# Convergence rates for the Trotter-Kato splitting

Simon Becker∗<EMAIL_ADDRESS>,  Niklas Galke†<EMAIL_ADDRESS>,  Robert Salzmann‡<EMAIL_ADDRESS>and  Lauritz van Luijk§
<EMAIL_ADDRESS>
###### Abstract.

We study convergence rates of the Trotter-Kato splitting

$e^{A+L}=\lim_{n\to\infty}\Big{(}e^{L/n}e^{A/n}\Big{)}^{n}$

in the strong operator topology. In the first part, we use complex
interpolation theory to treat generators $L$ and $A$ of contraction semigroups
on Banach spaces, with $L$ relatively $A$-bounded. In the second part, we
study unitary dynamics on Hilbert spaces and develop a new technique based on
the concept of energy constraints. Our results provide a complete picture of
the convergence rates for the Trotter splitting for all common types of
Schrödinger and Dirac operators, including singular, confining and magnetic
vector potentials, as well as molecular many-body Hamiltonians in dimension
$d=3$. Using the Brezis-Mironescu inequality, we derive convergence rates for
the Schrödinger operator with $V(x)=\pm|x|^{-a}$ potential. In each case, our
conditions are fully explicit.

∗ETH Zurich, Institute for Mathematical Research, Rämistrasse 101, 8092
Zurich, Switzerland

†Universitat Autònoma de Barcelona, Departament de Física, GIQ, 08193
Bellaterra, Barcelona, Spain

‡Univ Lyon, Inria, ENS de Lyon, LIP, 46 Allee d’Italie, 69364 Lyon Cedex 07,
France

§Leibniz University Hanover, Appelstrasse 5, 30167 Hannover, Germany

###### Contents

  1. 1 Introduction
    1. 1.1 Techniques
    2. 1.2 Related works
    3. 1.3 Outline of the article
  2. 2 Trotter convergence under relative boundedness
    1. 2.1 Elementary $\mathcal{O}(n^{-1})$ scaling for regular $L$
    2. 2.2 Sobolev towers and Favard spaces
    3. 2.3 Trotter convergence on Favard spaces
    4. 2.4 Strictly dissipative dynamics
    5. 2.5 Self-adjoint and anti self-adjoint generators on Hilbert spaces
  3. 3 Convergence rates for energy-limited unitary dynamics
    1. 3.1 Application to Schrödinger operators
    2. 3.2 Explicit example: Trotter splitting for the Harmonic oscillator
    3. 3.3 Application to magnetic Dirac operators
  4. 4 Numerical examples
    1. 4.1 Singular potentials
    2. 4.2 Confining potentials
  5. A Proofs of 2.2 and 2.17

##  1\. Introduction

We derive explicit convergence rates for the Trotter-Kato splitting formula,
which, under suitable assumptions on $A$ and $L$, states that

$e^{A+L}=\lim_{n\to\infty}\Big{(}e^{L/n}e^{A/n}\Big{)}^{n}$ (1.1)

in the strong operator topology. Originally developed by Lie for matrices, it
has been extended to operators by Trotter [T59], Chernoff [C68], and Kato
[K78]. Nelson [N64] clarified the connection between the splitting formula to
Feynman’s path integral for the Schrödinger dynamics. The formula is also of
key importance in numerical applications [Paz83, S94, JL00, Sec. 3.6], where
the Trotter formula is also often referred to as Strang splitting, as it, for
example, allows the separation of differential from multiplication operators
in applications in quantum mechanics. For applications, convergence rates are
important as they give a promise on the quality of the approximation. However,
uniform convergence rates, i.e., convergence rates in operator norm, are
typically not feasible since the Trotter product formula simply does not
converge uniformly in general [I03, Sec. 4]. This motivates the study of
convergence rates in the strong operator topology. That is, for a vector $x$,
we are interested in the speed of convergence

$\|{(e^{L/n}e^{A/n})}^{n}x-e^{A+L}x\|\to 0.$ (1.2)

While a convergence rate of $\mathcal{O}(n^{-1})$ is guaranteed in finite
dimensions, or more generally for bounded generators, this does not hold in
general: It was shown in [B+23a] that the splitting of the Coulomb Hamiltonian
$H=-\Delta-|x|^{-1}$, describing the hydrogen atom, converges with
$\mathcal{O}(n^{-1/4})$ on (eigenstates in) s-orbitals,
$\mathcal{O}(n^{-3/4})$ on p-orbitals and with $\mathcal{O}(n^{-1})$ on d- and
higher orbitals. Their technique of establishing strong convergence rates is
restricted to the Hilbert space setting and to eigenvectors of $A+L$. In this
article, we relax these restrictions by proving strong convergence rates on
sufficiently regular vectors also in the Banach space setting, recovering, in
particular, a form of $\mathcal{O}(n^{-1/4})$ convergence for the ground state
of the hydrogen atom.

Our article features two directions. In the first part of the article, Section
2, our guiding principle is to develop a theory to obtain convergence rates
under the assumption that $L$ is “small” compared to $A$. We gauge the
smallness using a relative boundedness condition of the operators. In this
case, the appropriate notion of regularity is captured using special
interpolation spaces defined through $A$, which are called _Favard spaces_ and
which we can characterize explicitly in many relevant cases. For Schrödinger
operators, for instance, in terms of standard Sobolev spaces. We demand that
$L$ does not decrease this form of regularity by too much, i.e., when applied
to a vector of high regularity, the output should remain regular, though
possibly to a lesser extent. After establishing
$\mathcal{O}(n^{-1})$-convergence rates for (1.2) for sufficiently regular
vectors $x$ and for suitable $L$, we prove
$\mathcal{O}(n^{-\delta})$-convergence rates with $0<\delta<1$ on less regular
vectors and for very general, but still relatively $A$-bounded $L$. The
guiding motive for this first part of the article is to understand the
convergence rates of the Trotter-splitting of Schrödinger operators with
singular potentials. For instance, we find for the three-dimensional Coulomb
potential

$\big{\|}\big{(}e^{-it|x|^{-1}/n}e^{-it\Delta/n}\big{)}^{n}\psi-e^{it(-\Delta-|x|^{-1})}\psi\big{\|}\lesssim
n^{-1/4+\varepsilon}\|\psi\|_{H^{2}({\mathbb{R}}^{3})}$ (1.3)

for any fixed $\varepsilon>0$ (cp. Corollary 2.22). Numerical evidence from
[B+23a] suggests that, on the ground state, the true convergence rate is
indeed $n^{-1/4}$. Thus, we expect our estimate to be essentially tight.
Proving an analytical lower bound, however, remains an open problem.

In the second part of the article, in Section 3, the guiding motive is to
understand Schrödinger operators with confining or trapping potentials. In
this case, the kinetic energy and potential energy are not in any sense small
with respect to one another. A standard example of such an operator is the
quantum harmonic oscillator. To treat the Trotter splitting of such operators,
we develop a new approach based on the concept of _energy constraints_ [W17,
S18, S20, vL24]. This approach differs from the first one due to the different
nature of the perturbation. As a prototypical example of the results obtained
with this approach, we mention the following convergence rate

$\big{\|}{\big{(}e^{-i\frac{t}{n}\Delta}e^{i\frac{t}{n}x^{2}}\big{)}^{n}\psi-e^{it(-\Delta+x^{2})}\psi}\big{\|}\leq\frac{6t^{2}}{n}\left\lVert(-\Delta+x^{2})\psi\right\rVert$
(1.4)

for the Trotter splitting of the harmonic oscillator on $L^{2}({\mathbb{R}})$
(cp. 3.15).

Taken together, the two approaches yield a fairly complete understanding of
Trotter convergence rates for closed quantum systems. The general nature of
the first approach allows us to apply it to a much larger class of dynamical
systems beyond just closed quantum systems. We outline this in the first few
sections of the article. The approach outlined in the second part of the
article is limited to unitary dynamics on Hilbert spaces.

###  1.1. Techniques

Both approaches for providing convergence rates for the Trotter-Kato product
formula (1.1) in Sections 2 and 3 follow similar proof techniques but diverge
at certain points as we explain in the following:

_Step 1: Key commutator bound._ In both Sections 2 and 3, the first step of
bounding the Trotter error is to use a telescoping sum argument, which gives
for $x$ of sufficient regularity

$\displaystyle\|{(e^{L/n}e^{A/n})}^{n}x-e^{A+L}x\|\leq\sup_{s,\tau\in[0,1]}\|[L,e^{sA/n}]x_{\tau}\|.$
(1.5)

Here, we can either have that $x_{\tau}$ is the original element $x$ evolved
by the joined dynamics, i.e. $x_{\tau}:=e^{\tau(A+L)}x,$ or by the product
dynamics, i.e. $x_{\tau}:=(e^{L/n}e^{A/n})^{\left\lfloor{\tau
n}\right\rfloor}x.$ We call this the _key commutator bound_ , which can be
found in Lemma 2.1 or by combining Lemma 3.6 and the relevant steps in the
proof of Theorem 3.5.

_Step 2: Stability of regularity._ Next, we need to ensure that the evolved
element $x_{\tau}$ has comparable regularity as the initial $x.$ The way we
quantify regularity depends on the context: In Section 2, we are in the regime
in which $A$ is the dominant generator as $L$ is relatively bounded with
$A$-bound $<1$. Hence, in this case, it is sensible to focus on notions of
regularity defined through the generator $A.$ In particular for that we
consider the Favard spaces $F_{\gamma}(A)$ which for $\gamma\in[0,1]$ are
exactly given by elements $x$ such that
$\|(e^{tA}-I)x\|=\mathcal{O}(t^{\gamma}).$ Hence, higher values of $\gamma$
correspond to higher regularity of the element $x.$ For $x\in F_{\gamma}(A)$
the relative boundedness assumptions on $L$ employed gives, by the stability
result on Favard spaces Lemma 2.6, that also $x_{\tau}\in F_{\gamma}(A)$ where
we chose $x_{\tau}=e^{\tau(A+L)}x$ in the above.

On the other hand, in Section 3, the role of $A$ and $L$ is symmetric. The
right notion of regularity is given by the _energy_ of a vector $x$ as
measured by a positive self-adjoint operator $G$, the ‘reference Hamiltonian’.
We are free to choose this reference Hamiltonian as long as the dynamics
generated $A$ and $L$ are energy-limited with respect to $G$ in the sense of
[vL24]. This ensures that $x_{\tau}$ above is of comparable regularity as the
initial element $x.$

_Step 3: Bounding the key commutator._ As the final step, we need to estimate
the key commutator, which appears on the right-hand side of (1.5). In Section
2 we write

$\displaystyle[L,e^{sA/n}]x_{\tau}=L(e^{sA/n}-I)x_{\tau}-(e^{sA/n}-I)Lx_{\tau}$
(1.6)

and bound each term individually. For the first term, we assume that the
singularity of $L$ is controlled when the operator is restricted to elements
of sufficient regularity. More, precisely we assume that $L$ is bounded on the
Favard space $F_{\alpha}(A)$ for some $0\leq\alpha<\gamma$ from which we see
that the first term decays as $\mathcal{O}(n^{-(\gamma-\alpha)}).$ For the
second term we assume that $L$ does not decrease the regularity of
$x_{\tau}\in F_{\gamma}(A)$ by too much. In particular if we still have
$Lx_{\tau}\in F_{\beta}(A)$ for some $0<\beta\leq\gamma$ the second term in
(1.6) decays as $\mathcal{O}(n^{-\beta}).$ Both assumptions on how $L$
interacts with the notion of regularity dictated by $A$ can explicitly be
verified for many interesting examples, e.g., Schrödinger operators with
Coulomb-type potentials and molecular Hamiltonians as shown at the end of
Section 2.5.1.

In Section 3, we choose a different approach to bound the right-hand side of
(1.5) by using

$\displaystyle\|[L,e^{sA/n}]x_{\tau}\|\leq\frac{s}{n}\sup_{u\in[0,s]}\|[L,A]x_{\tau,u}\|,$
(1.7)

where $x_{\tau,u}:=e^{uA/n}x_{\tau}.$ Under the above assumption of energy
limitedness and given that the commutator $[L,A]$ is bounded with respect to
$G^{1/2},$ the norm appearing on the right-hand side of (1.7) is indeed finite
and we find that the Trotter error decays as $\mathcal{O}(n^{-1}).$

###  1.2. Related works

Our article outlines an approach to obtain initial state-dependent convergence
rates for such splittings. Trotter schemes with applications to quantum
mechanics have been improved in various directions, for example, for bosonic
systems [M24], lattice systems [BL22], and in the context of the quantum Zeno
effect [EI05, MW19, BDS21, MR23]. Our approach is different in the sense that
we try to provide a framework that covers a wide range of, in some sense,
singular splittings that are not covered by most results in the literature.
While little about convergence rates beyond standard cases seems to be known
in the quantum setting, a more elaborate theory is available for strictly
dissipative dynamics; see [NSZ18, NSZ18b, NSZ19] and references therein. A
strength of our approach is that it unites several types of dynamics under one
umbrella approach. Finally, many works such as [M24] also consider
generalizations of the Trotter Formula and applications to non-autonomous
systems [NSZ20]. An overview of the current state of results can also be found
in [ZNI24]. While our method also allows for such generalizations, we decided
to focus on the most basic setting and will consider such applications in
future work. State-dependent error bounds for quantum systems have been
discussed in [AFL21, B+23b, B+23a, B+24, vL+24, vL24]. In particular, [B+23b,
B+23a] prove state-dependent convergence for the Trotter product for more
restrictive sets of vectors than what we are able to treat in this article.
The idea of applying energy constraints to obtain state-dependent convergence
rates was used before in the context of Lie group representations and so-
called quantum speed limits [BD20, B+21, vL+24].

###  1.3. Outline of the article

In the first part of the article, we study the case of generators $A$ and $L$
where $L$ is relatively $A$ bounded with $A$-bound $<1$.

  * -

In Section 2.1, we state in Theorem 2.2 a vanilla convergence result with
$\mathcal{O}(n^{-1})$ Trotter convergence rate under fairly general
assumptions on $L$ and $A$.

  * -

In Section 2.2, we introduce Favard spaces and discuss some of their
properties. These interpolation spaces capture the propagation of regularity
of data under the time evolution of contraction semigroups.

