that $u_{M}$ is a weak solution of (1.1) we follow the proof of [9,
Proposition 3.3] and set $u_{M,L}$ to be the solutions of

$(-\Delta+V+M)u_{M,L}=\mathbbm{1}_{B(0,L)}.$ (4.5)

Since $\mathbbm{1}_{B(0,L)}$ is in $L^{2}(\mathbb{R}^{d})$, Theorem 4.1
immediately implies that

$\displaystyle
u_{M,L}(x)=\int_{\mathbb{R}^{d}}G_{M}(x,y)\mathbbm{1}_{B(0,L)}(y)dy.$

Hence, by dominated convergence we have
$\lim_{L\rightarrow\infty}u_{M,L}(x)=u_{M}(x)$. As $u_{M,L}$ is a weak
solution of (4.5), one readily checks that $u_{M}$ is a solution of (2.2). The
fact that $1/u_{M}\in L^{\infty}_{\mathrm{loc}}(\mathbb{R}^{d})$. This follows
from the lower bound (4.3) together with the fact that the Green’s function
$G_{M}$ is nonnegative by Lemma 4.2. As $1/u_{M}\in
L_{\mathrm{loc}}^{\infty}(\mathbb{R}^{d})$ we get
$V+M+1/u_{M}\in{\mathcal{K}}_{d}^{\mathrm{loc}}$. The function $u_{M}$ is
nonnegative and is a distributional solution of
$(-\Delta+V+M+1/u_{M})u_{M}=0$. Thus, [43, Theorem 1.5] yields that $u_{M}$ is
continuous. Finally, $u_{M}\in H_{\mathrm{loc}}^{1}(\mathbb{R}^{d})$ follows
from a Caccioppoli-type inequality (see [44, Lemma 1.1]).

The cases $d\in\\{1,2\\}$ will now be covered by Hadamard’s method of descent.
Let’s start with $d=2$. We define
$\widetilde{V}(y_{1},y_{2},y_{3})=V(y_{1},y_{2})$ and check that
$\widetilde{V}$ satisfies the conditions of Proposition 2.3 for $d=3$. Once we
have done that, we can use the same proof as in [9, page 10] to conclude.
Similarly, one deals with the case $d=1$.

We first check that if $-\Delta_{\mathbb{R}^{d}}+V\geq M$, then we also have
$-\Delta_{\mathbb{R}^{3}}+\widetilde{V}\geq M$. To see this for $d=2$, we take
a Fourier transform in the variables and note that
$-\Delta_{\mathbb{R}^{3}}+\widetilde{V}$ is unitarily equivalent to the direct
integral

$\displaystyle\int_{\mathbb{R}}^{\oplus}(p^{2}+(-\Delta_{\mathbb{R}^{2}}+V))dp.$

Thus, by [49, Theorem XIII.85] we have
$\text{spec}(-\Delta_{\mathbb{R}^{3}}+\widetilde{V})=[\min\text{spec}(-\Delta_{\mathbb{R}^{2}}+V),\infty)$
and hence, establishes our first property.

We are left to show that for $d\in\\{1,2\\}$ if $V\in\mathcal{K}_{d}$
(respectively $\mathcal{K}_{d}^{\mathrm{loc}}$), then
$\widetilde{V}\in\mathcal{K}_{3}$ (respectively
$\mathcal{K}_{3}^{\mathrm{loc}}$). Let $d=2$, $x\in\mathbb{R}^{3}$ and
$0<\varepsilon\leq 1$. We compute

$\displaystyle\int_{B(x,\varepsilon)}\frac{|\widetilde{V}(y)}{|x-y|}dy$
$\displaystyle\leq
2\int_{B((0,0),\varepsilon)}|V(z_{1}+x_{1},z_{2}+x_{2})|\left(\frac{\pi}{2}+\ln\left(\frac{\varepsilon}{\sqrt{z_{1}^{2}+z_{2}^{2}}}\right)_{+}\right)dz_{2}dz_{3}$
$\displaystyle\leq(\pi+2)\int_{B((x_{1},x_{2}),\varepsilon)}\frac{|V(z_{1},z_{2})|}{|(x_{1},x_{2})-(z_{1},z_{1})|}dz_{1}dz_{2}.$

For $d=1$ we define $\widetilde{V}(y_{1},y_{2})=V(y_{1})$. We compute for
$x\in\mathbb{R}^{2}$ and $0<\varepsilon\leq 1$

$\displaystyle\int_{B(x,\varepsilon)}|\widetilde{V}(y)|\ln(|x-y|^{-1})dy$
$\displaystyle\leq\int_{B(0,\varepsilon)}|V(x_{1}+y_{1})|\left(\int_{-1}^{1}|\ln(|y_{2}|)|dy_{2}\right)dy_{1}$
$\displaystyle\leq
2\sup_{x_{1}\in\mathbb{R}}\int_{B(x_{1},\varepsilon)}|V(y_{1})|dy_{1}.$

Let $d=2$, let $\widetilde{V}(y_{1},y_{2},y_{3})=V(y_{1},y_{2})$ and let
$\widetilde{u}_{M}$ be the solution of

$\displaystyle(-\Delta_{\mathbb{R}^{3}}+\widetilde{V}+M)\widetilde{u}_{M}=1$

which we obtained in the case $d=3$. One readily checks that
$\widetilde{v_{M,L}}(x,t)=\widetilde{u_{M,L}}(x,t+\alpha)$ is a weak solution
of
$(-\Delta+\widetilde{V}+M)\widetilde{v_{M,L}}=\mathbbm{1}_{B((0,0,\alpha),L)}$.
Thus, we have

$\displaystyle\widetilde{u_{M,L}}(x,t+\alpha)=\int_{\mathbb{R}^{3}}G_{M}((x,t),y)\mathbbm{1}_{B((0,0,\alpha),L)}(y)dy.$

By dominated convergence we get

$\displaystyle\widetilde{u_{M}}(x,t+\alpha)=\lim_{L\rightarrow\infty}\widetilde{u_{M,L}}(x,t+\alpha)=\int_{\mathbb{R}^{3}}G_{M}((x,t),y)dy=\widetilde{u_{M}}(x,t).$

We define $u_{M}(x)=\widetilde{u_{M}}(x,0)$. One readily checks that $u_{M}$
inherits all the desired properties from $\widetilde{u_{M}}$. The same
strategy works for $d=1$.

Finally, we consider the case where $V_{+}\in\mathcal{K}_{d}$. In this case
the lower bound (4.3) holds true for all $x$ and hence
$\inf_{\mathbb{R}^{d}}u_{M}>0$. This implies in turn that $0<A_{M}<\infty$
since we have already established that $\sup_{\mathbb{R}^{d}}u_{M}<\infty$. ∎

Next we turn to the proof of Theorem 2.4.

###### Proof of Theorem 2.4.

(i) We adapt the proof in Lemma 3.3 to the potential $\frac{1}{u_{M}}-M$ and
define

$\displaystyle\mathcal{F}=\left\\{Q\in\mathcal{Q}_{|C_{c}\mu|^{-1/2}}\ :\
\sup_{Q}\left(\frac{1}{u_{M}}-M\right)\leq c\mu\right\\}$

Similarly, we pick a bump function $\chi_{Q}$ as before, but with
$\|\nabla\chi_{Q}\|_{L^{\infty}(\mathbb{R}^{d})}\leq 5|C_{c}\mu|^{1/2}$ and we
let

$\displaystyle\mathcal{H}=\mathrm{span}\left\\{\chi_{Q}u_{M}\ :\
Q\in\mathcal{F}\right\\},$

for which $\mathrm{dim}(\mathcal{H})=|\mathcal{F}|=n_{c}(\mu)$ holds. With
this, (3.2,3) become

$\displaystyle\int_{\mathbb{R}^{d}}\left(|\nabla(\chi_{Q}u_{M})|^{2}+V(\chi_{Q}u_{M})^{2}\right)$
$\displaystyle=\int_{Q}|\nabla\chi_{Q}|^{2}u_{M}^{2}+\int_{Q}\left(\frac{1}{u_{M}}-M\right)(\chi_{Q}u_{M})^{2}$
$\displaystyle\leq
5^{2}C_{c}|\mu|\int_{Q}u_{M}^{2}+c\mu\int_{\mathbb{R}^{d}}(\chi_{Q}u_{M})^{2}.$

Here, we use

$\int_{Q}u_{M}^{2}\leq|Q|\sup_{Q}u_{M}^{2}\leq|Q|A_{M}^{2}\inf_{{\mathbb{R}}^{d}}u_{M}^{2}\leq|Q|A_{M}^{2}\inf_{Q/2}u_{M}^{2}\leq
2^{d}(A_{M})^{2}\int_{Q}(\chi_{Q}u_{M})^{2}$

to conclude that

$\displaystyle\int_{\mathbb{R}^{d}}\left(|\nabla(\chi_{Q}u_{M})|^{2}+V(\chi_{Q}u_{M})^{2}\right)\leq\left(c-2^{d}(5A_{M})^{2}C_{c}\right)\mu\int_{\mathbb{R}^{d}}(\chi_{Q}u_{M})^{2}.$

Recalling that $C_{c}=\frac{c-1}{2^{d}(5A_{M})^{2}}$ we obtain that
$\mathcal{N}^{V}(\mu)\geq|\mathcal{F}|=n_{c}(\mu)$.

(ii) With the additional assumption (2.9), we have that

$n_{c}(\mu)\geq\left|\left\\{Q\in\mathcal{Q}_{(C_{c}|\mu|)^{-1/2}}\ :\
\inf_{Q}\left(\frac{1}{u_{M}}-M\right)\leq\frac{c\mu}{\widetilde{C}_{H}}\right\\}\right|.$

Just as in the proof of Lemma 3.1, this is now lower bounded by the measure of
the sublevel set.

(iii) Let $\mu\in\mathbb{R}$ and $\varepsilon>0$. For every $v$ in the form
domain of $-\Delta+V$ we have by (3.1)

$\displaystyle\int_{\mathbb{R}^{d}}$ $\displaystyle\left(|\nabla
v(x)|^{2}+(V(x)-\mu-\varepsilon)|v(x)|^{2}\right)dx$
$\displaystyle=\int_{\mathbb{R}^{d}}\left(\left|\nabla\left(\frac{v(x)}{u_{M}(x)}\right)\right|^{2}+\left(\frac{1}{u_{M}(x)}-M-\mu-\varepsilon\right)\left|\frac{v(x)}{u_{M}(x)}\right|^{2}\right)u_{M}(x)^{2}dx$
$\displaystyle\geq\left(\inf_{\mathbb{R}^{d}}u_{M}\right)^{2}\int_{\mathbb{R}^{d}}\left(\left|\nabla\left(\frac{v(x)}{u_{M}(x)}\right)\right|^{2}-A_{M}^{2}\left(\frac{1}{u_{M}(x)}-M-\mu-\varepsilon\right)_{-}\left|\frac{v(x)}{u_{M}(x)}\right|^{2}\right)dx.$

If the integral on the RHS of (2.11) is infinite, then there is nothing to
prove. Otherwise, we know from the standard CLR bound [33] that

$N:={\mathcal{N}}^{-A_{M}^{2}(1/u_{M}-M-\mu-\varepsilon)_{-}}\left(0\right)\leq
C_{\mathrm{CLR}}A_{M}^{d}\int_{\mathbb{R}^{d}}\left(\frac{1}{u_{M}(x)}-M-\mu-\varepsilon\right)_{-}^{d/2}dx$
(4.6)

and therefore, there exist linearly independent
$\varphi_{1},\dots,\varphi_{N}\in L^{2}(\mathbb{R}^{d})$ such that for $0\neq
w\in\\{f\in H^{1}(\mathbb{R}^{d})\ |\ \forall j\in\\{1,\dots,N\\}\ :\ \langle
f,\varphi_{j}\rangle_{L^{2}(\mathbb{R}^{d})}=0\\}$ we have

$\displaystyle\int_{\mathbb{R}^{d}}\left(\left|\nabla
w(x)\right|^{2}-A_{M}^{2}\left(\frac{1}{u_{M}(x)}-M-\mu-\varepsilon\right)_{-}\left|w(x)\right|^{2}\right)dx>0.$

Next, we define the codimension $N$ space

$\displaystyle{\mathcal{K}}=\left\\{v\in\mathrm{dom}((-\Delta+V)^{1/2})\ |\
\forall j\in\\{1,\dots,N\\}\ :\ \langle
v,\varphi_{j}u_{M}\rangle_{L^{2}(\mathbb{R}^{d})}=0\right\\}.$

For all $0\neq v\in{\mathcal{K}}\cap C_{c}^{\infty}(\mathbb{R}^{d})$ we have
by the above that

$\displaystyle\int_{\mathbb{R}^{d}}$ $\displaystyle\left(|\nabla
v(x)|^{2}+(V(x)-\mu-\varepsilon)|v(x)|^{2}\right)dx>0.$

As ${\mathcal{K}}\cap C_{c}^{\infty}(\mathbb{R}^{d})$ is dense
${\mathcal{K}}$, we conclude that

$\displaystyle\int_{\mathbb{R}^{d}}$ $\displaystyle\left(|\nabla
v(x)|^{2}+V(x)|v(x)|^{2}\right)dx>\mu\int_{\mathbb{R}^{d}}|v(x)|^{2}d$

for all non-zero $v\in{\mathcal{K}}$. Thus, by the min-max principle we get
that ${\mathcal{N}}^{V}(\mu)\leq N$. The claim follows from (4.6). ∎

##  5 The ground state energy

In this Section we will establish the relationship between the bottom of the
spectrum of $E_{0}=\inf\mathrm{spec}(-\Delta+V)$ and the infimum of the
shifted landscape function. We follow [28, Lemma 5], where a similar result is
proved for the torsion function in the case of nonnegative potentials.

###### Proof of Proposition 2.6.

Let $-\Delta_{L}$ denote the Dirichlet Laplacian on $B(0,L)$ and let $H_{L}$
be the operator $-\Delta_{L}+V+M$ on $L^{2}(B(0,L))$. Note that
$\sigma_{L}:=\inf\mathrm{spec}(H_{L})\geq\inf\mathrm{spec}(-\Delta+V+M)>0$.
Thus

$\displaystyle v_{L}=H_{L}^{-1}\mathbbm{1}_{B(0,L)}$

is well-defined. We remark that $v_{L}$ is nonnegative (integration by parts
yields that the Dirichlet Laplacian satisfies the Beurling-Deny condition [49,
Theorem XIII.50] and hence $e^{t\Delta_{L}}$ is positivity-preserving for
$t>0$ which implies via functional calculus that $H_{L}^{-1}$ is positivity-
preserving [49, Theorem XIII.44]).

Kato-class potentials are infinitesimally form-bounded by the Dirichlet
Laplacian (use [43, Theorem 4.7] to get that $V$ is is infinitesimally form-
bounded by $-\Delta$ on $\mathbb{R}^{d}$, which implies that the same holds
true on the ball) and as the Dirichlet Laplacian has compact resolvent ([49,
Theorem XIII.73 and its corollary]), so does $H_{L}$ (by [49, Theorem
XIII.68]). In particular, $\sigma_{L}$ is an eigenvalue. Let $\varphi_{L}$ be
an eigenfunction of $H_{L}$ with eigenvalue $\sigma_{L}$. We can choose
$\varphi_{L}>0$ (see [50, Theorem 11.8]). Furthermore, as $\varphi_{L}$ is
square-integrable and $B(0,L)$ has finite Lebesgue measure, we get that
$\varphi_{L}$ is integrable too. Thus, we get

$\displaystyle\int_{B(0,L)}\varphi_{L}=\int_{B(0,L)}\varphi_{L}(H_{L}v_{L})=\sigma_{L}\int_{B(0,L)}\varphi_{L}v_{L}\leq\sigma_{L}\|v_{L}\|_{L^{\infty}({\mathbb{R}}^{d})}\int_{B(0,L)}\varphi_{L}.$