  * -

In Section 2.3, we embed with our Theorem 2.7 the result of the previous
section into the broader framework of Favard spaces and convergence rates.

  * -

In Section 2.4 we apply the framework of Section 2.2 to the Trotter product of
generators of contraction semigroups. We start with so-called positive
generators, in which case one can easily define fractional powers and obtain
Corollary 2.13 on the convergence of the Trotter product.

  * -

In Section 2.5, we discuss applications for self-adjoint $A$.

  * -

In Section 2.5.1, we turn to unitary dynamics, which covers the realm of
Schrödinger dynamics in quantum mechanics. We provide an extensive discussion
of the Trotter splitting for molecular Hamiltonians with Coulomb singularity,
cf. Theorem 2.23.

In Section 3, we use a recently-developed framework of energy constraints to
obtain general convergence rates in 3.5 and 3.9, which we illustrate in the
subsequent Subsections:

  * -

In Section 3.1, we apply 3.5 to show a $\mathcal{O}(n^{-1})$ convergence for
all Schrödinger operators whose potentials have bounded second derivative.

  * -

In Section 3.2, we apply the findings of Section 3.1 to the harmonic
oscillator to obtain the convergence rate in (1.4).

  * -

In Section 3.3, we discuss applications to Dirac operators, such as magnetic
Dirac operators.

In Section 4, we illustrate our findings with some numerical experiments.

Notation. We denote the identity operator by $I$, write $a\lesssim b$ and
$a=\mathcal{O}(b)$ to indicate that there is a constant $C>0$ such that $a\leq
Cb$ and furthermore $a\sim b$ if $a\lesssim b$ and $b\lesssim a.$ We denote
the resolvent set by $\rho(A)$ and its complement, the spectrum, by
$\operatorname{Spec}(A).$ The space of bounded operators between Banach spaces
$X$ and $Y$ is denoted $B(X,Y)$ and the corresponding operator norm of a
linear operator $T:X\to Y$ is denoted by $\|T\|_{X\to Y}:=\sup_{\|x\|_{X}\leq
1}\|Tx\|_{Y}.$ We use the so-called Japanese bracket notation
$\langle\xi\rangle:=(1+|\xi|^{2})^{1/2}$ for vectors $\xi\in{\mathbb{R}}^{n}$.

Acknowledgements. We thank Tim Möbus and Alexander Hahn for many interesting
discussions on the topic. Moreover, we want to thank Daniel Burgarth,
Nilanjana Datta and Robin Hillier for organizing a workshop hosted by the ICMS
in Edinburgh in May 2023 during which this work has been initiated.

Funding. SB acknowledges support by the SNF Grant PZ00P2 216019. NG has been
supported by the Spanish MCIN (project PID2022-141283NB-I00) with the support
of FEDER funds, by the Spanish MCIN with funding from European Union
NextGenerationEU (grant PRTR-C17.I1) and the Generalitat de Catalunya, as well
as the Ministry of Economic Affairs and Digital Transformation of the Spanish
Government through the QUANTUM ENIA “Quantum Spain” project with funds from
the European Union through the Recovery, Transformation and Resilience Plan -
NextGenerationEU within the framework of the ”Digital Spain 2026 Agenda”. RS
acknowledges funding from the European Research Council (ERC Grant AlgoQIP,
Agreement No. 851716). LvL has been funded by the MWK Lower Saxony via the
Stay Inspired Program (Grant ID: 15-76251-2-Stay-9/22-16583/2022).

##  2\. Trotter convergence under relative boundedness

In this section, we obtain convergence rates for the Trotter-Kato product
formula (1.1) under the assumption that one of the generators, say $L$, is
relatively bounded by the other: We say $L$ is _relatively $A$-bounded_ if
$D(A)\subseteq D(L)$ and there exist $a,b\geq 0$ such that for all $x\in D(A)$
we have

$\displaystyle\|Lx\|\leq a\|Ax\|+b\|x\|.$ (2.1)

We call the infimum over all $a$ for which there is a $b\geq 0$ such that the
above inequality holds the _$A$ -bound_ of $L.$

For the remainder of this section, we consider situations in which $L$ is
relatively $A$-bounded with $A$-bound $<1.$ The first consequence of this
assumption is that for $L$ and $A$ being generators of $C_{0}$ contraction
semigroups111We call a $C_{0}$ semigroup $(T_{t})_{t\geq 0}$ a contraction
semigroup if all maps $T_{t}$ are contractions, i.e., $\|T_{t}\|\leq 1.$, also
$A+L$ with domain $D(A+L)=D(A)$ defines a $C_{0}$ contraction semigroup [EN00,
Thm. 2.7]. The joint dynamics $e^{t(A+L)}$ therefore leaves $D(A)$ invariant.
As $A$ is closed [EN00, Thm. 1.4], $D(A)$ becomes a Banach space when equipped
with the graph norm $\|x\|_{D(A)}=\|x\|+\|Ax\|$ and $e^{t(A+L)}:D(A)\to D(A)$
is a bounded operator by the closed graph theorem. The latter can be seen
explicitly by noting that since the $A$-bound of $L$ is strictly smaller than
1, the graph norms of $A$ and $A+L$ are equivalent: In fact, for $x\in D(A)$,
we have

$\begin{split}\left\lVert
Ax\right\|\leq\left\lVert(A+L)x\right\rVert+\left\lVert
Lx\right\rVert\leq\left\lVert(A+L)x\right\rVert+a\|Ax\|+b\|x\|\end{split}$

and hence

$\displaystyle\|Ax\|\leq\frac{1}{1-a}\Big{(}\|(A+L)x\|+b\|x\|\Big{)}$

and furthermore

$\begin{split}\left\lVert(A+L)x\right\|\leq\left\lVert
Ax\right\rVert+\left\lVert Lx\right\rVert\leq(1+a)\left\lVert
Ax\right\rVert+b\|x\|\end{split}$

which gives $\|x\|_{D(A)}\sim\|x\|_{D(A+L)}.$222To conclude the boundedness of
$e^{t(A+L)}:D(A)\to D(A)$ use
$\|e^{t(A+L)}x\|_{D(A)}\lesssim\|e^{t(A+L)}x\|_{D(A+L)}$ together with
$\|(A+L)e^{t(A+L)}x\|=\|e^{t(A+L)}(A+L)x\|\leq\|(A+L)x\|\lesssim\|x\|_{D(A)}$
where we used that $e^{t(A+L)}$ is a contraction.

In this regime, obtaining convergence rates in the Trotter-Kato formula
reduces to studying the commutator $[L,e^{sA/n}],$ which from now on we shall
refer to as the _key commutator._ This is the content of the following lemma:

######  Lemma 2.1 (Key commutator bound).

Let $X$ be a Banach space and $L$ and $A$ be generators of strongly continuous
contraction semigroups on $X$ such that $L$ is relatively $A$-bounded, with
$A$-bound $<1$. Then for all $n\in\mathbb{N}$, $t\geq 0$ and $x\in D(A)$ we
have

$\left\|\left(\left(e^{tL/n}e^{tA/n}\right)^{n}-e^{t(A+L)}\right)x\right\|\leq
t\sup_{s,\tau\in[0,t]}\left\|[L,e^{sA/n}]e^{\tau(A+L)}x\right\|.$

###### Proof of Lemma 2.1.

By the semigroup property, it is easy to check that we can write the operator
difference of interest as a telescoping sum, i.e.

$\displaystyle\left(e^{tL/n}e^{tA/n}\right)^{n}-e^{t(A+L)}=\sum_{j=0}^{n-1}\left(e^{tL/n}e^{tA/n}\right)^{j}\left(e^{tL/n}e^{tA/n}-e^{t(A+L)/n}\right)e^{t(n-j-1)(A+L)/n}.$

Therefore, we see for $x\in D(A)$ using that $e^{tA/n}$ and $e^{tL/n}$ are
contractions

$\displaystyle\left\|\left(\left(e^{tL/n}e^{tA/n}\right)^{n}-e^{t(A+L)}\right)x\right\|\leq\sum_{j=0}^{n-1}\left\|\left(e^{tL/n}e^{tA/n}-e^{t(A+L)/n}\right)e^{t(n-j-1)(A+L)/n}x\right\|.$

Now, using that by the relative boundedness assumption $D(A+L)=D(A),$ we can
write for all $y\in D(A)\subseteq D(L)$ 333Note that the integral in (2) is
well-defined as a Bochner integral as the integrand is continuous, which is
true by $L$ being relatively $A$-bounded and the graph norms of $A$ and $A+L$
being equivalent.

$\displaystyle\left(e^{tL/n}e^{tA/n}-e^{t(A+L)/n}\right)y$
$\displaystyle=\int_{0}^{1}\frac{d}{ds}\left(e^{stL/n}e^{stA/n}e^{(1-s)t(A+L)/n}\right)y\,ds$
$\displaystyle=\frac{t}{n}\int_{0}^{1}\left(e^{stL/n}Le^{stA/n}e^{(1-s)t(A+L)/n}-e^{stL/n}e^{stA/n}Le^{(1-s)t(A+L)/n}\right)y\,ds$
$\displaystyle=\frac{t}{n}\int_{0}^{1}e^{stL/n}[L,e^{stA/n}]e^{(1-s)t(A+L)/n}y\,ds.$
(2.2)

Plugging this into the telescopic sum, we find

$\displaystyle\left\|\left(\left(e^{tL/n}e^{tA/n}\right)^{n}-e^{t(A+L)}\right)x\right\|$
$\displaystyle\leq\frac{t}{n}\sum_{j=0}^{n-1}\left\|\int_{0}^{1}e^{stL/n}[L,e^{stA/n}]e^{(n-j-s)t(A+L)/n}x\,ds\right\|$
$\displaystyle\leq
t\sup_{s,\tau\in[0,t]}\left\|[L,e^{sA/n}]e^{\tau(A+L)}x\right\|.$

∎

###  2.1. Elementary $\mathcal{O}(n^{-1})$ scaling for regular $L$

In this section, we provide a first simple derivation of convergence rates for
the Trotter product formula using the key commutator bound established in
Lemma 2.1. We aim for a point-wise estimate with the Trotter product being
applied to a fixed Banach space element of sufficient regularity, say $x\in
D(A^{2}).$ Then, for generators $L$, which do not decrease this regularity by
too much (with the precise meaning given in the following), we derive a
$\mathcal{O}(n^{-1}$ convergence rate of the Trotter product.

Denoting the semigroup generated by $A$ by $T_{t}=e^{tA}$ and furthermore
$x_{\tau}:=e^{\tau(A+L)}x,$ we observe that we can decompose the key
commutator in Lemma 2.1 as

$[L,T_{s/n}]x_{\tau}=L(T_{s/n}-I)x_{\tau}-(T_{s/n}-I)Lx_{\tau}.$ (2.3)

To estimate both terms on the right-hand side, we use here that for $y\in
D(A)$, we have

$\displaystyle\left\|(T_{s/n}-I)y\right\|=\frac{s}{n}\left\|\int_{0}^{1}T_{s/n}Ay\,dr\right\|\leq\frac{s}{n}\left\|Ay\right\|.$
(2.4)

Hence, the first term in (2.3) can easily be estimated under the additional
assumption that the joined dynamics $e^{\tau(A+L)}$ leaves $D(A^{2})$
invariant and therefore

$\displaystyle x_{\tau}\in D(A^{2}).$ (2.5)

In this case using that $L$ is relatively $A$-bounded as (2.1), we see

$\displaystyle\|L(T_{s/n}-I)x_{\tau}\|\leq
a\|A(T_{s/n}-I)x_{\tau}\|+b\|(T_{s/n}-I)x_{\tau}\|\leq\frac{s}{n}\left(a\|A^{2}x_{\tau}\|+b\|Ax_{\tau}\|\right).$

Furthermore, the second term on the right-hand side of (2.3) can also be
estimated, assuming that

$\displaystyle Lx_{\tau}\in D(A),$ (2.6)

in which case we have

$\|(T_{s/n}-I)Lx_{\tau}\|\leq\frac{s}{n}\|ALx_{\tau}\|.$

Both assumption (2.5) and (2.6) are fulfilled if $L$ satisfies
$LD(A^{2})\subseteq D(A)$ and furthermore

$\displaystyle\|ALy\|\leq a^{\prime}\|A^{2}y\|+b^{\prime}\|y\|$ (2.7)

for some $0\leq a^{\prime}<1$ and $b^{\prime}\geq 0$ and all $y\in D(A^{2}).$
In fact (2.7) together with [P18, Lem. IV.3] gives that
$D((A+L)^{2})=D(A^{2})$ and therefore that the restriction
$e^{\tau(A+L)}|_{D(A^{2})}:D(A^{2})\to D(A^{2})$ defines a well-defined,
bounded operator from which we can conclude that $x_{\tau}\in D(A^{2})$ and
$Lx_{\tau}\in D(A)$ using $x\in D(A^{2}).$ Combining all of the above, we
arrive at the following result, which we can think of as a straightforward
operator theoretic generalization of the $\mathcal{O}(n^{-1})$ convergence
rate for matrices.

######  Theorem 2.2 (Perturbative $\mathcal{O}(n^{-1})$-Trotter).

Let $X$ be a Banach space and $(L,D(L))$ and $(A,D(A))$ with $D(A)\subseteq
D(L)$ generators of contraction semigroups. Assume there exist $0\leq a<1$ and
$b\geq 0$ such that

$\|Lx\|\leq a\|Ax\|+b\|x\|$ (2.8)

for all $x\in D(A).$ Moreover, assume that $L\,D(A^{2})\subseteq D(A)$ and
that there exist $0\leq a^{\prime}<1$ and $b^{\prime}\geq 0$ such that

$\displaystyle\|ALx\|\leq a^{\prime}\|A^{2}x\|+b^{\prime}\|x\|$ (2.9)

for all $x\in D(A^{2}).$ Then for all $n\in\mathbb{N}$, $t\geq 0$ and $x\in
D(A^{2})$ we have

$\displaystyle\left\|\left(\left(e^{tL/n}e^{tA/n}\right)^{n}-e^{t(A+L)}\right)x\right\|\leq\frac{t^{2}}{n}\left(c_{2}\|A^{2}x\|+c_{1}\|Ax\|+c_{0}\|x\|\right)$
(2.10)

for some $c_{i}\geq 0$ depending only on $a,b,a^{\prime}$ and $b^{\prime}$. If
we can choose $b=0$, then $c_{1}=0$. If additionally $b^{\prime}=0$ then also
$c_{0}=0$.

A formal proof of this result in which the constants $c_{0},c_{1},c_{2}$ are
expressed in terms of $a,b,a^{\prime},b^{\prime}$ can be found in Appendix A.

In many circumstances, the requirements of Theorem 2.2 are too strong for many
practically relevant generators $A$ and $L.$ In particular, as we see below,
the assumption $LD(A^{2})\subseteq D(A)$ is not satisfied in the case of
Schrödinger operators with singular potentials, e.g.,
$X=L^{2}({\mathbb{R}}^{3}),$ $A=-i\Delta$ and multiplication operator $L=iV$
corresponding to the potential $V(x)=\pm|x|^{-1}.$ To also be able to treat
these cases, we extend the argument of Theorem 2.2 using milder regularity
assumptions. For that, we use the so-called _Favard spaces_ of $A,$ which are
interpolation spaces between the underlying Banach space $X$ and the domains
$D(A^{k})$ for $k\in{\mathbb{N}}.$ It turns out those provide a more nuanced
regularity analysis compared to the setting of Theorem 2.2 leading to
convergence rates of the Trotter product given by fractional powers of
$n^{-1}.$

In the next section, we review the theory of Sobolev towers and Favard spaces
of generators of $C_{0}$ contraction semigroups. After that, in Section 2.3,
we provide an extension of Theorem 2.2 involving Favard spaces (see Theorem
2.7).

###  2.2. Sobolev towers and Favard spaces

For $A$ being the generator of a strongly continuous contraction semigroup on
some Banach space $X$ the corresponding Sobolev tower is defined as follows:
For $k\in{\mathbb{N}}_{0}$ consider the space $D(A^{k})$, where $D(A^{0})=X,$
equipped with the graph norms $\|x\|_{D(A^{k})}=\|x\|+\|A^{k}x\|.$ The graph
norm is equivalent to the sum of all $\|A^{l}x\|$ for $l\leq k$ [P18, Lem.
II.10], i.e.

$\displaystyle\|x\|_{D(A^{k})}\lesssim\sum_{l=0}^{k}\|A^{l}x\|\lesssim\|x\|_{D(A^{k})}.$
(2.11)

Since $A^{k}$ is a closed operator [P18, Lemma II.8, Lemma II.9], $D(A^{k})$
is a Banach space. Furthermore, for $(T_{t})_{t\geq 0}$ being the strongly
continuous contraction semigroup generated by $A,$ we have that the
restrictions $T_{t}|_{D(A^{k})}$ define strongly continuous semigroups with
generator given by the restriction $A|_{D(A^{k+1})}.$

Inspired by the fact that $A$ satisfies
$\sup_{\lambda>0}\|\lambda(\lambda-A)^{-1}\|<\infty$ [EN00, Thm. 3.5], we can
define complex interpolation spaces between the different levels of the
Sobolev tower: For that, let $\alpha\in(0,\infty)$ and furthermore
$r_{\alpha}\in(0,1]$ and $k_{\alpha}\in{\mathbb{N}}_{0}$ be the unique numbers
such that $\alpha=r_{\alpha}+k_{\alpha}.$ One then obtains complex
interpolation spaces
$(D(A^{k_{\alpha}}),D(A^{k_{\alpha}+1}))_{r_{\alpha},\infty},$ commonly
referred to as _Favard spaces_ , of the form [L09, Prop. 3.1]

$\displaystyle F_{\alpha}\equiv F_{\alpha}(A)\equiv F_{\alpha}(A;X)$
$\displaystyle:=\Big{\\{}x\in
D(A^{k_{\alpha}});\,|x|_{F_{\alpha}}:=\sup_{\lambda>0}\|\lambda^{r_{\alpha}}A(\lambda-A)^{-1}x\|_{D(A^{k_{\alpha}})}<\infty\Big{\\}}.$

Here, for later convenience, we introduced the notation $F_{\alpha}(A;X)$
explicitly denoting the Banach space $X$ from which the Favard spaces are
constructed. These spaces naturally become Banach spaces when equipped with an
interpolation norm that is equivalent to
$\|x\|_{F_{\alpha}}:=\|x\|_{D(A^{k_{\alpha}})}+|x|_{F_{\alpha}}$. In
particular, for $\alpha\in(0,1]$, the Favard spaces $F_{\alpha}$ are
interpolation spaces between the Banach space $X$ and the domain $D(A).$
Furthermore, for $\alpha\geq 1$ the Favard space $F_{\alpha}$ can equivalently
be understood as a Favard space of order $r_{\alpha}\in(0,1]$ but with
$D(A^{k_{\alpha}})$ as the underlying Banach space instead of $X,$ i.e.

$\displaystyle F_{\alpha}(A;X)=F_{r_{\alpha}}(A;D(A^{k_{\alpha}})).$ (2.12)

######  Example 2.3.

We consider the heat semigroup on $X=C_{b}({\mathbb{R}}^{d})$, see [BF19]. Its
generator $A=\Delta$ has a domain $D(\Delta)=C^{2}_{b}({\mathbb{R}})$ for
$d=1$, while for $d\geq 2$ the domain is given by

$D(\Delta)=\\{f\in C_{b}({\mathbb{R}}^{d})\cap
W^{2,p}({\mathbb{R}}^{d}),\text{ for all }p\in[1,\infty)\text{ and }\Delta
f\in C_{b}({\mathbb{R}}^{d})\\}.$

Then for $\alpha\in(0,1)\setminus\\{1/2\\}$ the Favard spaces are the Hölder
spaces

$F_{\alpha}=C_{b}^{2\alpha}({\mathbb{R}}^{d})$

of bounded continuous functions that are $2\alpha$ Hölder continuous and for
$\alpha=1/2$

$F_{1/2}=\left\\{f\in
C_{b}({\mathbb{R}}^{d});\sup_{x,y}\frac{|f(x)+f(y)-2f((x+y)/2)|}{|x-y|}<\infty\right\\}.$

######  Remark 2.4.

One can define Sobolev towers and Favard spaces more generally for operators
$A$ only assuming $(0,\infty)\subseteq\rho(A)$ and
$\sup_{\lambda>0}\|\lambda(\lambda-A)^{-1}\|\leq C$ for some $C>0.$ Under this
assumption, all statements in this section remain true apart from (2.17) below
and the ones explicitly involving the semigroup $T_{t}.$ Note that in the
special case $C=1$ and $A$ being densely defined, $A$ is immediately closed
since $\rho(A)\neq\emptyset$ and hence this assumption is equivalent to $A$
being the generator of a strongly continuous contraction semigroup [EN00, Thm.
3.5].

Note that for all $k\in{\mathbb{N}}_{0}$ and $k<\alpha\leq\beta\leq k+1$ we
have $D(A^{k+1})\subseteq F_{k+1}\subseteq F_{\beta}\subseteq
F_{\alpha}\subseteq D(A^{k}).$ Furthermore, for all $0<\alpha\leq\beta$ and
$x\in F_{\beta}$ we have the upper bound444To see this, we use the fact that
$\|x\|_{F_{\alpha}}$ is equivalent to the norm in (2.16). Consider then first
the case $r_{\beta}\geq r_{\alpha}.$ In that case, we have
$\sup_{t>0}\|t^{-r_{\alpha}}(T_{t}-I)x\|_{D(A^{k_{\alpha}})}\leq\sup_{0<t\leq
1}\|t^{-r_{\beta}}(T_{t}-I)x\|_{D(A^{k_{\alpha}})}+2\|x\|_{D(A^{k_{\alpha}})}\lesssim\|x\|_{F_{\beta}}$
where we used (2.11) to bound
$\|x\|_{D(A^{k_{\alpha}})}\lesssim\|x\|_{D(A^{k_{\beta}})}.$ If on the other
hand $r_{\beta}<r_{\alpha}$ then $k_{\beta}\geq k_{\alpha}+1$ and the bound
follows by $\sup_{0<t\leq
1}\|t^{-r_{\alpha}}(T_{t}-I)x\|_{D(A^{k_{\alpha}})}\lesssim\sup_{0<t\leq
1}\|t^{1-r_{\alpha}}x\|_{D(A^{k_{\alpha}+1})}\leq\|x\|_{D(A^{k_{\beta}})}.$

$\displaystyle\|x\|_{F_{\alpha}}\lesssim\|x\|_{F_{\beta}}.$ (2.13)

The importance of these spaces is in their relationship to the asymptotic
behavior of associated semigroups. Indeed, we have the equivalent
characterization [L09, Prop. 5.7]

$F_{\alpha}=\Big{\\{}x\in
D(A^{k_{\alpha}});\,\sup_{t>0}\|t^{-r_{\alpha}}(T_{t}x-x)\|_{D(A^{k_{\alpha}})}<\infty\Big{\\}}.$
(2.14)

In particular, for $x\in F_{\alpha}$ we have

$\|T_{t}x-x\|_{D(A^{k_{\alpha}})}={\mathcal{O}(\|x\|_{F_{\alpha}}}\,t^{r_{\alpha}}).$
(2.15)

In this context, it is useful to note that the Favard norm
$\|x\|_{F_{\alpha}}$ is equivalent to

$\displaystyle\|x\|_{D(A^{k_{\alpha}})}+\sup_{t>0}\|t^{-r_{\alpha}}(T_{t}-I)x\|_{D(A^{k_{\alpha}})}.$
(2.16)

Furthermore, in the special case of analytic semigroups, the Favard spaces are
equivalently described by the elements $x\in X$ for which the
$\|x\|_{F_{\alpha}}$ equivalent norm

$\|x\|_{D(A^{k_{\alpha}})}+\sup_{t\in(0,1)}t^{1-r_{\alpha}}\|AT_{t}x\|_{D(A^{k_{\alpha}})}$

is finite.

The following lemma generalises (2.15) from the norms on $X$ and $D(A^{k})$ to
the Favard norms.

######  Lemma 2.5.

Let $A$ be the generator of a strongly continuous contraction semigroup
$\left(T_{t}\right)_{t\geq 0}$ and $\alpha,\gamma\in(0,\infty)$ such that
$\alpha\leq\gamma.$ Then, for $\gamma^{\prime}=\min\\{\gamma-\alpha,1\\}$

$\|T_{t}-I\|_{F_{\gamma}\to
F_{\alpha}}=\mathcal{O}\left(t^{\gamma^{\prime}}\right).$

###### Proof.

For the proof, we use that

$\displaystyle\|T_{t}-I\|_{F_{\gamma}\to
F_{\alpha}}\lesssim\|T_{t}-I\|_{F_{\gamma}\to
D(A^{k_{\alpha}})}+\sup_{s>0}\|s^{-r_{\alpha}}(T_{s}-I)(T_{t}-I)\|_{F_{\gamma}\to
D(A^{k_{\alpha}})}.$ (2.17)

First consider the case $\gamma-\alpha<1.$ In that case
$k_{\alpha}=k_{\gamma}$ and hence $\gamma-\alpha=r_{\gamma}-r_{\alpha}\geq 0.$
To bound the second term in (2.17) we use

$\displaystyle\sup_{s>0}\|s^{-r_{\alpha}}(T_{s}-I)(T_{t}-I)\|_{F_{\gamma}\to
D(A^{k_{\alpha}})}$ $\displaystyle\lesssim\sup_{0<s\leq
t}s^{r_{\gamma}-r_{\alpha}}\|T_{t}-I\|_{D(A^{k_{\alpha}})}+t^{r_{\gamma}}\sup_{s>t}s^{-r_{\alpha}}\|T_{s}-I\|_{D(A^{k_{\alpha}})}\leq
4t^{r_{\gamma}-r_{\alpha}},$

where for the last inequality we have used that $\left(T_{t}\right)_{t\geq 0}$
restricted to $D(A^{k_{\alpha}})$ defines a contraction semigroup. As for the
first term in (2.17) we have $\|T_{t}-I\|_{F_{\gamma}\to
D(A^{k_{\alpha}})}\lesssim
t^{r_{\gamma}}=\mathcal{O}\left(t^{\gamma-\alpha}\right)$ for $t\to 0$
immediately from definition, we have shown $\|T_{t}-I\|_{F_{\gamma}\to
F_{\alpha}}=\mathcal{O}\left(t^{\gamma-\alpha}\right).$

Next we consider the case $\gamma-\alpha\geq 1.$ For that note that for all
$x\in D(A^{k_{\alpha}+1})\subseteq F_{\gamma}$ we have

$\displaystyle\|(T_{t}-I)x\|_{D(A^{k_{\alpha}})}=\left\|\int^{t}_{0}T_{s}Ax\,ds\right\|_{D(A^{k_{\alpha}})}\leq
t\|Ax\|_{D(A^{k_{\alpha}})}\lesssim t\|x\|_{D(A^{k_{\alpha}+1})},$ (2.18)

where for the second inequality, we have used (2.11). Using this and focusing
on the second part in (2.17) we see

$\sup_{s>0}\|s^{-r_{\alpha}}(T_{s}-I)(T_{t}-I)\|_{F_{\gamma}\to
D(A^{k_{\alpha}})}\\\ \lesssim
t\sup_{s>0}\|s^{-r_{\alpha}}(T_{s}-I)\|_{F_{\gamma}\to
D(A^{k_{\alpha}+1})}\lesssim
t\sup_{s>0}\|s^{-r_{\alpha}}(T_{s}-I)\|_{F_{\alpha+1}\to
D(A^{k_{\alpha}+1})}\lesssim t,$

where we used $F_{\gamma}\subseteq F_{\alpha+1}$ together with (2.13). For the
first term in (2.17) we can use (2.18) directly which gives
$\|T_{t}-I\|_{F_{\gamma}\to D(A^{k_{\alpha}})}\lesssim t.$ Combining both
yields $\|T_{t}-I\|_{F_{\gamma}\to F_{\alpha}}\lesssim t$ and finishes the
proof.

∎

Finally, we recall a basic but useful property of Favard spaces that shows
that they are stable under relatively bounded perturbations.