This implies

$\displaystyle 1\leq\sigma_{L}\|v_{L}\|_{L^{\infty}({\mathbb{R}}^{d})}.$

Note that $(-\Delta+V+M)(u_{M}-v_{L})=0$ on $B(0,L)$ and $u_{M}-v_{L}=u>0$ on
$\partial B(0,L)$. Thus, by the weak maximum principle (see [53, Theorem 8.1])
we have $u_{M}>v_{L}$ on $B(0,L)$ and therefore,

$1\leq\sigma_{L}\|u_{M}\|_{L^{\infty}({\mathbb{R}}^{d})}.$ (5.1)

Note that
$\mathrm{dom}(H_{L}^{1/2})\subseteq\mathrm{dom}(H_{K}^{1/2})\subseteq\mathrm{dom}((-\Delta+V+M)^{1/2})$
for all $K\geq L\geq 0$. Thus, $L\mapsto\sigma_{L}$ is a decreasing function
and so

$\displaystyle\lim_{L\rightarrow\infty}\sigma_{L}\geq\inf\mathrm{spec}(-\Delta+V+M).$

Since $C_{c}^{\infty}(\mathbb{R}^{d})$ is dense in the form domain of
$-\Delta+V+M$ (see [54, Theorem 8.2.1], there is a sequence
$(\xi_{n})_{n\in\mathbb{N}}\subseteq C_{c}^{\infty}(\mathbb{R}^{d})$,
normalized in $L^{2}(\mathbb{R}^{d})$, such that

$\displaystyle\lim_{n\rightarrow\infty}\langle\xi_{n},H\xi_{n}\rangle_{L^{2}(\mathbb{R}^{d})}=\inf\mathrm{spec}(-\Delta+V+M).$

Now, for every $n$ there exists $L_{n}$ such that
$\xi_{n}\in\mathrm{dom}(H_{L_{n}}^{1/2})$ and so
$\langle\xi_{n},H\xi_{n}\rangle_{L^{2}(\mathbb{R}^{d})}\geq\sigma_{L_{n}}$. It
follows that

$\displaystyle\lim_{L\rightarrow\infty}\sigma_{L}=\inf\mathrm{spec}(-\Delta+V+M).$

Taking the limit $L\rightarrow\infty$ in (5.1) yields (2.14). ∎

We note that for nonnegative potentials, one can use the same proof as in [13,
Theorem 1], respectively [28, Lemma 5] to get the upper bound

$E_{0}\leq(4+d\log(2))\inf_{\mathbb{R}^{d}}\frac{1}{u_{M}}-M.$

Under additional assumption on the heat kernel of $-\Delta+V$ the dimensional
constant can be further improved [19, Theorem 1.5].

The difference between the bottom of the spectrum of $-\Delta+V$ and the
infimum of $1/u_{M}-M$ may a priori be large. However, the bound just proven
allows us to set up the iteration procedure described in Corollary 2.7 which
we now show to converges to the bottom of the spectrum.

###### Proof of Corollary 2.7.

By (2.14) we have that the function

$\displaystyle F:\quad(-E_{0},\infty)$
$\displaystyle\rightarrow(-E_{0},\infty)$ $\displaystyle M$
$\displaystyle\mapsto M-\inf_{\mathbb{R}^{d}}\frac{1}{u_{M}}$

is well defined. Furthermore, $u_{M}$ is positive and bounded above and so

$\displaystyle F(M)=M-\inf_{\mathbb{R}^{d}}\frac{1}{u_{M}}<M.$

This readily implies that the sequence $\\{M_{n}\\}_{n}$ is convergent with
$M_{\infty}=\lim_{n\rightarrow\infty}M_{n}\geq-E_{0}$. We claim that $F$ is
continuous, which implies that $F(M_{\infty})=M_{\infty}$. If
$M_{\infty}>-E_{0}$, then it is in the domain of $F$ and so
$F(M_{\infty})<M_{\infty}$, which is a contradiction. The continuity of $F$
follows from $u_{M+\delta}\leq u_{M}$ for all $\delta\geq 0$ (see [52, Lemma
2.1]) and the fact that $\lim_{\delta\rightarrow
0^{+}}u_{M+\delta}(x)=u_{M}(x)$ for all $x\in\mathbb{R}^{d}$ (using [52,
Theorem 1.3] and the continuity of the shifted landscape functions).

Finally,

$\displaystyle-E_{0}<M_{n}-\inf_{\mathbb{R}^{d}}\frac{1}{u_{M_{n}}}<M_{n}$

for all $n$. Since $M_{n}\to-E_{0}$, we immediately conclude that
$\lim_{n\to\infty}\inf_{\mathbb{R}^{d}}\frac{1}{u_{M_{n}}}=0$, which implies
(2.15). ∎

##  6 Examples

Since the constants that appear in the previous sections all depend on either
Harnack constants $C_{HM}$ or $A_{M}$, it is a priori unclear how sharp the
estimates in terms of the landscape function really are. In this section, we
provide a partial answer to that question by considering special cases. First
of all, we show that for radially symmetric potentials of the form
$-|v|^{-\rho}$ (in particular the Coulomb potential) whose eigenvalues
accumulate at the bottom of the essential spectrum $\mu=0$, the sharp constant
in the bound (2.11) can be computed so that it becomes asymptotically exact.
Secondly, we turn to the other end of the discrete spectrum and analyze the
first few eigenvalues appearing in case of the spherical well potential
$-\varepsilon\mathbbm{1}_{\mathbb{R}^{d}\setminus B(0,\delta)}$ for
$\varepsilon\ll 1$: The landscape function is explicit and its minimum
approximates the ground state energy to order $\varepsilon$.

###  6.1 Asymptotics at the bottom of the essential spectrum

The asymptotic for power law potentials is known and we first recall a general
result [55] in $d\geq 3$; the result extends to lower dimensions with
appropriate assumptions. For other variants see also [49, Theorem XIII.82] or
(with an oscillatory prefactor) [56, Theorem 3.1, Theorem 3.2].

######  Theorem 6.1.

Let $d\geq 3$. Let $1<R<\infty$ and $V:\mathbb{R}^{d}\rightarrow\mathbb{R}$
such that $V$ is $C^{1}$ for $|x|\geq R$ with
$\lim_{|x|\rightarrow\infty}V(x)=0$. There exists
$q:\mathbb{R}\rightarrow(-\infty,0]$ such that

$V(x)=q(|x|)+O(|x|^{-2})\quad\text{as}\quad|x|\rightarrow\infty.$ (6.1)

Moreover, $q$ is $C^{5}$ for $r\geq R$ and

$M_{1}r^{-1}\leq|q^{(k)}(r)/q^{(k-1)}(r)|\leq M_{2}r^{-1}\quad(k\leq 5)$ (6.2)

for all $r\geq R$ and for some $0<M_{1},M_{2}<\infty$. Finally, we assume that
there exists $\alpha>0$ such that $d+\alpha\geq 2$ and such that

$\int_{|x|\leq R}|V(x)|^{(d+\alpha)/2}dx<\infty.$ (6.3)

Then, as $\mu\rightarrow 0^{-}$

${\mathcal{N}}^{V}(\mu)=(1+o(1))\left[(2\sqrt{\pi})^{d}\Gamma\left(\frac{d}{2}+1\right)\right]^{-1}\int_{\mathbb{R}^{d}}\left(\mu-V(x)\right)_{+}^{d/2}dx.$
(6.4)

###### Proof of Proposition 2.8.

Our goal is to establish pointwise estimates for $u_{M}$ that match the
assumptions (6.1,6.2). In particular, we will construct a radial function $q$
such that $|u_{M}(x)-q(|x|)|\leq C_{M}/(1+|x|^{2})$.

By standard elliptic theory, we know that $u_{M}\in
C^{\infty}(\mathbb{R}^{d}\setminus\\{0\\})$. The strategy is to compute
successive approximations of $u_{M}$ in terms of powers of the potential. We
first note that

$\displaystyle(-\Delta+V+M)(u_{M}-\frac{1}{M}+\frac{V}{M^{2}})=-\frac{\Delta
V}{M^{2}}+\frac{V^{2}}{M^{2}},$

where $\Delta V$ decays fast, while $V^{2}$ does not. This formula suggests
the correct expansion away from the origin where the potential is singular.
Let $N\in\mathbb{N}$ be such that $N\rho<2\leq(N+1)\rho$. Then

$(-\Delta+V+M)\bigg{(}u_{M}-\sum_{j=0}^{N}\frac{(-V)^{j}}{M^{j+1}}\bigg{)}=-\sum_{j=0}^{N}\Delta\bigg{(}\frac{(-V)^{j}}{M^{j+1}}\bigg{)}+(-1)^{N+1}\frac{V^{N+1}}{M^{N+1}},$
(6.5)

since $(V+M)\sum_{j=0}^{N}\frac{(-V)^{j}}{M^{j+1}})$ is telescopic. By
assumption, the r.h.s. decays as $O(1/|x|^{2})$. Although the resolvent is a
bounded operator from $L^{\infty}(\mathbb{R}^{d})$ to itself, this is not
quite sufficient to conclude, both because of the singularity at the origin
and the insufficient decay at infinity. We shall therefore introduce cutoffs
in neighbourhoods of the origin and of infinity.

We choose a radial function $\chi\in C_{c}^{\infty}(\mathbb{R}^{d})$ with
$0\leq\chi\leq 1$, $\chi(x)\equiv 1$ for $1/2\leq|x|\leq 3/2$ and
$\chi(x)\equiv 0$ for $|x|\in[0,1/4]\cup[2,\infty)$. Let
$x_{0}\in\mathbb{R}^{d}\setminus\\{0\\}$ and let

$\tilde{\chi}(x)=\chi(|x_{0}|^{-1}x).$

With this, we define

$\displaystyle\widetilde{u_{M}}=(-\Delta+V\tilde{\chi}+M)^{-1}\tilde{\chi}.$

Let moreover

$\displaystyle u_{M,L}=(-\Delta+V+M)^{-1}\mathbbm{1}_{B(0,L)}.$

Now,

$(-\Delta+V\tilde{\chi}+M)\bigg{(}\widetilde{u_{M}}-\sum_{j=0}^{N}\frac{(-V)^{j}\tilde{\chi}^{j}}{M^{j+1}}\bigg{)}=-\sum_{j=1}^{N}\Delta\bigg{(}\frac{(-V)^{j}\tilde{\chi}^{j}}{M^{j+1}}\bigg{)}-\frac{(-V)^{N+1}\tilde{\chi}^{N+1}}{M^{N+1}},$
(6.6)

which readily implies that

$\bigg{\|}\widetilde{u_{M}}-\sum_{j=0}^{N}\frac{(-V)^{j}\tilde{\chi}^{j}}{M^{j+1}}\bigg{\|}_{L^{\infty}(\mathbb{R}^{d})}\leq\frac{C}{|x_{0}|^{2}},$
(6.7)

where $C$ only depend on $\rho,d,M,\|\chi\|_{C^{2}(\mathbb{R}^{d})}$ and
$\|(-\Delta+V+M)^{-1}\|_{\infty,\infty}$, but not on $x_{0}$.
We now need to estimate the difference
$u_{M,L}(x_{0})-\widetilde{u_{M}}(x_{0})$. We have

$(-\Delta+V\tilde{\chi}+M)(u_{M,L}-\widetilde{u_{M}})=V(\tilde{\chi}-1)u_{M,L}+\mathbbm{1}_{B(0,L)}-\tilde{\chi}$
(6.8)

which is nonnegative whenever $L\geq 2|x_{0}|$. Since
$(-\Delta+V\tilde{\chi}+M)^{-1}$ is positivity-preserving, this implies that
$u_{M,L}-\widetilde{u_{M}}\geq 0$ for $L\geq 2|x_{0}|$. Furthermore, if
additionally $|x_{0}|\geq 2$, then we have

$\displaystyle(-\Delta+V\tilde{\chi}+M)(u_{M,L}-\widetilde{u_{M}})=0$

on the annulus $B(0,|x_{0}|+1)\setminus B(0,|x_{0}|-1)$. Note that
$u_{M,L}-\widetilde{u_{M}}$ is radial and, by the weak maximum principle,
monotone in the radius and so

$|u_{M,L}(x_{0})-\widetilde{u_{M}}(x_{0})|\leq|B(0,1/2)|^{-1/2}\|u_{M,L}-\widetilde{u_{M}}\|_{L^{2}(B(x_{0},1))}.$
(6.9)

Hence, we are left to obtain a local $L^{2}$ estimate in a neighbourhood of
$x_{0}$. For this we will use yet another cut-off function. Namely, we choose
$\Xi\in C_{c}^{\infty}(\mathbb{R})$ such that $\frac{1}{e^{2}}\leq\Xi(r)\leq
1$ for $|r|\leq 2$, $\Xi(r)=1$ for $|r|\leq 1$ and $\Xi(r)=e^{-r}$ for $r>2$.
Then we define

$\displaystyle\Xi_{\varepsilon}(x)=\Xi(\varepsilon(|x-x_{0}|)),$

where we will choose $0<\varepsilon\leq 1$ later to be sufficiently small. We
compute

$\displaystyle\nabla\Xi_{\varepsilon}(x)=\Xi^{\prime}(\varepsilon|x-x_{0}|)\varepsilon\frac{x-x_{0}}{|x-x_{0}|}$

and

$\displaystyle\Delta\Xi_{\varepsilon}(x)=\Xi^{\prime\prime}(\varepsilon|x-x_{0}|)\varepsilon^{2}+\varepsilon\frac{d-1}{|x-x_{0}|}\Xi^{\prime}(\varepsilon|x-x_{0}|).$

Thus, for $|x-x_{0}|\geq 2/\varepsilon$ we have

$\displaystyle|\nabla\Xi_{\varepsilon}(x)|+|\Delta\Xi_{\varepsilon}(x)|\leq\varepsilon\left[1+\varepsilon\left(1+\frac{d-1}{2}\right)\right]\Xi_{\varepsilon}(x).$

As $1/e^{2}\leq\Xi(r)\leq 1$ for $|r|\leq 2$, we get for all
$x\in\mathbb{R}^{d}$

$|\nabla\Xi_{\varepsilon}(x)|+|\Delta\Xi_{\varepsilon}(x)|\leq\varepsilon\left[e^{2}\left(\|\Xi^{\prime}\|_{L^{\infty}([0,2])}+\|\Xi^{\prime\prime}\|_{L^{\infty}([0,2])}\right)+\left(2+\frac{d-1}{2}\right)\right]\Xi_{\varepsilon}(x)=C\varepsilon\Xi_{\varepsilon}(x),$
(6.10)

with $C$ independent of $\varepsilon,x_{0},x$.

With these estimates in hand, we go back to (6.8). We will write
$\|\cdot\|_{p,q}$ for the operator norm
$\|\cdot\|_{{\mathcal{L}}(L^{p}(\mathbb{R}^{d}),L^{q}(\mathbb{R}^{d}))}$.
Furthermore, we will repeatedly use the fact (see [52, Lemma 2.1]) that for
all $V\leq W\leq 0$ and all $f\in L^{2}(\mathbb{R}^{d})$,

$(-\Delta+W+M)^{-1}|f|\leq(-\Delta+V+M)^{-1}|f|,$

where the inequality has to be understood pointwise almost everywhere. This
implies in particular the operator monotonicity of
$V\mapsto(-\Delta+V+M)^{-1}$ in $L^{2}({\mathbb{R}}^{d})$. In turn, the same
holds by density for $\|(-\Delta+V+M)^{-1}\|_{1,2}$.