######  Lemma 2.6 (Stability of Favard spaces).

Let $A_{0}$ and $A$ be generators of strongly continuous contraction
semigroups. Let $B=A-A_{0}$ be relatively $A_{0}$ bounded with $A_{0}$-bound
given by $a\geq 0.$ Then $F_{\alpha}(A_{0})\subset F_{\alpha}(A)$ for all
$\alpha\in(0,1]$ and in particular if $a<1$ then
$F_{\alpha}(A_{0})=F_{\alpha}(A).$

###### Proof.

It suffices to show the inclusion $F_{\alpha}(A_{0})\subset F_{\alpha}(A)$,
since for $a<1$, we also have for some $b>0$

$\|A_{0}x\|\leq\|Ax\|+\|Bx\|\leq\|Ax\|+a\|A_{0}x\|+b\|x\|$

and thus

$\|Bx\|\leq a\|A_{0}x\|+b\|x\|\leq\frac{a}{1-a}\|Ax\|+\frac{b}{1-a}\|x\|.$

This implies $F_{\alpha}(A)\subset F_{\alpha}(A_{0})$ by the first part.

Since $A$ and $A_{0}$ generate strongly continuous contraction semigroups, we
have $(0,\infty)\subset\rho(A_{0}),\rho(A)$ and
$\sup_{\lambda>0}\|\lambda(\lambda-
A_{0})^{-1}\|,\sup_{\lambda>0}\|\lambda(\lambda-A)^{-1}\|\leq 1.$ We observe
that we have the simple estimate

$\|A(\lambda-A)^{-1}x\|\leq\|x\|+\|\lambda(\lambda-A)^{-1}x\|\leq 2\|x\|.$

This is enough to conclude that for fixed $\delta>0$

$\sup_{\lambda\in(0,\delta)}\|\lambda^{\alpha}A(\lambda-A)^{-1}x\|\leq
2\delta^{\alpha}\|x\|.$ (2.19)

On the other hand, using the resolvent identity

$\|A(\lambda-A)^{-1}x\|\leq\|A(\lambda-
A_{0})^{-1}x\|+\|A(\lambda-A)^{-1}B(\lambda-A_{0})^{-1}x\|.$ (2.20)

For the first term on the right-hand side of (2.20), relative boundedness
yields

$\|A(\lambda-A_{0})^{-1}x\|\leq(1+a)\|A_{0}(\lambda-
A_{0})^{-1}x\|+b\|(\lambda-A_{0})^{-1}x\|.$

For the second term on the right-hand side of (2.20), we find

$\begin{split}\|A(\lambda-A)^{-1}B(\lambda-
A_{0})^{-1}x\|&\leq\|A(\lambda-A)^{-1}\|\|B(\lambda-A_{0})^{-1}x\|\\\
&\leq\Big{(}1+\|\lambda(\lambda-A)^{-1}\|\Big{)}\|B(\lambda-A_{0})^{-1}x\|\\\
&\leq 2\|B(\lambda-A_{0})^{-1}x\|\\\ \end{split}$

Using $A_{0}$-boundedness of $B$ we find for the last expression in the
previous line

$\|B(\lambda-A_{0})^{-1}x\|\leq a\|A_{0}(\lambda-A_{0})^{-1}x\|+c\|(\lambda-
A_{0})^{-1}x\|.$

For $\lambda>\delta$, we have

$\|\lambda^{\alpha}(\lambda-
A_{0})^{-1}x\|\leq\frac{\|x\|}{\delta^{1-\alpha}}.$

Thus, we have shown

$\sup_{\lambda>\delta}\|\lambda^{\alpha}A(\lambda-A)^{-1}x\|\lesssim\sup_{\lambda>\delta}\|\lambda^{\alpha}A_{0}(\lambda-
A_{0})^{-1}x\|+\|x\|$

which together with $\eqref{eq:bound1}$ implies the claim. ∎

###  2.3. Trotter convergence on Favard spaces

In this section, we use the theory of Favard spaces outlined in Section 2.2 to
extend Theorem 2.2 by interpolating initial data regularity $D(A^{2})$ all the
way down to $X.$ In particular, we consider the case $x\in F_{\gamma}$ for
some $0<\gamma<2$ and $F_{\gamma}$ denoting the corresponding Favard space of
$A.$

For that, we follow a similar argument as the one outlined in Section 2.1: The
starting point of the derivation is again the key commutator bound obtained in
Lemma 2.1 using the fact that we can decompose

$[L,T_{s/n}]x_{\tau}=L(T_{s/n}-I)x_{\tau}-(T_{s/n}-I)Lx_{\tau},$ (2.21)

with $T_{s/n}=e^{sA/n}$ and $x_{\tau}=e^{\tau(A+L)}x.$ Here, instead of
relying on the naive estimate (2.4) we employ the refined version provided in
Lemma 2.5:

For the first term on the right-hand side of (2.21) we assume that
$F_{\alpha}\subseteq D(L)$ for some $\alpha<\min\\{1,\gamma\\}$ or in other
words, by the closed graph theorem, that the restriction
$L|_{F_{\alpha}}:F_{\alpha}\to X$ is bounded. This assumption, in particular,
implies that $L$ is infinitesimally $A$-bounded [EN00, Lem. 2.13] and hence
functions analogously to the relative boundedness assumption (2.8) in Theorem
2.2. Assuming furthermore that the joined dynamics $e^{\tau(A+L)}$ leaves the
Favard space $F_{\gamma}$ invariant which implies that

$\displaystyle x_{\tau}\in F_{\gamma}.$ (2.22)

This property together with Lemma 2.5 imply that

$\left\|L(T_{s/n}-I)x_{\tau}\right\|\leq\|L\|_{F_{\alpha}\to
X}\left\|(T_{s/n}-I)x_{\tau}\right\|_{F_{\alpha}}=\mathcal{O}_{\|L\|_{F_{\alpha}\to
X}\|x_{\tau}\|_{F_{\gamma}}}\left(\left(\frac{s}{n}\right)^{\gamma^{\prime}}\right).$
(2.23)

with $\gamma^{\prime}=\min\\{1,\gamma-\alpha\\}.$

The second term on the right-hand side of (2.21) can easily be estimated once
we additionally assume

$\displaystyle Lx_{\tau}\in F_{\beta}$ (2.24)

for some $\beta\leq\gamma$ in which case we have

$\left\|(T_{s/n}-I)Lx_{\tau}\right\|=\mathcal{O}_{\|Lx_{\tau}\|_{F_{\beta}}}\left(\left(\frac{s}{n}\right)^{\beta}\right).$
(2.25)

As seen in the proof of Theorem 2.7 below, the invariance of $F_{\gamma}$
under $e^{\tau(A+L)}$ and hence (2.22) as well as (2.24) both follow when
assuming that $LF_{\gamma}\subseteq F_{\beta}$ with
$k_{\gamma}<\beta\leq\gamma,$ where $k_{\gamma}\in\\{0,1\\}$ is such that
$\gamma-k_{\gamma}\in(0,1].$ In the case $k_{\gamma}=1,$ this assumption
implies that $L$ is also infinitesimally $A$-bounded on the graph space $D(A)$
and can hence be can be seen as the analog of (2.9) in Theorem 2.2.

We are now ready to state and formally prove our main theorem of this section
in the following:

######  Theorem 2.7.

Let $X$ be a Banach space and $(L,D(L))$ and $(A,D(A))$ be generators of
strongly continuous contraction semigroups. Moreover, assume that the Favard
spaces of $A$ satisfy $F_{\alpha}\subseteq D(L)$ and $LF_{\gamma}\subseteq
F_{\beta}$ for some positive numbers $\alpha<\min\\{1,\gamma\\}$ and
$k_{\gamma}<\beta\leq\gamma<2,$ where $k_{\gamma}\in\\{0,1\\}$ is such that
$\gamma-k_{\gamma}\in(0,1].$ Then for all $n\in\mathbb{N}$ and $t\geq 0$ we
have

$\displaystyle\left\|\left(e^{tL/n}e^{tA/n}\right)^{n}-e^{t(A+L)}\right\|_{F_{\gamma}\to
X}=\mathcal{O}\left(\frac{t^{1+\delta}}{n^{\delta}}\right),$ (2.26)

where $\delta=\min\\{1,\beta,\gamma-\alpha\\}.$

######  Remark 2.8.

In the special case $\gamma=1,$ using that $D(A)\subseteq F_{1},$ the same
argument as for Theorem 2.7 gives under the condition that
$F_{\alpha}\subseteq D(L)$ and $LD(A)\subseteq F_{\beta}$ for some
$0<\alpha,\beta<1$ that

$\displaystyle\left\|\left(e^{tL/n}e^{tA/n}\right)^{n}-e^{t(A+L)}\right\|_{D(A)\to
X}=\mathcal{O}\left(\frac{t^{1+\delta}}{n^{\delta}}\right),$

where $\delta=\min\\{\beta,1-\alpha\\}.$

######  Remark 2.9.

Theorem 2.2 can be seen as an extreme point of the above Theorem 2.7 extending
it to the case $\gamma=2$ and $\beta=\alpha=1.$ In this case the Favard spaces
$F_{2}$ and $F_{1}$ are replaced by $D(A^{2})$ and $D(A)$ respectively.
Furthermore, relative $A$-boundedness of $L$ with bound strictly smaller than
1 with respect to the norm on $X$ and the graph norm on $D(A)$, i.e., (2.8)
and (2.9), needs to be assumed additionally.

###### Proof of Theorem 2.7.

Since the map $L:F_{\alpha}\to X,$ is well defined and closed (as $L:D(L)\to
X$ is closed) we see by the closed graph theorem that it is also bounded and
furthermore, using $\alpha<1,$ that $L$ is infinitesimally $A$-bounded with
respect to the norm on $X$ [EN00, Lem. 2.13], i.e., for all $\varepsilon>0$
and $x\in D(A)$ we have

$\displaystyle\|Lx\|\leq\varepsilon\|Ax\|+b_{\varepsilon}\|x\|$ (2.27)

for some $b_{\varepsilon}\geq 0.$

We want to repeat this argument to show that $L$ is also infinitesimally
$A$-bounded with respect to the graph norm on $D(A^{k_{\gamma}}).$ In the case
$k_{\gamma}=0$ we have already done this (2.27), so we focus on
$k_{\gamma}=1$: From $LF_{\gamma}\subseteq F_{\beta}$ and the closed graph
theorem we get that $L:F_{\gamma}\to F_{\beta}$ is bounded. Using
$k_{\gamma}<\beta$ and hence $F_{\beta}\subseteq D(A^{k_{\gamma}}),$ this in
particular implies boundedness of $L:F_{\gamma}\to D(A^{k_{\gamma}})$.
Furthermore, using $\gamma<k_{\gamma}+1$ and [EN00, Lem. 2.13] on the Banach
space $D(A^{k_{\gamma}}),$ we see that $L$ is infinitesimally $A$-bounded with
respect to the graph norm, i.e., for all $\varepsilon>0$ and $x\in
D(A^{k_{\gamma}+1})\subseteq F_{\gamma}\subseteq D(L)$ we have

$\displaystyle\|Lx\|_{D(A^{k_{\gamma}})}\leq\varepsilon\|Ax\|_{D(A^{k_{\gamma}})}+c_{\varepsilon}\|x\|_{D(A^{k_{\gamma}})}$
(2.28)

for some $c_{\varepsilon}\geq 0.$

Since $(L,D(L))$ generates a strongly continuous contraction semigroup and is
hence dissipative, we can use (2.27) and [EN00, Theorem 2.7] which gives that
$(A+L,D(A))$ generates a strongly continuous contraction semigroup as well.
Furthermore, again from (2.27) we see that the graph norms of $A+L$ and $A$
are equivalent. This shows that the restriction $e^{t(A+L)}|_{D(A)}$ defines a
strongly continuous semigroup on the graph space $D(A),$ which satisfies
$\|e^{t(A+L)}\|_{D(A)}\lesssim\|e^{t(A+L)}\|_{D(A+L)}\leq 1$ for all $t\geq
0.$ Its generator is given by $A+L$ restricted on $D((A+L)^{2}).$555In the
case of $k_{\gamma}=1$ we even have $D((A+L)^{2})=D(A^{2})$ which follows
using (2.27) and (2.28) and [P18, Lem. IV.3]. By [EN00, Theorem 3.8], it has
the set of positive numbers in its resolvent set and satisfies
$\sup_{\lambda>0}\|\lambda(\lambda-A+L)^{-1}\|_{D(A)}\lesssim 1.$

We can hence employ the stability result of Favard spaces, Lemma 2.6 666Note,
that Lemma 2.6, is applicable in this context as $F_{\gamma}(A)$ and
$F_{\gamma}(A+L)$ can be seen as Favard spaces of order $r_{\gamma}\in(0,1]$
but with underlying Banach space being $D(A^{k_{\gamma}})$ instead of $X.$ on
$D(A^{k_{\gamma}})$ as underlying Banach space together with (2.28), which
gives $F_{\gamma}(A+L)=F_{\gamma}(A).$ Furthermore, we have that $e^{t(A+L)}$
is bounded uniformly over $t\geq 0$ with respect to the Favard norm of
$F_{\gamma}\equiv F_{\gamma}(A)$ using the closed graph theorem for the
operator $e^{t(A+L)}:F_{\gamma}\to F_{\gamma}.$

The key commutator bound in Lemma 2.1 gives for all $x\in D(A)$

$\left\|\left(\left(e^{tL/n}e^{tA/n}\right)^{n}-e^{t(A+L)}\right)x\right\|\leq
t\sup_{s,\tau\in[0,t]}\left\|[L,e^{sA/n}]e^{\tau(A+L)}x\right\|.$ (2.29)

In fact, since $L:F_{\alpha}\to X$ is bounded and furthermore
$F_{\gamma}\subseteq\overline{D(A)}^{\|\,\cdot\,\|_{F_{\alpha}}}$, which holds
by $\gamma<\alpha$ and [EN00, Prop. 5.14., Theo. 5.15], we get (2.29) for all
$x\in F_{\gamma}.$ Using this and the above we see