Commuting the multiplication operator $\Xi_{\varepsilon}$ through the
resolvent yields

$\Xi_{\varepsilon}(u_{M,L}-\widetilde{u_{M}})=(-\Delta+V\tilde{\chi}+M)^{-1}\Xi_{\varepsilon}(V(\tilde{\chi}-1)u_{M,L}+\mathbbm{1}_{B(0,L)}-\tilde{\chi})\\\
\quad+(-\Delta+V\tilde{\chi}+M)^{-1}\left[-2\nabla\cdot(\nabla\Xi_{\varepsilon})+(\Delta\Xi_{\varepsilon})\right](u_{M,L}-\widetilde{u_{M}}).$
(6.11)

Using (6.10) we can estimate the last term in $L^{2}$ by

$\|(-\Delta+V\tilde{\chi}+M)^{-1}\left[-2\nabla\cdot(\nabla\Xi_{\varepsilon})+(\Delta\Xi_{\varepsilon})\right](u_{M,L}-\widetilde{u_{M}})\|_{L^{2}(\mathbb{R}^{d})}\\\
\leq\left[2\|(-\Delta+V\tilde{\chi}+M)^{-1}\nabla\|_{2,2}+\|(-\Delta+V\tilde{\chi}+M)^{-1}\|_{2,2}\right]C\varepsilon\|\Xi_{\varepsilon}(u_{M,L}-\widetilde{u_{M}})\|_{L^{2}(\mathbb{R}^{d})}.$
(6.12)

Since $V\leq V\tilde{\chi}\leq 0$, and with the fact that $V$ is form bounded
by $-\Delta$ with relative bound zero, we conclude that

$\displaystyle\|(-\Delta+V\tilde{\chi}+M)^{-1}\nabla\|_{2,2}\leq\|(-\Delta+V+M)^{-1/2}\|_{2,2}\|(-\Delta+V+M)^{-1/2}\nabla\|_{2,2}<\infty.$

Hence, the term of (6.12) in $[\cdots]$ is bounded and there is
$0<\varepsilon\leq 1$ such that

$\|(-\Delta+V\tilde{\chi}+M)^{-1}\left[-2\nabla\cdot(\nabla\Xi_{\varepsilon})+(\Delta\Xi_{\varepsilon})\right](u_{M,L}-\widetilde{u_{M}})\|_{L^{2}(\mathbb{R}^{d})}\leq\frac{1}{2}\|\Xi_{\varepsilon}(u_{M,L}-\widetilde{u_{M}})\|_{L^{2}(\mathbb{R}^{d})}.$

With this, (6.11) yields

$\|\Xi_{\varepsilon}(u_{M,L}-\widetilde{u_{M}})\|_{L^{2}(\mathbb{R}^{d})}\leq
2\|(-\Delta+V\tilde{\chi}+M)^{-1}\Xi_{\varepsilon}(V(\tilde{\chi}-1)u_{M,L}+\mathbbm{1}_{B(0,L)}-\tilde{\chi})\|_{L^{2}(\mathbb{R}^{d})}$

We now split
$V(\tilde{\chi}-1)u_{M,L}=V(\tilde{\chi}-1)(\mathbbm{1}_{\mathbb{R}^{d}\setminus
B(0,1)}+\mathbbm{1}_{B(0,1)})u_{M,L}$. Using that

$|V(\tilde{\chi}-1)\mathbbm{1}_{\mathbb{R}^{d}\setminus
B(0,1)}u_{M,L}+\mathbbm{1}_{B(0,L)}-\tilde{\chi}|\leq\mathbbm{1}_{\mathbb{R}^{d}\setminus
B(x_{0},|x_{0}|/2)},$

the definition of $\tilde{\chi}$ and the fact that $(-\Delta+V+M)^{-1}$ is
bounded from $L^{1}(\mathbb{R}^{d})$ to $L^{2}(\mathbb{R}^{d})$ (see [47,
Theorem B.2.2.]) we conclude that

$\displaystyle\|\Xi_{\varepsilon}(u_{M,L}-\widetilde{u_{M}})\|_{L^{2}(\mathbb{R}^{d})}$
$\displaystyle\leq
2\|(-\Delta+V+M)^{-1}\|_{2,2}\|\Xi_{\varepsilon}\mathbbm{1}_{\mathbb{R}^{d}\setminus
B(x_{0},|x_{0}|/2)}\|_{L^{2}(\mathbb{R}^{d})}$
$\displaystyle\quad+2\|(-\Delta+V+M)^{-1}\|_{1,2}\|\Xi_{\varepsilon}u_{M,L}\|_{L^{\infty}(B(0,1))}\|V\|_{L^{1}(B(0,1))}$
$\displaystyle\leq Ce^{-\varepsilon|x_{0}|/4},$

because of our choice of $\Xi_{\varepsilon}$, where $C$ does not depend on
$x_{0}$ or $L$. Using (6.9), the fact that
$1/e^{2}\leq\Xi_{\varepsilon}(x)\leq 1$ for $x\in B(x_{0},1)$, we finally
obtain for $|x_{0}|\geq 4/\varepsilon$

$\displaystyle|u_{M,L}(x_{0})-\widetilde{u}(x_{0})|\leq
Ce^{-\varepsilon|x_{0}|/4},$

where $\varepsilon,C$ do not depend on $L$ or $x_{0}$ (but on $M$). Combining
this with (6.7) and noting that $\tilde{\chi}=1$ in a neighbourhood of $x_{0}$
we get for $|x_{0}|$ sufficiently large (independent of $L$)

$\displaystyle|u_{M,L}(x_{0})-\sum_{j=0}^{N}\frac{(-V(x_{0}))^{j}}{M^{j+1}}|\leq\frac{C}{|x_{0}|^{2}},$

with $C$ independent of $L,x_{0}$.

As $u_{M,L}$ converges pointwise to $u_{M}$ for $L\rightarrow\infty$, we
conclude that

$\displaystyle|u_{M}(x)-\sum_{j=0}^{N}\frac{(-V(x))^{j}}{M^{j+1}}|\leq\frac{C}{|x|^{2}},$

for all $|x|$ sufficiently large. Recalling now that $-V(x)=|x|^{-\rho}$, this
readily implies that there exists a polynomial $P$ with real coefficients of
degree at most $N$ and $P(0)=0=P^{\prime}(0)$ such that

$\frac{1}{u_{M}(x)}-M+V(x)-P(|x|^{-\rho})=O(|x|^{-2})$ (6.13)

for $|x|\rightarrow\infty$. Moreover, the boundedness of $1/u_{M}$ tells us
that $\frac{1}{u_{M}(x)}-M$ satisfies (6.3).

Altogether, we obtain first that

${\mathcal{N}}^{V}(\mu)=(1+o(1))\left[(2\sqrt{\pi})^{d}\Gamma\left(\frac{d}{2}+1\right)\right]^{-1}\int_{\mathbb{R}^{d}}\left(\mu-V(x)\right)_{+}^{d/2}dx$

as $\mu\to 0^{-}$, since $V$ clearly satisfies the assumptions of Theorem 6.1.
By (6.13), this further implies that

${\mathcal{N}}^{V}(\mu)=(1+o(1))\left[(2\sqrt{\pi})^{d}\Gamma\left(\frac{d}{2}+1\right)\right]^{-1}\int_{\mathbb{R}^{d}}\left(\mu-\left(\frac{1}{u_{M}(x)}-M\right)\right)_{+}^{d/2}dx.$

Finally, (6.13) also implies that Theorem 6.1 applies to $\frac{1}{u_{M}}-M$
and therefore that

${\mathcal{N}}^{V}(\mu)=(1+o(1)){\mathcal{N}}^{1/u_{M}-M}(\mu),$

which concludes the proof. ∎

In the case of the Coulomb potential, the proof above yields that the
effective potential asymptotically equals the original potential itself,
namely

$\frac{1}{u_{M}(x)}-M=-\frac{1}{|x|}+O(|x|^{-2}).$

###  6.2 Ground state energy for the square well

We now turn to $V(x)=-\varepsilon\mathbbm{1}_{B(0,\delta)}$ for
$\varepsilon,\delta>0$ and focus on the case $d=1$. The ground state energy of
$-\Delta+\varepsilon\mathbbm{1}_{\mathbb{R}\setminus(-\delta,\delta)}$ is, for
$\varepsilon$ small enough, given by the smallest positive solution of the
transcendental equation

$\displaystyle\sqrt{\varepsilon-x}=\sqrt{x}\tan(\sqrt{x}\delta).$

This can be rewritten as

$\displaystyle\varepsilon=F(x),\quad\text{where}\quad
F(x)=x\left(1+\tan(\sqrt{x}\delta)^{2}\right).$

The function $F$ has an analytic extension at the origin such that
$F(x)=x+x^{2}\delta^{2}+O(x^{3})$, and hence

$\displaystyle
x=F^{-1}(\varepsilon)=\varepsilon-\delta^{2}\varepsilon^{2}+O(\varepsilon^{3}).$

This implies that as $\varepsilon\to 0^{+}$

$\displaystyle\inf\mathrm{spec}(-\Delta-\varepsilon\mathbbm{1}_{(-\delta,\delta)})=\inf\mathrm{spec}(-\Delta+\varepsilon\mathbbm{1}_{\mathbb{R}\setminus(-\delta,\delta)})-\varepsilon=-\delta^{2}\varepsilon^{2}+O(\varepsilon^{3}).$

Next we compute the shifted landscape function, which is an even solution of

$-f^{\prime\prime}(x)+cf(x)=1$

where the constant $c$ depends on $M$ and whether $x$ is inside or outside of
the well.

We start with the case $M>\varepsilon$, in which case

$\displaystyle
u_{M}(x)=\begin{cases}\frac{1}{M-\varepsilon}+a_{2}\cosh(\sqrt{M-\varepsilon}x),&|x|<\delta,\\\
\frac{1}{M}+b_{1}e^{-\sqrt{M}|x|}+b_{2}\cosh(\sqrt{M}x),&|x|>\delta.\end{cases}$

As $u_{M}$ is bounded, we get $b_{2}=0$. The other constants are set by the of
$u_{M}$ differentiability at $|x|=\delta$:

$\displaystyle a_{2}$
$\displaystyle=\frac{\frac{1}{M}-\frac{1}{M-\varepsilon}}{\cosh(\sqrt{M-\varepsilon}\delta)+\frac{\sqrt{M-\varepsilon}}{\sqrt{M}}\sinh(\sqrt{M-\varepsilon}\delta)}$
$\displaystyle b_{1}$
$\displaystyle=-\frac{\sqrt{M-\varepsilon}}{\sqrt{M}}\sinh(\sqrt{M-\varepsilon}\delta)e^{\sqrt{M}\delta}a_{2}.$

The maximum of $u_{M}$ is attained at the origin and so

$\inf_{\mathbb{R}}\frac{1}{u_{M}}-M=\varepsilon\frac{M-\varepsilon}{M}\frac{1}{\cosh(\sqrt{M-\varepsilon}\delta)+\frac{\sqrt{M-\varepsilon}}{\sqrt{M}}\sinh(\sqrt{M-\varepsilon}\delta)}-\varepsilon+o(\varepsilon).$

Hence, for any $M>\epsilon$ — in particular arbitrarily large! — the first
approximation of the ground state energy given by Corollary 2.7 is already of
order $\varepsilon$.

For $M=\varepsilon$, the computation in [9, Section 6] yields

$\displaystyle\inf_{\mathbb{R}}\frac{1}{u_{\varepsilon}}-\varepsilon=\frac{-\delta\varepsilon^{3/2}-\frac{\delta^{2}\varepsilon^{2}}{2}}{1+\delta\sqrt{\varepsilon}+\frac{\delta^{2}\varepsilon}{2}}.$

Hence, for small $\varepsilon$, this yields an approximation of order
$\varepsilon^{3/2}$.

Finally if $M<\varepsilon$, then

$\displaystyle
u_{M}(x)=\begin{cases}\frac{1}{M-\varepsilon}+a\cos(\sqrt{\varepsilon-M}x),&|x|<\delta,\\\
\frac{1}{M}+be^{-\sqrt{M}|x|},&|x|>\delta.\end{cases}$

Differentiability yields now

$\displaystyle a$
$\displaystyle=\frac{\frac{1}{M}-\frac{1}{M-\varepsilon}}{\cos(\sqrt{\varepsilon-M}\delta)-\frac{\sqrt{\varepsilon-M}}{\sqrt{M}}\sin(\sqrt{\varepsilon-M}\delta)}$
(6.14) $\displaystyle b$
$\displaystyle=\frac{\sqrt{\varepsilon-M}}{\sqrt{M}}\sin(\sqrt{\varepsilon-M}\delta)e^{\sqrt{M}\delta}a$
(6.15)

This allows us to exhibit explicitly the blowup of (2.15) in Corollary 2.7.
For this we first note that $\varepsilon+E_{0}$ is the ground state energy of
the operator
$-\Delta+\varepsilon\mathbbm{1}_{(-\infty,-\delta)\cup(\delta,\infty)}(x)$ and
hence

$\displaystyle\sqrt{-E_{0}}=\sqrt{\varepsilon+E_{0}}\tan(\sqrt{\varepsilon+E_{0}}\delta).$

It follows from this and (6.14) that $\lim_{M\to(-E_{0})^{+}}a=\infty$ and the
same for $b$ by (6.15). Hence $\lim_{M\to(-E_{0})^{+}}u_{M}(x)=\infty$, or in
other words for the effective potential

$\lim_{M\to(-E_{0})^{+}}\left(\frac{1}{u_{M}(x)}-M\right)=E_{0}$

for all $x\in{\mathbb{R}}$.

##  7 Upper and lower Lieb-Thirring bounds

In this section we recall the standard argument for the derivation of the
Lieb-Thirring inequality from estimates on the eigenvalue counting function.
First we rewrite the sum of eigenvalues in terms of an integral.

######  Lemma 7.1.

For any $\gamma>0$ we have

$\mathrm{tr}((-\Delta+V)_{-}^{\gamma})=\gamma\int_{0}^{\infty}\lambda^{\gamma-1}\mathcal{N}^{V}(-\lambda)d\lambda.$
(7.1)

###### Proof.

If $\mathcal{N}^{V}(-\lambda)=\infty$ for any $\lambda>0$, then both
expression in (7.1) are infinity. On the other hand, if
$\mathcal{N}^{V}(-\lambda)<\infty$ for all $\lambda>0$, then the negative
spectrum consists of countably many eigenvalues of finite multiplicity. We
label them as $\lambda_{1}\leq\lambda_{2}\leq...<0$. Then we can write

$\displaystyle\mathrm{tr}((-\Delta+V)_{-}^{\gamma})=\sum_{j}|\lambda_{j}|^{\gamma}.$

Since $\mathcal{N}^{V}$ is a piecewise constant function, we have for any
$N\in\mathbb{N}$

$\gamma\int_{|\lambda_{N}|}^{\infty}\lambda^{\gamma-1}\mathcal{N}^{V}(-\lambda)d\lambda=\sum_{j=1}^{N-1}|\lambda_{j}|^{\gamma}+(N-1)|\lambda_{N}|^{\gamma},$

and the remainder term vanishes if $\sum_{j}|\lambda_{j}|^{\gamma}<\infty$,
concluding the proof. ∎

With this, it now a simple calculation to derive two-sided Lieb-Thirring
bounds from the estimates (2.4).