$\displaystyle\left\|\left(e^{tL/n}e^{tA/n}\right)^{n}-e^{t(A+L)}\right\|_{F_{\gamma}\to
X}\leq
t\sup_{s,\tau\in[0,t]}\left\|[L,e^{sA/n}]e^{\tau(A+L)}\right\|_{F_{\gamma}\to
X}$ $\displaystyle\lesssim
t\sup_{s\in[0,t]}\left\|[L,e^{sA/n}]\right\|_{F_{\gamma}\to X}\leq
t\sup_{s\in[0,t]}\left\|L\left(e^{sA/n}-I\right)\right\|_{F_{\gamma}\to
X}+t\sup_{s\in[0,t]}\left\|\left(e^{sA/n}-I\right)L\right\|_{F_{\gamma}\to
X}.$ (2.30)

For the first term in (2.3) we can use Lemma 2.5 together with the fact that
$L:F_{\alpha}\to X$ is bounded which gives

$\displaystyle\sup_{s\in[0,t]}\left\|L\left(e^{sA/n}-I\right)\right\|_{F_{\gamma}\to
X}\lesssim\sup_{s\in[0,t]}\left\|e^{sA/n}-I\right\|_{F_{\gamma}\to
F_{\alpha}}=\mathcal{O}\left(\left(\frac{t}{n}\right)^{\gamma^{\prime}}\right)$

where $\gamma^{\prime}=\min\\{1,\gamma-\alpha\\}.$ To finish the proof we use
the boundedness of $L:F_{\gamma}\to F_{\beta}$ to bound the second term in
(2.3) as

$\displaystyle\sup_{s\in[0,t]}\left\|\left(e^{sA/n}-I\right)L\right\|_{F_{\gamma}\to
X}\lesssim\sup_{s\in[0,t]}\left\|e^{sA/n}-I\right\|_{F_{\beta}\to
X}=\mathcal{O}\left(\left(\frac{t}{n}\right)^{\beta^{\prime}}\right),$

where $\beta^{\prime}=\min\\{1,\beta\\}.$ ∎

In the following subsections, we apply the result of Theorem 2.7 for special
cases of generators for which we have good control over the corresponding
Favard spaces. The first case, in Section 2.4, considers $-A$ to be a positive
operator on some Banach space $X.$ In the second case, in Section 2.5, we
consider $X$ to be a Hilbert space and $A$ either self-adjoint with $-A$ being
positive semidefinite or $A$ being anti self-adjoint (i.e., $iA$ self-adjoint)
and hence $A$ being the generator of a unitary group. In all of these cases,
we see that the Favard spaces of $A$ are closely linked to the domains of the
fractional powers of $A$ (or, more precisely, of $|A|$). This leads us to
formulate the abstract result of Theorem 2.7 for these special cases in a more
concrete way in Corollaries 2.13, 2.14, 2.18, 2.21 respectively.

###  2.4. Strictly dissipative dynamics

Positive operators on general Banach spaces are defined as follows:

######  Definition 2.10 (Positive operator).

A closed operator $B$ is called positive on a Banach space $X$ if
$(-\infty,0]\subseteq\rho(B)$ and
$\|(\lambda-B)^{-1}\|=\mathcal{O}((1-\lambda)^{-1})$ for
$\lambda\in(-\infty,0].$

######  Remark 2.11.

If $B$ is a self-adjoint operator on a Hilbert space, then positivity in the
usual sense, i.e., $\inf_{x\in D(B);\|x\|=1}\langle Bx,x\rangle>0$ is
equivalent to positivity as in the above definition.

The relevance of this definition for us is by the following simple
observation.

######  Lemma 2.12.

Let $A$ be a generator of a strongly continuous contraction semigroup, then
$-A$ is positive if $0\in\rho(A).$

###### Proof.

Since $0\in\rho(A),$ we have for some $\varepsilon>0$ by openness of the
resolvent set and continuity of the resolvent in $\lambda$ that
$[-\varepsilon,\varepsilon]\subseteq\rho(A)$ and

$\|(\lambda+A)^{-1}\|=\mathcal{O}(1)\text{ for }|\lambda|\leq\varepsilon.$

Furthermore, we have by [EN00, Thm. 3.5], the bound
$\|\lambda(\lambda-A)^{-1}\|\leq 1$ for $\lambda>0$ for generators of
contraction semigroups. Combining both, we see

$\|(\lambda+A)^{-1}\|=\mathcal{O}((1-\lambda)^{-1})\text{ for
}\lambda\in(-\infty,0].$

∎

If one assumes that $-A$ is positive, one can define $A^{-\alpha}$ for
$\alpha>0$, and the operator is given by the explicit formula [EN00, Corr.
5.28]

$A^{-\alpha}=\frac{1-e^{-2\pi i\alpha}}{2\pi
i}\int_{0}^{\infty}s^{-\alpha}(s-A)^{-1}\ ds\text{ for }\alpha\in(0,1).$

The operator $A^{\alpha}$ is then defined as the inverse of $A^{-\alpha}$ with
$D(A^{\alpha})=\operatorname{ran}(A^{-\alpha}).$ For $\alpha<\beta$, we also
have $D(A^{\beta})\subseteq F_{\beta}\subseteq D(A^{\alpha})$, where the first
inclusion can be found in [L09, Prop. 4.7] and the proof of the second
inclusion follows from [EN00, Prop. 5.33] 777The proof assumes a negative
growth bound of the semigroup since the book only defines Favard spaces in
this context. However, a careful inspection of the proof shows that it applies
to our setting, as well.

$\|x\|+\|A^{\alpha}x\|\lesssim\|x\|_{F_{\beta}}\lesssim\|x\|+\|A^{\beta}x\|.$
(2.31)

Using this, we can immediately deduce the following result from Theorem 2.7:

######  Corollary 2.13.

Let $X$ be a Banach space and $L$ and $A$ be generators of strongly continuous
contraction semigroups on $X$ with $0\in\rho(A)$, $D(A^{\alpha})\subseteq
D(L)$, and $LD(A)\subseteq D(A^{\beta})$ for some $\alpha,\beta\in(0,1).$
Then, for fixed $\varepsilon>0$, the Trotter-Kato product satisfies

$\displaystyle\left\|\left(\left(e^{tL/n}e^{tA/n}\right)^{n}-e^{t(A+L)}\right)\right\|_{D(A)\to
X}=\mathcal{O}_{\varepsilon}\left(\frac{t^{1+\delta}}{n^{\delta}}\right),$

where $\delta=\min\\{\beta,1-\alpha-\varepsilon\\}.$

###### Proof.

By (2.31) we have that $LD(A)\subseteq D(A^{\beta})\subseteq F_{\beta}.$
Furthermore, since $D(A^{\alpha})\subseteq D(L)$ we have, again by (2.31) that
for any $\varepsilon>0$ fixed also $F_{\alpha+\varepsilon}\subseteq D(L).$
Hence, the conditions of Theorem 2.7 or Remark 2.8 are fulfilled which
finishes the proof. ∎

Assuming $X$ to be a Hilbert space, see [L09, Corr. 4.30], we have equality
$D(A^{\beta})=F_{\beta}$ . This way, we get rid of the $\varepsilon>0$
dependence in Theorem 2.13 and one finds

######  Corollary 2.14.

Let $X$ be a Hilbert space and $L$ and $A$ be generators of strongly
continuous contraction semigroups on $X$ with $0\in\rho(A)$,
$D(A^{\alpha})\subset D(L)$, and $LD(A)\subset D(A^{\beta})$ for some
$\alpha,\beta\in(0,1).$ Then, the Trotter-Kato product satisfies

$\displaystyle\left\|\left(\left(e^{tL/n}e^{tA/n}\right)^{n}-e^{t(A+L)}\right)\right\|_{D(A)\to
X}=\mathcal{O}\left(\frac{t^{1+\delta}}{n^{\delta}}\right)$

where $\delta=\min\\{\beta,1-\alpha\\}.$

Our framework in the above theorem allows, for example, for $A$ to be the
Dirichlet-Laplacian on a sufficiently nice bounded domain
$\Omega\subset\mathbb{R}^{n}$, since
$0\notin\operatorname{Spec}(\Delta_{H^{2}(\Omega)\cap H_{0}^{1}(\Omega)}).$
However, it does not allow us to study, for instance, basic operators such as
$A=\Delta$ on $\mathbb{R}^{n}.$ We will take care of this in the next two
subsections.

###  2.5. Self-adjoint and anti self-adjoint generators on Hilbert spaces

In the case of self-adjoint or anti self-adjoint $A$, one can use the spectral
theorem to easily characterize the Favard spaces. This is the content of the
following Lemma:

######  Lemma 2.15.

Let $-A$ be self-adjoint and positive semi-definite on a Hilbert space $X$,
then in the sense of the functional calculus for all $\alpha>0$

$F_{\alpha}(A)=D(|A|^{\alpha})$

and the Favard norm is equivalent to the graph norm of $|A|^{\alpha}.$

Let $K$ be self-adjoint, then for all $\alpha>0$

$F_{\alpha}(iK)=D(|K|^{\alpha})$

and the corresponding Favard norm is equivalent to the graph norm of
$|K|^{\alpha}.$

######  Remark 2.16.

The specific form of the Favard spaces presented in Lemma 2.15 enables us to
provide stronger connections between different levels of the Sobolev tower:
Let $-A$ be self-adjoint and positive semidefinite. Then for all $\alpha>0$
the graph space $D(|A|^{\alpha})$ naturally becomes a Hilbert space when
equipped with the scalar product $\langle
x,y\rangle_{\alpha}:=\langle|A|^{\alpha}x,|A|^{\alpha}y\rangle+\langle
x,y\rangle.$ It is easy to verify that the restriction
$-A^{\prime}:=-A|_{D(|A|^{\alpha+1})}$ defines again a self-adjoint, positive
semi-definite operator on $D(|A|^{\alpha})$ with fractional powers
$|A^{\prime}|^{\beta}$ being defined on the domain
$D(|A^{\prime}|^{\beta})=D(|A|^{\alpha+\beta}).$ Hence, using Lemma 2.15, we
see that the Favard spaces of $A^{\prime}$ satisfy

$\displaystyle
F_{\beta}(A^{\prime};D(|A|^{\alpha}))=D(|A|^{\alpha+\beta})=F_{\alpha+\beta}(A;X),$
(2.32)

which can be seen as a refined version of (2.12).

Let now $L$ be closed and such that $F_{\alpha}(A;X)\subseteq D(L)$ and
$LF_{\gamma}(A;X)\subseteq F_{\beta}(A;X)$ for some positive numbers
$\alpha<1$ and $\gamma-1<\beta\leq\gamma<2.$ Using (2.32) instead of (2.12),
we can argue similarly as in the proof of Theorem 2.7 but replacing
$D(A^{k_{\gamma}})$ with $D(|A|^{\beta})$ in the case $\gamma\geq 1$ that
$e^{t(A+L)}:F_{\gamma}(A;X)\to F_{\gamma}(A;X)$ is a well-defined bounded
operator. This enables us in Corollary 2.18 to lift the restriction
$\beta>k_{\gamma}$ needed in Theorem 2.7 and allow for all
$\beta\geq\gamma-1.$ The same argument can also be employed for anti self-
adjoint generators, i.e., $A=iK$ for $K$ being self-adjoint, giving the same
extended range on the parameter $\beta$ in Corollary 2.21.

###### Proof of Lemma 2.15.

Let $r\in(0,1)$ and $m\in{\mathbb{N}}.$ Using the functional calculus, we find
with $E_{|A|}$ being the spectral measure of $|A|=-A$

$\|\lambda^{r}A^{m}(\lambda-A)^{-1}x\|^{2}=\int_{0}^{\infty}\lambda^{2r}\frac{t^{2m}}{(\lambda+t)^{2}}d\langle
E_{|A|}(t)x,x\rangle.$

The right-hand side is maximized for $\lambda=\frac{{r}}{1-r}t$ which shows
that

$\begin{split}\sup_{\lambda>0}\|\lambda^{r}A^{m}(\lambda-A)^{-1}x\|^{2}&=\left(\frac{r}{1-r}\right)^{2r}\int_{0}^{\infty}\frac{t^{2(m+r)}}{t^{2}(1+\frac{r}{1-r})^{2}}d\langle
E_{|A|}(t)x,x\rangle\\\
&=r^{2r}(1-r)^{2-2r}\int_{0}^{\infty}t^{2(r+m-1)}d\langle
E_{|A|}(t)x,x\rangle\\\ &\sim\||A|^{r+m-1}x\|^{2}.\end{split}$

Noting that the maximising $\lambda$ is independent of $m$, this shows in
particular for all $\alpha>0,$ with $r_{\alpha}\in(0,1)$ and
$k_{\alpha}\in{\mathbb{N}}_{0}$, the unique numbers such that
$\alpha=r_{\alpha}+k_{\alpha},$ that

$\displaystyle\sup_{\lambda>0}\|\lambda^{r_{\alpha}}A(\lambda-A)^{-1}x\|_{D(A^{k_{\alpha}})}\sim\||A|^{r_{\alpha}}x\|+\||A|^{\alpha}x\|$

Noting that $\||A|^{r_{\alpha}}x\|\leq\|x\|+\||A|^{\alpha}x\|,$ this gives by
definition of the Favard norm
$\|x\|_{F_{\alpha}(A)}\sim\|x\|_{D(|A|^{\alpha})}.$

For the second part of the Lemma, using again the functional calculus for
$r\in(0,1)$ and $m\in{\mathbb{N}}$

$\begin{split}\|\lambda^{r}K^{m}(\lambda-
iK)^{-1}x\|^{2}&=\int_{-\infty}^{\infty}\lambda^{2r}\frac{t^{2m}}{\lambda^{2}+t^{2}}d\langle
E_{K}(t)x,x\rangle\end{split}$

The right-hand side is maximized for $\lambda=\frac{\sqrt{r}}{\sqrt{1-r}}t$
which shows that

$\begin{split}\sup_{\lambda>0}\|\lambda^{r}K^{m}(\lambda-
iK)^{-1}x\|^{2}&=\left(\frac{r}{1-r}\right)^{r}\int_{-\infty}^{\infty}\frac{t^{2(m+r)}}{\frac{r}{1-r}t^{2}+t^{2}}d\langle
E_{K}(t)x,x\rangle\\\
&=r^{r}(1-r)^{1-r}\int_{-\infty}^{\infty}t^{2(r+m-1)}d\langle
E_{K}(t)x,x\rangle\\\ &\sim\||K|^{r+m-1}x\|^{2}.\end{split}$

Following the same argument as above this shows
$\|x\|_{F_{\alpha}(iK)}\sim\|x\|_{D(|K|^{\alpha})}.$ ∎

The two results presented in Lemma 2.15 can be unified in the following way:
For normal operators $N$, defining contraction semigroups, we define commuting
self-adjoint operators [S12, Prop. 5.30]
$\operatorname{Re}(N):=\overline{(N+N^{*})}/2$ and
$\operatorname{Im}(N):=\overline{(N-N^{*})}/(2i)$ such that
$N=\operatorname{Re}(N)+i\operatorname{Im}(N).$ For $N$ to generate a
contraction semigroup, we demand that $\operatorname{Re}(N)\leq 0.$ Then, we
recall that
$T_{t}=e^{tN}:=e^{t\operatorname{Re}(N)}e^{it\operatorname{Im}(N)}=e^{it\operatorname{Im}(N)}e^{t\operatorname{Re}(N)}.$
We then have the following generalization of Lemma 2.15, which we prove in the
appendix.