###### Proof of Corollary 2.2.

Lemma 7.1 and the upper bound (2.4) yield immediately that

$\mathrm{tr}((-\Delta+V)_{-}^{\gamma})=\gamma\int_{0}^{|E_{0}|}\lambda^{\gamma-1}\mathcal{N}^{V+\delta+|E_{0}|}(-\lambda+\delta+|E_{0}|)d\lambda\\\
\leq
C_{0,d}^{\frac{d}{2}}\gamma\int_{0}^{|E_{0}|}\lambda^{\gamma-1}(-\lambda+\delta+|E_{0}|)^{\frac{d}{2}}\left|\left\\{x\in\mathbb{R}^{d}:\frac{1}{u_{M}(x)}\leq
C_{0,d}(-\lambda+\delta+|E_{0}|)\right\\}\right|d\lambda.$ (7.2)

Now since $\gamma\geq 1$, we use that
$\lambda^{\gamma-1}\leq(\lambda+\delta)^{\gamma-1}$. We split the integral in
two. On $[0,\frac{|E_{0}|}{2}]$, we use that
$|E_{0}|-\lambda+\delta\leq|E_{0}|+\delta\leq\frac{|E_{0}|+\delta}{\delta}(\lambda+\delta)$,
while $|E_{0}|-\lambda+\delta\leq\frac{|E_{0}|}{2}+\delta\leq\lambda+\delta$
holds on $[\frac{|E_{0}|}{2},|E_{0}|]$. Writing the condition
$\frac{1}{u_{M}(x)}\leq C_{0,d}(-\lambda+\delta+|E_{0}|)$ as
$\lambda+\delta\leq|E_{0}|+2\delta-\frac{1}{C_{0,d}u_{M}(x)}$ and changing
variables to $s=\lambda+\delta$ yields that if $\gamma\geq 1$, then

$\displaystyle\mathrm{tr}((-\Delta+V)_{-}^{\gamma})$ $\displaystyle\leq
C_{0,d}^{\frac{d}{2}}\gamma\left(1+\frac{|E_{0}|}{\delta}\right)^{\frac{d}{2}}\int_{\delta}^{|E_{0}|+\delta}s^{\gamma-1+\frac{d}{2}}\left|\left\\{x\in\mathbb{R}^{d}:|E_{0}|+2\delta-\frac{1}{C_{0,d}u_{M}(x)}\geq
s\right\\}\right|$ $\displaystyle\leq
C_{0,d}^{\frac{d}{2}}\frac{\gamma}{\gamma+\frac{d}{2}}\left(1+\frac{|E_{0}|}{\delta}\right)^{\frac{d}{2}}\int_{{\mathbb{R}}^{d}}\left(|E_{0}|+2\delta-\frac{1}{C_{0,d}u_{M}(x)}\right)_{+}^{\frac{d}{2}+\gamma}dx$

by the layer cake representation.

Similarly, for the lower bound, we start as in (7.2) but with $c_{0,d}$
instead of $C_{0,d}$. Here,
$|E_{0}|-\lambda+\delta\geq\frac{|E_{0}|}{2}+\delta>\lambda$ on
$[0,\frac{|E_{0}|}{2}]$ while
$|E_{0}|-\lambda+\delta\geq\delta\geq\frac{\delta}{|E_{0}|}\lambda$ on
$[\frac{|E_{0}|}{2},|E_{0}|]$. We conclude that for $\gamma>0$,

$\displaystyle\mathrm{tr}((-\Delta+V)_{-}^{\gamma})$
$\displaystyle\geq\min\left\\{1,\frac{\delta}{|E_{0}|}\right\\}^{\frac{d}{2}}c_{0,d}^{d/2}\gamma\int_{0}^{E_{0}}\lambda^{\frac{d}{2}+\gamma-1}\left|\left\\{x\in\mathbb{R}^{d}:\lambda\leq|E_{0}|+\delta-\frac{1}{c_{0,d}u_{M}(x)}\right\\}\right|d\lambda$
$\displaystyle=\min\left\\{1,\frac{\delta}{|E_{0}|}\right\\}^{\frac{d}{2}}c^{d/2}\frac{\gamma}{\gamma+\frac{d}{2}}\int_{\mathbb{R}^{d}}\left(|E_{0}|+\delta-\frac{1}{c_{0,d}u_{M}(x)}\right)_{-}^{\frac{d}{2}+\gamma}dx.$

For the last equality we used the bound (2.4) to conclude that the measure
vanishes for all $\lambda>|E_{0}|$ since $\mathcal{N}^{V}(-\lambda)=0$ for
those $\lambda$. ∎

###### Proof of Corollary 2.5.

We follow the same strategy as above, using (2.10,2.11) instead of (2.4). For
the lower bound, we immediately have

$\displaystyle\mathrm{tr}((-\Delta+V)_{-}^{\gamma})$
$\displaystyle=\gamma\int_{0}^{|E_{0}|}\lambda^{\gamma-1}{\mathcal{N}}^{V}(-\lambda)d\lambda$
$\displaystyle\geq
C_{c}^{\frac{d}{2}}\gamma\int_{0}^{|E_{0}|}\lambda^{\gamma-1+\frac{d}{2}}\left|\left\\{x\in{\mathbb{R}}^{d}:\lambda\leq\frac{\widetilde{C}_{H}}{c}\left(M-\frac{1}{u_{M}(x)}\right)\right\\}\right|$
$\displaystyle=C_{c}^{\frac{d}{2}}\frac{\gamma}{\gamma+\frac{d}{2}}\bigg{(}\frac{\widetilde{C}_{H}}{c}\bigg{)}^{\gamma+\frac{d}{2}}\int_{{\mathbb{R}}^{d}}\left(\frac{1}{u_{M}(x)}-M\right)_{-}^{\gamma+\frac{d}{2}}dx.$

For the upper bound, we follow the proof of Theorem 2.4(iii) to get

${\mathcal{N}}^{V}(\mu)\leq{\mathcal{N}}^{-A_{M}^{2}(1/u_{M}-M)_{-}}\left(\mu+\varepsilon\right).$

With this,

$\mathrm{tr}((-\Delta+V)_{-}^{\gamma})\leq\gamma\int_{0}^{|E_{0}|}\lambda^{\gamma-1}{\mathcal{N}}^{-A_{M}^{2}(1/u_{M}-M)_{-}}\left(-\lambda+\varepsilon\right)d\lambda$

Since the counting function is càdlàg by definition (2.1), we let
$\varepsilon\to 0^{+}$ and conclude that

$\mathrm{tr}((-\Delta+V)_{-}^{\gamma})\leq\mathrm{tr}((-\Delta-
A_{M}^{2}(1/u_{M}-M)_{-})_{-}^{\gamma})$

and we use the standard Lieb-Thirring inequality to conclude. ∎

#### Acknowledgements.

The three authors were supported by NSERC of Canada. S.B. would like to thank
S. Mayboroda and D. Arnold for inspiring discussions.