######  Proposition 2.17.

Let $N$ be a normal operator on a Hilbert space with $\operatorname{Re}(N)\leq
0$ as above, then for $\alpha\in(0,1)$

$D(|N|^{\alpha})=F_{\alpha}(\operatorname{Re}(N))\cap
F_{\alpha}(\operatorname{Im}(N))=F_{\alpha}(N).$

Using this characterization of the Favard spaces, we obtain the following
Corollary 2.18 from Theorem 2.7 and Remark 2.16. It can be seen as the
analogue of Corollary 2.14 without the requirement that $0\in\rho(A).$

######  Corollary 2.18.

Let $X$ be a Hilbert space and $L$ and $A$ be generators of strongly
continuous contraction semigroups on $X$ with $-A$ self-adjoint and positive
semidefinite. Moreover assume that $D(|A|^{\alpha})\subseteq D(L)$, and
$LD(|A|^{\gamma})\subseteq D(|A|^{\beta})$ for some positive numbers
$\alpha<\min\\{1,\gamma\\}$ and $\gamma-1<\beta\leq\gamma<2.$ Then for all
$n\in\mathbb{N}$ and $t\geq 0$ the Trotter-Kato product satisfies

$\displaystyle\left\|\left(\left(e^{tL/n}e^{tA/n}\right)^{n}-e^{t(A+L)}\right)\right\|_{D(|A|^{\gamma})\to
X}=\mathcal{O}\left(\frac{t^{1+\delta}}{n^{\delta}}\right),$ (2.33)

where $\delta=\min\\{1,\beta,\gamma-\alpha\\}.$

####  2.5.1. Unitary dynamics and singular potentials

In this section, we use the characterisation of the Favard spaces in Lemma
2.15 to formulate the result of Theorem 2.7 in the case of unitary dynamics on
a Hilbert space. Hence, we shall now assume that $X$ is a Hilbert space and
that $A=iK$, with $K$ being self-adjoint.

######  Example 2.19 (Schrödinger operators).

For the case of $K=-\Delta\geq 0$ on $X=L^{2}(\mathbb{R}^{d})$, which is self-
adjoint on the domain $D(K)=H^{2}(\mathbb{R}^{d}),$ Lemma 2.15 gives that the
Favard spaces $F_{\alpha}\equiv F_{\alpha}(iK)$ are precisely given by the
Sobolev spaces $H^{2\alpha}(\mathbb{R}^{d})$ for all $\alpha>0$

######  Example 2.20 (Dirac operators).

Let $d=2$ for simplicity, then the two-dimensional Dirac operator reads in
terms of Pauli matrices $\sigma_{i}$

$K=\sum_{i=1}^{2}\sigma_{i}D_{x_{i}}+\sigma_{3}m.$

By the standard Foldy-Wouthuysen transform, the operator $K$ is unitarily
equivalent to

$\operatorname{diag}(\sqrt{(-\Delta)+m^{2}},-\sqrt{(-\Delta)+m^{2}}).$

From this it follows that $F_{\alpha}\equiv
F_{\alpha}(iK)=H^{\alpha}(\mathbb{R}^{2};{\mathbb{C}}^{2}).$

Using Lemma 2.15 we obtain the following corollary from Theorem 2.7 and Remark
2.16 for the case of $A=iK$ and $L=iH$ for some self-adjoint operators $K$ and
$H$:

######  Corollary 2.21.

Let $X$ be a Hilbert space and $H$ and $K$ be self-adjoint. Moreover assume
that $D(|K|^{\alpha})\subseteq D(H)$, and $HD(|K|^{\gamma})\subseteq
D(|K|^{\beta})$ for some positive numbers $\alpha<\min\\{1,\gamma\\}$ and
$\gamma-1<\beta\leq\gamma<2.$ Then for all $n\in\mathbb{N}$ and
$t\in{\mathbb{R}}$ the Trotter-Kato product satisfies

$\displaystyle\left\|\left(\left(e^{itH/n}e^{itK/n}\right)^{n}-e^{it(H+K)}\right)\right\|_{D(|K|^{\gamma})\to
X}=\mathcal{O}\left(\frac{|t|^{1+\delta}}{n^{\delta}}\right),$ (2.34)

where $\delta=\min\\{1,\beta,\gamma-\alpha\\}.$

Application to singular potentials. In the following we apply Corollary 2.21
for the case of $X=L^{2}(\mathbb{R}^{d})$, multiplication operator $H=V$ with
singular potential $V(x)=|x|^{-a},$888Note that the assumptions on $H$ in
Corollary 2.21 are independent of the sign of $H.$ Hence, all arguments here
also work for the attractive singular potentials $V(x)=-|x|^{-a}.$ for some
$a>0,$ $K=-\Delta\geq 0.$ Here, from Lemma 2.15 and Example 2.19, we have that
$F_{\alpha}(iK)=D(K^{\alpha})=H^{2\alpha}({\mathbb{R}}^{d})$ for all
$\alpha>0.$

The relative boundedness condition in Corollary 2.21, i.e., the condition
$H^{2\alpha}({\mathbb{R}}^{d})\subseteq D(V)$ for some suitable
$\alpha\in(0,1),$ can be obtained as follows: Assume that the potential
satisfies $V=V_{1}+V_{2}\in
L^{p}({\mathbb{R}}^{d})+L^{\infty}({\mathbb{R}}^{d})$ for some $p\geq 1.$ Then
we have for $1/2=1/p+1/q$

$\|V\varphi\|_{2}\leq\|V_{1}\|_{p}\|\varphi\|_{q}+\|V_{2}\|_{\infty}\|\varphi\|_{2}.$
(2.35)

By the Sobolev embedding, we have $H^{k}(\mathbb{R}^{d})\subseteq
L^{q}(\mathbb{R}^{d})$ for parameters

$\frac{1}{2}-\frac{k}{d}=\frac{1}{q}$

and hence we can choose $k=\frac{d(q-2)}{2q}$ and $\alpha=\frac{d(q-2)}{4q}.$
For the special case of $p=2$ and $q=\infty,$ we find $\alpha=\frac{d}{4}.$
For the potential $V(x)=|x|^{-a}$, we can choose $p=2$ as long as $a<d/2,$ in
which case we have shown that $H^{d/2}({\mathbb{R}}^{d})\subseteq D(V).$

To verify the second condition in Corollary 2.21 for singular potentials,
i.e., $VD((-\Delta)^{\gamma})\subseteq D((-\Delta)^{\beta})$ for some suitable
$k_{\gamma}<\beta\leq\gamma<2$, we can invoke the Brezis-Mironescu inequality
[BM01, Corr.4] which states that for $s\in(1,\infty)$ and
$\beta=\frac{s-1}{2}$

$\|(-\Delta)^{\beta}(fg)\|_{2}\lesssim\|f\|_{\infty}\|g\|_{H^{s-1}}+\|g\|_{2s}\|f\|_{H^{s}}^{1-1/s}\|f\|_{\infty}^{1/s}.$
(2.36)

We use this for $g=V_{1}$ being the singular part of the potential
$V(x)=|x|^{-a}$ as the remainder can be chosen to be smooth and hence
trivially satisfies the desired condition. In order for $V(x)=|x|^{-a}$ to be
in $L^{2s}_{\text{loc}}(\mathbb{R}^{d})$ (and hence $V_{1}\in
L^{2s}(\mathbb{R}^{d})$) we require that $a<\frac{d}{2s}.$ Thus, as above
$a<\frac{d}{2}$ is necessary for this condition to apply. More generally, for
this choice of potential $V\in H^{s-1}_{\text{loc}}(\mathbb{R}^{d})$ if and
only if $a<\tfrac{d}{2}-(s-1).$ With this choice and furthermore $f=\varphi\in
H^{2\gamma}(\mathbb{R}^{d})=D((-\Delta)^{\gamma}),$ we find from (2.36) for
$\gamma\geq\tfrac{d}{4},$ and hence by Sobolev embedding $\varphi\in
L^{\infty}(\mathbb{R}^{d}),$ and $s\leq 2\gamma$

$\|(-\Delta)^{\beta}(V_{1}\varphi)\|_{2}\lesssim\|\varphi\|_{\infty}\|V_{1}\|_{H^{s-1}}+\|V_{1}\|_{2s}\|\varphi\|_{H^{s}}^{1-1/s}\|\varphi\|_{\infty}^{1/s}.$
(2.37)

Hence, collecting all constraints on $s$ and using $\beta=\tfrac{s-1}{2},$ we
see $VD((-\Delta)^{\gamma})\subseteq D((-\Delta)^{\beta})$ for all
$\beta<\min\\{\tfrac{d-2a}{4a},\tfrac{d-2a}{4},\gamma-\tfrac{1}{2}\\}$

For the case $a=1,d=3,$ i.e., for the Coulomb potential, and furthermore
$\gamma=1$, we can therefore choose any $\beta<1/4.$ Thus, Corollary 2.21
gives in the Coulomb case $V(x)=\pm|x|^{-1}$ a Trotter convergence as

$\left\|\left(\left(e^{itV/n}e^{-it\Delta/n}\right)^{n}-e^{it(-\Delta+V)}\right)\right\|_{H^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})}=\mathcal{O}(n^{-1/4+\varepsilon}).$

for all $t\in{\mathbb{R}}$ and $\varepsilon>0$ fixed. More generally, the
above derivations give the following:

######  Corollary 2.22.

Let $V(x)=\pm|x|^{-a}$ with $0<a<d/2$ and $\gamma\in(0,2).$ Then the Trotter
formula converges as

$\left\|\left(\left(e^{itV/n}e^{-it\Delta/n}\right)^{n}-e^{it(-\Delta+V)}\right)\right\|_{H^{2\gamma}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})}=\mathcal{O}\left(n^{d/4-\gamma}+n^{-\beta}+n^{-1}\right)$

for all
$\gamma-1<\beta<\min\\{\tfrac{d-2a}{4a},\tfrac{d-2a}{4},\gamma-\tfrac{1}{2}\\}.$

Molecular Hamiltonians. We consider a molecule with $N$ electrons of mass
$m_{\mathrm{e}}$ and charge $-e$. In addition, we have $M$ nuclei of masses
$m_{j}$ and charges $Z_{j}e$ for $j=1,...,M$ described by the many-body
Schrödinger operator

$H=-\sum_{j=1}^{N}\frac{\Delta_{x_{j}}}{2m_{\mathrm{e}}}-\sum_{j=1}^{M}\frac{\Delta_{y_{j}}}{2m_{j}}+V(x,y)$

where

$V(x,y)=\frac{1}{2}\sum_{i\neq
j}\frac{e^{2}}{|x_{i}-x_{j}|}-\sum_{i,j}\frac{e^{2}Z_{j}}{|x_{i}-y_{j}|}+\frac{1}{2}\sum_{i\neq
j}\frac{e^{2}}{|y_{i}-y_{j}|}.$ (2.38)

In the many-body setting, we can repeat the steps leading to the result of
Corollary 2.22 by working with (Sobolev-)Bochner spaces instead of standard
Sobolev spaces.

To streamline the notation we shall no longer discriminate between $x$ and $y$
variables and just consider $x\in\mathbb{R}^{3(N+M)}.$

We shall allow for Trotter splittings

$A:=i\Delta-iV_{1}\text{ and }L:=-iV_{2}$ (2.39)

where $V_{1}$ and $V_{2}$ is any choice of summands in (2.38) such that
$V=V_{1}+V_{2}.$ This includes, of course, the Trotter splitting into kinetic
and potential energy, but also allows us to consider the kinetic energy
together with the nucleus-nucleus and nucleus-electron interaction, separated
from the electron-electron interaction. We have this freedom since combining
Lemma 2.6 with Lemma 2.15, we see that the Favard spaces $F_{\alpha}(A)$ are
precisely the Sobolev spaces $H^{2\alpha}(\mathbb{R}^{N+M})$ for all
$\alpha\in(0,1)$ and are independent of $V_{1},$ since $V_{1}$ is relatively
$-\Delta$ bounded.

We shall show that:

######  Theorem 2.23.

For the Trotter-splitting (2.39), there is for any $\varepsilon>0$ a constant
$C_{\varepsilon,N,M}$ such that

$\left\|\left(\left(e^{tL/n}e^{tA/n}\right)^{n}-e^{t(A+L)}\right)\right\|_{H^{2}(\mathbb{R}^{3(N+M)})\to
L^{2}(\mathbb{R}^{3(N+M)})}\leq C_{\varepsilon,N,M}n^{-1/4+\varepsilon}.$

###### Proof.

The potential $V$ consists of linear combinations of Coulomb potentials
$V_{\operatorname{sp}}(x_{i}-x_{j})$ in terms of the single particle Coulomb
potential

$V_{\operatorname{sp}}(x):=|x|^{-1}\text{ for
}x\in\mathbb{R}^{3}\setminus\\{0\\}.$

We start by proving the $V_{\text{sp}}:H^{3/2}(\mathbb{R}^{3(N+M)})\to
L^{2}(\mathbb{R}^{3(N+M)})$ boundedness. Let $\varphi\in
H^{2}(\mathbb{R}^{3(N+M)})$, we have the natural identification of the multi-
variable $L^{2}$ as a single particle $L^{2}$ Bochner space

$\|V_{\text{sp}}\varphi\|_{L^{2}(\mathbb{R}^{3(N+M)})}=\|V_{\text{sp}}\varphi\|_{L^{2}(\mathbb{R}^{3};X)},$

where we defined $X:=L^{2}(\mathbb{R}^{3(N+M-1)})$. We also notice that
$\varphi\in H^{2}(\mathbb{R}^{3};X)$ which follows easily using the Fourier
representation of the Sobolev space and the simple inequality
$\langle\xi_{1}\rangle\leq\langle(\xi_{1},..,\xi_{3(N+M)})\rangle.$ This
allows us to repeat the estimate in (2.35) in the Bochner space and we obtain
using the same composition of
$V_{\text{sp}}=V_{\text{sp},1}+V_{\text{sp},2}\in
L^{2}(\mathbb{R}^{3})+L^{\infty}(\mathbb{R}^{3})$ as in the subsection before

$\begin{split}\|V_{\text{sp}}\varphi\|_{L^{2}(\mathbb{R}^{3};X)}&\leq\|V_{\text{sp},1}\|_{L^{2}(\mathbb{R}^{3})}\|\varphi\|_{L^{\infty}(\mathbb{R}^{3};X)}+\|V_{\text{sp},2}\|_{L^{\infty}(\mathbb{R}^{3})}\|\varphi\|_{L^{2}(\mathbb{R}^{3};X)}\\\
&\lesssim\|V_{\text{sp},1}\|_{L^{2}(\mathbb{R}^{3})}\|\varphi\|_{H^{3/2}(\mathbb{R}^{3};X)}+\|V_{\text{sp},2}\|_{L^{\infty}(\mathbb{R}^{3})}\|\varphi\|_{L^{2}(\mathbb{R}^{3};X)}\\\
&\lesssim\|V_{\text{sp},1}\|_{L^{2}(\mathbb{R}^{3})}\|\varphi\|_{H^{3/2}(\mathbb{R}^{3(N+M)})}+\|V_{\text{sp},2}\|_{L^{\infty}(\mathbb{R}^{3})}\|\varphi\|_{L^{2}(\mathbb{R}^{3};X)}\\\
&\lesssim(\|V_{\text{sp},1}\|_{L^{2}(\mathbb{R}^{3})}+\|V_{\text{sp},2}\|_{L^{\infty}(\mathbb{R}^{3})})\|\varphi\|_{H^{3/2}(\mathbb{R}^{3(N+M)})}.\end{split}$
(2.40)

Next, we show that $V_{\text{sp}}:H^{2}(\mathbb{R}^{3(N+M)})\to
H^{s-1}(\mathbb{R}^{3(N+M)})$ where $s<3/2.$ It suffices to consider the
singular part of the potential $V_{\text{sp},1}$.

Using Peetre’s inequality, which implies that for $t\geq 0$

$\langle(\xi_{1},..,\xi_{m},\xi_{m+1},...,\xi_{n})\rangle^{t}\leq
2^{t}\langle(\xi_{1},..,\xi_{m})\rangle^{t}\langle(\xi_{m+1},...,\xi_{n})\rangle^{t},$

we have for $Y:=H^{s-1}(\mathbb{R}^{3(N+M-1)})$ and $s<3/2$ together with the
Brezis-Mironescu inequality that

$\begin{split}\|V_{\text{sp},1}\varphi\|_{H^{s-1}(\mathbb{R}^{3(N+M)})}&\lesssim\|V_{\text{sp},1}\varphi\|_{H^{s-1}(\mathbb{R}^{3};Y)}\\\
&\lesssim\|\varphi\|_{H^{3/2}(\mathbb{R}^{3};Y)}\|V_{\text{sp},1}\|_{H^{s-1}(\mathbb{R}^{3})}+\|V_{\text{sp},1}\|_{L^{2s}(\mathbb{R}^{3})}\|\varphi\|_{H^{s}(\mathbb{R}^{3};Y)}^{1-1/s}\|\varphi\|_{H^{3/2}(\mathbb{R}^{3};Y)}^{1/s}\\\
&\lesssim\|\varphi\|_{H^{2}(\mathbb{R}^{3(N+M)})}(\|V_{\text{sp},1}\|_{H^{s-1}(\mathbb{R}^{3})}+\|V_{\text{sp},1}\|_{L^{2s}(\mathbb{R}^{3})}),\end{split}$
(2.41)

where we used in the last line that for $\alpha,\beta\geq 0$

$\langle(\xi_{1},..,\xi_{m})\rangle^{\alpha}\langle(\xi_{m+1},...,\xi_{n})\rangle^{\beta}\leq\langle(\xi_{1},..,\xi_{m},\xi_{m+1},...,\xi_{n})\rangle^{\alpha+\beta}.$

This extends the estimate (2.37) of the previous section to the many-particle
Hilbert space.

It remains to argue that we can lift the bounds (2.40) and (2.41) to the two-
body interactions appearing in the many-body Hamiltonian, instead of the
single particle interaction $V_{\operatorname{sp}}.$ To see this, notice that
the two-particle interaction
$W(x_{1},x_{2}):=\sqrt{2}V_{\text{sp}}(x_{1}-x_{2})$ – without loss of
generality, we consider now the first two particles – is given by
$W(x):=(V\circ U)(x),$ where we introduced the unitary map

$U(x):=2^{-1/2}(x_{1}-x_{2},x_{1}+x_{2},x_{3},...,x_{N+M}).$

Using the Fourier representation of the Sobolev spaces, we find that the
Sobolev norms are invariant under composition by a unitary map

$\begin{split}\|\langle-\Delta\rangle^{\beta}(s\circ
U)\|_{L^{2}({\mathbb{R}}^{3(N+M)})}&=\|\hat{s}\circ
U\|_{L^{2}({\mathbb{R}}^{3(N+M)};\langle\xi\rangle^{2\beta}d\xi)}\\\
&=\|\hat{s}\|_{L^{2}({\mathbb{R}}^{3(N+M)};\langle\xi\rangle^{2\beta}d\xi)}\\\
&=\|\langle-\Delta\rangle^{\beta}s\|_{L^{2}({\mathbb{R}}^{3(N+M)})}.\end{split}$

This property allows us to lift bounds (2.40) and (2.41) to

$\begin{split}\|W\varphi\|_{L^{2}(\mathbb{R}^{3};X)}&=\|(V_{\text{sp}}\cdot(\varphi\circ
U^{*}))\circ U\|_{L^{2}(\mathbb{R}^{3};X)}=\|(V_{\text{sp}}\cdot(\varphi\circ
U^{*}))\|_{L^{2}(\mathbb{R}^{3};X)}\\\
&\lesssim(\|V_{\text{sp},1}\|_{L^{2}(\mathbb{R}^{3})}\|+\|V_{\text{sp},2}\|_{L^{\infty}(\mathbb{R}^{3})}\|)\|\varphi\circ
U^{*}\|_{H^{3/2}(\mathbb{R}^{3(N+M)})}\\\
&\lesssim(\|V_{\text{sp},1}\|_{L^{2}(\mathbb{R}^{3})}\|+\|V_{\text{sp},2}\|_{L^{\infty}(\mathbb{R}^{3})}\|)\|\varphi\|_{H^{3/2}(\mathbb{R}^{3(N+M)})}\end{split}$
(2.42)

and

$\begin{split}\|W_{1}\varphi\|_{H^{s-1}(\mathbb{R}^{3(N+M)})}&=\|(V_{\text{sp},1}\cdot(\varphi\circ
U^{*}))\circ
U\|_{H^{s-1}(\mathbb{R}^{3(N+M)})}=\|V_{\text{sp},1}\cdot(\varphi\circ
U^{*})\|_{H^{s-1}(\mathbb{R}^{3(N+M)})}\\\ &\lesssim\|\varphi\circ
U^{*}\|_{H^{2}(\mathbb{R}^{3(N+M)})}(\|V_{\text{sp},1}\|_{H^{s-1}(\mathbb{R}^{3})}+\|V_{\text{sp},1}\|_{L^{2s}(\mathbb{R}^{3})})\\\
&\lesssim\|\varphi\|_{H^{2}(\mathbb{R}^{3(N+M)})}(\|V_{\text{sp},1}\|_{H^{s-1}(\mathbb{R}^{3})}+\|V_{\text{sp},1}\|_{L^{2s}(\mathbb{R}^{3})}).\end{split}$
(2.43)

The identities (2.42) and (2.43) complete the proof, and the result follows
from Theorem 2.7 and the characterization of Favard spaces in terms of Sobolev
spaces as explained before the statement of the result. ∎

In summary, we have shown that for very general input states, we can get
arbitrarily close to the $\mathcal{O}(n^{-1/4})$ rate that was observed for
the ground state of the single-particle Coulomb Hamiltonian in [B+23a], even
for many-body Coulomb Hamiltonians.

##  3\. Convergence rates for energy-limited unitary dynamics

In this section, we present a method for obtaining convergence rates inspired
directly by quantum theory. Unlike the previous section, which focused on
possibly singular but small perturbations, this approach is motivated to study
regular but large perturbations tailored towards applications in quantum
mechanics. Our approach is based on the “operator $E$-norm” introduced by
Shirokov [S20] and the notion of “energy-limited dynamics” introduced in
[vL24].

We fix a separable Hilbert space $\mathcal{H}$ and a positive self-adjoint
operator with $\inf\operatorname{Spec}G=0$. We may think about $\mathcal{H}$
as the Hilbert space describing a quantum system and of $G$ as an energy-
observable (a reference Hamiltonian) for this system. For vectors
$\psi\in\mathcal{H}$, the _energy-expectation value_ $\mathbf{E}(\psi)$ is
given by

$\mathbf{E}(\psi):={\|G^{1/2}\psi\|}^{2}$ (3.1)

if $\psi\in D(G^{1/2})$ and by $\mathbf{E}(\psi):=\infty$ otherwise.

In the following, we use the language of Sobolev towers, which we introduced
in Subsection 2.2 and will briefly recall for the reader’s convenience the
setting they are used in this section. For $n\in{\mathbb{N}}$, we set
$\mathcal{H}_{n}=D(G^{n/2})$ which is a Hilbert space in the graph inner
product $\langle\psi,\phi\rangle_{n}=\langle\psi,\phi\rangle+\langle
G^{n/2}\psi,G^{n/2}\phi\rangle$. Note that $\mathcal{H}_{1}$ is precisely the
space of finite-energy vectors because

$\mathbf{E}(\psi)={\|\psi\|}_{1}^{2}-{\|\psi\|}^{2}.$ (3.2)

For negative $n$, $\mathcal{H}_{n}$ is defined as the completion of
$\mathcal{H}$ with respect to the inner product
$\langle\psi,\phi\rangle_{n}=\langle\psi,(1+G^{n})\phi\rangle$. We have
continuous embeddings

$\ldots\supset\mathcal{H}_{-1}\supset\mathcal{H}\supset\mathcal{H}_{1}\supset\ldots.$
(3.3)

The dual space of $\mathcal{H}_{n}$ is naturally identified with
$\mathcal{H}_{-n}$. Thus, the adjoint $A^{*}$ of a bounded operator $A\in
B(\mathcal{H}_{n},\mathcal{H}_{m})$ is naturally a bounded operator
$A^{*}:\mathcal{H}_{-m}\to\mathcal{H}_{-n}$. An operator $A\in
B(\mathcal{H}_{n},\mathcal{H}_{-n})$ is called hermitian if $A=A^{*}$ [RS75,
p. 192]. Note that relatively $G^{1/2}$-bounded operators on $D(G^{1/2})$ are
precisely the elements of $B(\mathcal{H}_{1},\mathcal{H})$. Similarly,
hermitian quadratic forms $a$ on $\mathcal{H}$ with form domain
$Q(a)=\mathcal{H}_{1}$ for which $M>0$ exists such that $\pm a\leq M(G+1)$,
i.e.,

$\pm a(\psi,\psi)\leq M{\|\psi\|}_{1}^{2},\qquad\psi\in\mathcal{H}_{1},$ (3.4)

may be regarded as hermitian operators $A\in
B(\mathcal{H}_{1},\mathcal{H}_{-1})$ via the equation

$a(\psi,\phi)=\langle\phi,A\psi\rangle_{-1},\qquad\psi,\phi\in\mathcal{H}_{1}.$
(3.5)

Following [S20], the operator $E$-norm is defined as:

######  Definition 3.1 ($E$-norm, Shirokov [S20]).

The _operator $E$-norm_ ${\|{\,\cdot\,}\|}_{E}^{G}$ is defined on the space
$B(\mathcal{H}_{1},\mathcal{H})$ of $G^{1/2}$-bounded operators for $E>0$ by

${\|A\|}_{E}^{G}=\sup\big{\\{}{\|A\psi\|};{\|\psi\|}\leq 1,\
\mathbf{E}(\psi)\leq E\big{\\}}.$ (3.6)

We remark that the optimization may be restricted to a core ${\mathcal{D}}$ of
$G^{1/2}$, i.e., a $\|\cdot\|_{1}$-dense subspace
${\mathcal{D}}\subset\mathcal{H}_{1}$, [vL24]. It is shown in [S20] that the
operator $E$-norm is equivalent to the operator norm in
$B(\mathcal{H}_{1},\mathcal{H})$. In particular, operator $E$-norms at
different energy constraints $E>0$ are equivalent. In fact, Shirokov showed
[S20]:

${\|{\,\cdot\,}\|}_{E}^{G}\leq{\|{\,\cdot\,}\|}_{E^{\prime}}^{G}\leq\sqrt{\frac{E^{\prime}}{E}}\,{\|{\,\cdot\,}\|}_{E}^{G},\qquad
E^{\prime}\geq E>0.$ (3.7)

Since we are typically interested in upper bounds on the operator $E$-norm,
the following equivalent definition via a minimization (see [vL24, Prop.
2.19]) will be useful:

${\|A\|}_{E}^{G}=\min\big{\\{}\sqrt{\lambda E+E_{0}}:\lambda,E_{0}\geq 0,\
A^{*}A\leq\lambda G+E_{0}\big{\\}}.$ (3.8)