## References

  * [1] E.H. Lieb and W. Thirring.  Inequalities for the moments of the eigenvalues of Schrödinger equations and their relations to Sobolev inequalities.  Studies in Mathematical Physics: Essays in Honor of Valentine Bergmann, pages 269–303, 1976.
  * [2] S.M. Birman and M.Z. Solomyak.  Asymptotic behavior of the spectrum of differential equations.  Journal of Soviet Mathematics, 12:247–283, 1979.
  * [3] R.L. Frank, A. Laptev, and T. Weidl.  Schrödinger operators: Eigenvalues and Lieb–Thirring inequalities, volume 200.  Cambridge University Press, 2022.
  * [4] U.W. Schmincke.  On Schrödinger’s factorization method for Sturm-Liouville operators.  Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 80(1-2):67–84, 1978.
  * [5] A. Grigor’yan, Y. Netrusov, and S.-T. Yau.  Eigenvalues of elliptic operators and geometric applications.  Surveys in differential geometry, 9(1):147–217, 2004.
  * [6] D. Damanik and C. Remling.  Schrödinger operators with many bound states.  Duke Mathematical Journal, 136(1):51 – 80, 2007.
  * [7] O. Safronov and B. Vainberg.  Estimates for negative eigenvalues of a random Schrödinger operator.  Proceedings of the American Mathematical Society, 136(11):3921–3929, 2008.
  * [8] S. Molchanov and B. Vainberg.  Bargmann type estimates of the counting function for general Schrödinger operators.  Journal of Mathematical Sciences, 184:457–508, 2012.
  * [9] S. Bachmann, R. Froese, and S. Schraven.  Counting eigenvalues of Schrödinger operators using the landscape function.  Journal of Spectral Theory, 13(4):1445–1472, 2023.
  * [10] B. Poggi.  Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fields.  arXiv preprint arXiv:2107.14103, 2021.
  * [11] L.E. Payne.  Bounds for solutions of a class of quasilinear elliptic boundary value problems in terms of the torsion function.  Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 88(3-4):251–265, 1981.
  * [12] R. Sperb.  Bounds for the first eigenvalue of the elastically supported membrane on convex domains.  Zeitschrift für angewandte Mathematik und Physik ZAMP, 54:879–902, 2003.
  * [13] M. van den Berg and T. Carroll.  Hardy inequality and ${L}^{p}$ estimates for the torsion function.  Bulletin of the London Mathematical Society, 41(6):980–986, 2009\.
  * [14] T. Giorgi and R.G. Smits.  Principal eigenvalue estimates via the supremum of torsion.  Indiana University Mathematics Journal, pages 987–1011, 2010.
  * [15] M. van den Berg.  Estimates for the torsion function and Sobolev constants.  Potential Analysis, 36(4):607–616, 2012.
  * [16] M. van den Berg and D. Bucur.  On the torsion function with Robin or Dirichlet boundary conditions.  Journal of Functional Analysis, 266(3):1647–1666, 2014.
  * [17] A. Aghajani, A. Mosleh Tehrani, and N. Ghoussoub.  Pointwise lower bounds for solutions of semilinear elliptic equations and applications.  Advanced Nonlinear Studies, 14(4):839–856, 2014.
  * [18] M. van den Berg.  Spectral bounds for the torsion function.  Integral Equations and Operator Theory, 88:387–400, 2017.
  * [19] H. Vogt.  ${L}^{\infty}$-estimates for the torsion function and ${L}^{\infty}$-growth of semigroups satisfying Gaussian bounds.  Potential Analysis, 51:37–47, 2019.
  * [20] M. Filoche and S. Mayboroda.  Universal mechanism for Anderson and weak localization.  Proceedings of the National Academy of Sciences, 109(37):14761–14766, 2012.
  * [21] D.N. Arnold, G. David, M. Filoche, D. Jerison, and S. Mayboroda.  Computing spectra without solving eigenvalue problems.  SIAM Journal on Scientific Computing, 41(1):B69–B92, 2019.
  * [22] G. David, M. Filoche, and S. Mayboroda.  The landscape law for the integrated density of states.  Advances in Mathematics, 390:107946, 2021.
  * [23] P. Desforges, S. Mayboroda, S. Zhang, G. David, D.N. Arnold, W. Wang, and M. Filoche.  Sharp estimates for the integrated density of states in Anderson tight-binding models.  Physical Review A, 104(1):012207, 2021.
  * [24] D. Arnold, M. Filoche, S. Mayboroda, W. Wang, and S. Zhang.  The landscape law for tight binding Hamiltonians.  Communications in Mathematical Physics, 396(3):1339–1391, 2022\.
  * [25] D.N. Arnold, G. David, M. Filoche, D. Jerison, and S. Mayboroda.  Localization of eigenfunctions via an effective potential.  Communications in Partial Differential Equations, 44(11):1186–1216, 2019.
  * [26] M. Filoche, S. Mayboroda, and T. Tao.  The effective potential of an M-matrix.  Journal of Mathematical Physics, 62(4), 2021.
  * [27] W. Wang and S. Zhang.  The exponential decay of eigenfunctions for tight-binding Hamiltonians via landscape and dual landscape functions.  In Annales Henri Poincaré, volume 22, pages 1429–1457. Springer, 2021.
  * [28] M. van den Berg, D. Bucur, and T. Kappeler.  On efficiency and localisation for the torsion function.  Potential Analysis, 57(4):571–600, 2022.
  * [29] D. Mugnolo.  Pointwise eigenvector estimates by landscape functions: Some variations on the Filoche–Mayboroda–van den Berg bound.  Mathematische Nachrichten, 2023.
  * [30] S. Steinerberger.  Localization of quantum states and landscape functions.  Proceedings of the American Mathematical Society, 145(7):2895–2907, 2017.
  * [31] E.H. Lieb.  Bounds on the eigenvalues of the Laplace and Schrödinger operators.  Bulletin of the American Mathematical Society, 82(5):751–753, 1976\.
  * [32] G.V. Rozenblum.  Distribution of the discrete spectrum of singular differential operators.  Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 1:75–86, 1976\.
  * [33] M. Cwikel.  Weak type estimates for singular values and the number of bound states of Schrödinger operators.  Annals of Mathematics, 106:93–100, 1977.
  * [34] J.G. Hoskins, H. Quan, and S. Steinerberger.  Magnetic Schrödinger operators and landscape functions.  Communications in Partial Differential Equations, 49(1-2):1–14, 2024.
  * [35] J. Lu, C. Murphey, and S. Steinerberger.  Fast localization of eigenfunctions via smoothed potentials.  Journal of Scientific Computing, 90:1–18, 2022.
  * [36] M. Otelbaev.  Bounds for eigenvalues of singular differential operators.  Mathematical notes of the Academy of Sciences of the USSR, 20(6):1038–1042, 1976.
  * [37] M. Otelbaev.  Imbedding theorems for spaces with a weight and their application to the study of the spectrum of a Schrödinger operator.  Trudy Matematicheskogo Instituta imeni V.A. Steklova, (Procedings of the Steklov Mathematical Institute), 150:265–305, 1979.
  * [38] Z. Shen.  Eigenvalue asymptotics and exponential decay of eigenfunctions for Schrödinger operators with magnetic fields.  Transactions of the American Mathematical Society, 348(11):4465–4488, 1996.
  * [39] Z. Shen.  On bounds of $N(\lambda)$ for a magnetic Schrödinger operator.  Duke Mathematical Journal, 94(3):479–507, 1998.
  * [40] L. Bugliaro, J. Fröhlich, G.M. Graf, J. Stubbe, and C. Fefferman.  A Lieb-Thirring bound for a magnetic Pauli Hamiltonian.  Communications in Mathematical Physics, 187:567–582, 1997.
  * [41] Z. Shen.  On moments of negative eigenvalues for the Pauli operator.  Journal of Differential Equations, 149(2):292–327, 1998.
  * [42] T.D. Do and L.X. Truong.  Spectral asymptotics for generalized Schrödinger operators.  Annales Fennici Mathematici, 48(2):703–727, 2023.
  * [43] M. Aizenman and B. Simon.  Brownian motion and Harnack inequality for Schrödinger operators.  Communications on Pure and Applied Mathematics, 35(2):209–273, 1982\.
  * [44] F. Chiarenza, E. Fabes, and N. Garofalo.  Harnack’s inequality for Schrödinger operators and the continuity of solutions.  Proceedings of the American Mathematical Society, 98(3):415–425, 1986.
  * [45] L. Read.  Lieb—Thirring type bounds for perturbed Schrödinger operators with single-well potentials.  Journal of Mathematical Physics, 64(6), 2023.
  * [46] Q. Han and F. Lin.  Elliptic partial differential equations, volume 1.  American Mathematical Soc., 2011.
  * [47] B. Simon.  Schrödinger semigroups.  Bulletin of the American Mathematical Society, 7(3):447–526, 1982\.
  * [48] G. David, A. Gloria, and S. Mayboroda.  The landscape function on $\mathbb{R}^{d}$.  arXiv preprint arXiv:2307.11182, 2023.
  * [49] M. Reed and B. Simon.  IV: Analysis of Operators, volume 4.  Elsevier, 1978.
  * [50] E.H. Lieb and M. Loss.  Analysis, volume 14.  American Mathematical Society, 2001.
  * [51] L.E. Payne and H.F. Weinberger.  An optimal Poincaré inequality for convex domains.  Archive for Rational Mechanics and Analysis, 5(1):286–292, 1960\.
  * [52] M.A. Perelmuter.  Positivity preserving operators and one criterion of essential self-adjointness.  Journal of Mathematical Analysis and Applications, 82(2):406–419, 1981.
  * [53] D. Gilbarg and N.S. Trudinger.  Elliptic partial differential equations of second order, volume 224\.  Springer, 1977.
  * [54] E.B. Davies.  Spectral theory and differential operators, volume 42.  Cambridge University Press, 1995.
  * [55] F.H. Brownell and C.W. Clark.  Asymptotic distribution of the eigenvalues of the lower part of the Schrödinger operator spectrum.  Journal of Mathematics and Mechanics, pages 31–70, 1961.
  * [56] G. Raikov.  Discrete spectrum for Schrödinger operators with oscillating decaying potentials.  Journal of Mathematical Analysis and Applications, 438(2):551–564, 2016.