The operator inequality in (3.8) is in the sense of quadratic forms on
$\mathcal{H}_{1}=D(G^{1/2})$. The reason that we are interested in the
operator $E$-norm is that convergence in $E$-norm implies strong
convergence.999In fact, Shirokov showed that the operator $E$-norm induces the
strong operator topology on bounded subsets of $B(\mathcal{H})$ if the
reference Hamiltonian has compact resolvent [S20]. For instance, consider
self-adjoint operators $A$ and $B$ on $\mathcal{H}$ such that $A+B$ is
essentially self-adjoint, then the Trotter product converges by [T59, K78],
and convergence rates in $E$-norm

${\|{(e^{itA/n}e^{itB/n})}^{n}-e^{it\overline{(A+B)}}\|}_{E}^{G}\leq\varepsilon(t,n,E)$
(3.9)

immediately yield state-dependent convergence rates: If
$\psi\in\mathcal{H}_{1}$ is a unit vector, then

${\|{(e^{itA/n}e^{itB/n})}^{n}\psi-e^{it\overline{(A+B)}}\psi\|}\leq\varepsilon(t,n,\mathbf{E}(\psi)).$
(3.10)

To prove such convergence rates for Trotter products in the operator $E$-norm,
we need the dynamics to respect the energy scale determined by $G$. For this,
we follow [vL24]. The maximal output energy given an energy-constraint $E>0$
is described by [W17]

$f_{U}(E):=\sup\big{\\{}\mathbf{E}(U\psi);{\|\psi\|}\leq 1,\
\mathbf{E}(\psi)\leq E\big{\\}},\qquad E>0.$ (3.11)

The function $f_{U}$ is a concave nondecreasing function
${\mathbb{R}}^{+}\to[0,\infty]$. It is connected to the operator $E$-norm via

$f_{U}(E)=\Big{(}{\|G^{1/2}U\|}_{E}^{G}\Big{)}^{2}.$ (3.12)

Unitary operators for which $f_{U}$ is finite are called energy-limited [W17,
vL24, S19]. Thus, a unitary $U$ is energy-limited if and only if it restricts
to an operator $B(\mathcal{H}_{1},\mathcal{H}_{1})$.

######  Definition 3.2 (Energy-limited dynamics [vL24]).

A unitary one-parameter group $U(t)$ is _energy-limited_ if there exist
‘stability constants’ $\omega,E_{0}\geq 0$ such that

$f_{U(t)}(E)\leq e^{\omega{|t|}}(E+E_{0})-E_{0},\qquad t\in{\mathbb{R}},\
E>0.$ (3.13)

A collection of unitary one-parameter groups $U_{j}(t)$, $j\in I$, is _jointly
energy-limited_ if there exist joint stability constants $\omega,E_{0}$ such
that (3.13) holds for all $U_{j}(t)$.

Note that the right-hand side of (3.13) is a semigroup of affine functions of
$E$ for times $t>0$. Energy-limitedness guarantees that the energy expectation
value $E(t):=\mathbf{E}(U(t)\psi)$ depends continuously on $t$ for vectors
$\psi\in\mathcal{H}_{1}$. In fact, it guarantees the continuity bound [vL24]

${|E(t+s)-E(s)|}\leq\omega{|t|}(E(s)+E_{0}{\|\psi\|}^{2})+\mathcal{O}({|t|}^{2}).$
(3.14)

It is easy to see that every f Energy-limited unitary one-parameter groups are
fully characterized by the following structure theorem:

######  Theorem 3.3 ([vL24, Thm. 3.7]).

Let $H$ be a self-adjoint operator and let $U(t)=e^{-itH}$. Let
$\omega,E_{0}\geq 0$. Then $U(t)$ is energy-limited with stability constants
$\omega,E_{0}$ if and only if

  1. (i)

$\mathcal{H}_{1}$ is invariant under $U(t)$ and the restriction
$U_{0}(t):=U(t)\upharpoonright\mathcal{H}_{1}$ is a strongly continuous one-
parameter group of bounded operators on $\mathcal{H}_{1}$.

  2. (ii)

The inequality $\pm i[G,H]\leq\omega(G+E_{0})$ holds in the sense that

$|\langle G^{1/2}\psi,G^{1/2}H\psi\rangle-\langle
G^{1/2}H\psi,G^{1/2}\psi\rangle|\leq\omega({\|G^{1/2}\psi\|}^{2}+E_{0}{\|\psi\|}^{2})$
(3.15)

for all $\psi\in\mathcal{H}_{1}\cap D(H)$ such that $H\psi\in\mathcal{H}_{1}$.

These necessary and sufficient conditions for energy-limitedness are rather
hard to check in practice. The following result of Fröhlich [F77] (cp. [vL24,
Thm. 3.8]) gives sufficient conditions for energy-limitedness that are much
easier to check. Recall that an operator $H$ is $G$-bounded if and only if $H$
is in $B(\mathcal{H}_{2},\mathcal{H})$.

######  Theorem 3.4 (Fröhlich [F77]).

Let $G_{0}\geq 0$ be a positive self-adjoint operator with
$\inf\operatorname{Spec}G_{0}=0$ and set $G=(G_{0}+E_{0})^{2}-E_{0}^{2}$. Let
$H$ be a $G_{0}$-bounded self-adjoint operator with unitary group
$U(t)=e^{itH}$ and let ${\mathcal{D}}$ be a core for $G_{0}$. Let
$\omega,\nu\geq 0$ and $E_{0}>0$.

  1. (i)

If the operator inequality $\pm i[H,G_{0}]\leq\omega(G_{0}+E_{0})$ holds for
some $\omega,E_{0}>0$ in the sense of quadratic forms on ${\mathcal{D}}$,
i.e.,

${|\langle H\psi,G_{0}\psi\rangle-\langle
G_{0}\psi,H\psi\rangle|}\leq\langle\psi,\omega(G_{0}+E_{0})\psi\rangle,\qquad\psi\in{\mathcal{D}},$
(3.16)

then $U(t)$ is energy-limited with stability constants $\omega,E_{0}$ with
respect to $G_{0}$.

  2. (ii)

If $-[H,G_{0}]^{2}\leq\nu^{2}(G_{0}+E_{0})^{2}$ holds in the sense
that101010The left-hand side of (3.17) is defined as the supremum over
$|\langle H\phi,G_{0}\psi\rangle-\langle G_{0}\phi,H\psi\rangle|$ where $\phi$
ranges over all unit vectors in $\mathcal{H}_{2}$.

${\|[H,G_{0}]\psi\|}\leq\nu{\|(G_{0}+E_{0})\psi\|},\qquad\psi\in\mathcal{H}_{2},$
(3.17)

then $U(t)$ is energy-limited with respect to the reference Hamiltonian $G$
and has stability constants $2\nu,E_{0}^{2}$ with respect to $G$.

###### Proof.

The result follows from Fröhlich’s work [F77] applied to $N=G_{0}+E_{0}$
(Nelson’s assumption $N\geq 1$ can be relaxed to $N\geq c1$ for some $c>0$
without causing any trouble in his proofs). Unfortunately, this result is
stated with several additional assumptions that are not used in the proof (as
he remarks in [F77, Rem. 1]). To obtain item (i), we apply [vL24, Thm. B.1],
which is proved by adapting Fröhlich’s argument, to the operators $N=G+E_{0}$
and $A=\pm iH$ yields item (i) under the assumptions stated above. We now come
to item (ii). By [F77, Lem. A.1 and Rem. 1] (again applied to $N=G+E_{0}$ and
$A=iH$), it holds that ${\|NU(t)\psi\|}^{2}\leq e^{2\nu t}{\|N\psi\|}^{2}$
which proves the claim. ∎

Readers familiar with Nelson’s commutator theorem [RS75, Thm. X.37] may
recognize that the assumption of self-adjointness is redundant: If
${\mathcal{D}}$ is a core for $G$, every symmetric $G$-bounded operator $H$
satisfying (3.16) on ${\mathcal{D}}$ is essentially self-adjoint and satisfies
the same inequality on ${\mathcal{D}}=\mathcal{H}_{2}$.

We now come to the main theorem of this section, which proves convergence
rates for the Trotter product of energy-limited unitary groups in the operator
$E$-norm.

######  Theorem 3.5.

Let $A$ and $B$ be self-adjoint operators generating energy-limited unitary
one-parameter groups and let $\omega,E_{0}$ be joint stability constants. Let
${\mathcal{D}}$ be a core for $G^{1/2}$ such that

$(t,s)\mapsto e^{itA}e^{isB}\psi,\quad\text{and}\quad(t,s)\mapsto
e^{itB}e^{isA}\psi$ (3.18)

are both in $C^{2}({\mathbb{R}}^{2},\mathcal{H})$ for all
$\psi\in{\mathcal{D}}$. If the commutator $[A,B]:{\mathcal{D}}\to\mathcal{H}$
is $G^{1/2}$-bounded, then $A+B$ is essentially self-adjoint on
${\mathcal{D}}$ and

$\|\big{(}e^{itA/n}e^{itB/n}\big{)}^{n}-e^{i\overline{(A+B)}}\|_{E}^{G}\leq\frac{t^{2}}{2n}{\|[A,B]\|}_{f_{2t}(E)}^{G}$
(3.19)

for all $n\in\mathbb{N}$, where $f_{t}(E)=E+(e^{\omega t}-1)(E+E_{0})$.

We remark that the $C^{2}$ assumption in 3.5 ensures that the commutator
$[A,B]$ makes sense as an operator on the core ${\mathcal{D}}$. Furthermore,
as seen in Step 2 in the proof of Theorem 3.5, essential self-adjointness of
$A+B$ on ${\mathcal{D}}$ is a direct consequence of the convergence of the
Trotter product. This is in sharp contrast to the setup in Section 2 in which
the fact that the sum of the generators is again a generator of a strongly
continuous contraction semigroup or even unitary group followed already
directly from the employed relative boundedness assumptions.

Recall that an operator $K$ on a Hilbert space generates a contraction
semigroup if and only if it is maximally dissipative, i.e., $K$ is dissipative
and admits no (proper) dissipative extension [ACE23].

######  Lemma 3.6.

Let $A,B$ be self-adjoint operators generating energy-limited unitary dynamics
with joint stability constants $\omega,E_{0}$. Let $iC$ be a maximally
dissipative operator. Then

$\|(e^{itA/n}e^{itB/n})^{n}-e^{itC}\|_{E}^{G}\leq
n{\|e^{itA/n}e^{itB/n}-e^{itC/n}\|}_{f_{2t-2t/n}(E)}^{G},$ (3.20)

where $f_{t}(E)=e^{\omega t}E+(e^{\omega t}-1)E_{0}$ and $t>0$. If $C$ is
self-adjoint and $e^{itC}$ is energy-preserving, i.e., commutes with $G$ for
all $t\in{\mathbb{R}}$, we have

$\|(e^{itA/n}e^{itB/n})^{n}-e^{itC}\|_{E}^{G}\leq
n{\|e^{itA/n}e^{itB/n}-e^{itC/n}\|}_{E}^{G}.$ (3.21)

###### Proof.

Set $V(t)=e^{itA}e^{itB}$. Then $f_{V(t)}(E)\leq f_{2t}(E)$ and

$\displaystyle{\|V(t/n)^{n}-e^{itC}\|}_{E}^{G}$
$\displaystyle=\bigg{\|}{\sum_{j=1}^{n}e^{i(t(j-1)/n)C}(V(t/n)-e^{itC/n})V(t/n)^{n-j}}\bigg{\|}_{E}^{G}$
$\displaystyle\leq\sum_{j=1}^{n}{\|(V(t/n)-e^{itC/n})V(t/n)^{n-j}\|}_{E}^{G}$
$\displaystyle\leq\sum_{j=1}^{n}{\|V(t/n)-e^{itC/n}\|}_{f_{2t\frac{n-j}{n}}(E)}^{G}$
$\displaystyle\leq n{\|V(t/n)-e^{itC/n}\|}_{f_{2t-2t/n}(E)}^{G},$

where we used the semigroup property of $f_{t}$ and that $f_{t}(E)$ is
monotone in $t$. The second claim follows similarly by sorting the dynamics
generated by $C$ in the first equation to the right instead of the left. ∎

######  Lemma 3.7.

Under the assumptions of 3.5, we have

${\|e^{-itA}Be^{itA}-B\|}_{E}^{G}\leq t{\|[A,B]\|}_{f_{t}(E)}^{G}.$ (3.22)

###### Proof.

Let $\psi,\phi$ be unit vectors in ${\mathcal{D}}$ and let
$\langle\psi,G\psi\rangle\leq E$. Then

$\displaystyle\langle\phi,(e^{-itA}Be^{itA}-B)\psi\rangle$
$\displaystyle=\int_{0}^{t}\frac{d}{ds}\langle
e^{isA}\phi,Be^{isA}\psi\rangle\,ds$ $\displaystyle=\int_{0}^{t}\langle
Ae^{isA}\phi,Be^{isA}\psi\rangle-\langle e^{isA}\phi,BAe^{isA}\psi\rangle\,ds$
$\displaystyle=\int_{0}^{t}\langle e^{isA}\phi,[A,B]e^{isA}\psi\rangle\,ds$

Optimizing over unit vectors $\phi,\psi\in{\mathcal{D}}$ with
$\langle\psi,G\psi\rangle\leq E$, we get

${\|e^{-itA}Be^{itA}-B\|}_{E}^{G}\leq\int_{0}^{t}{\|[A,B]\|}_{f_{s}(E)}^{G}\,ds\leq
t{\|[A,B]\|}_{f_{t}(E)}^{G}.$

∎

We need the following Lemma on dissipative extensions of symmetric operators
(see [ACE23, Thm. 2.5]):

######  Lemma 3.8.

Let $T$ be a symmetric operator on a separable Hilbert space $\mathcal{H}$.
Let $S\supset T$ be an extension such that $iS$ is dissipative, i.e., an
extension such that $\operatorname{Im}\langle\psi,S\psi\rangle\leq 0$,
$\psi\in D(S)$, then it follows that

$T\subset S\subset T^{*}\quad\text{and}\quad T\subset S^{*}\subset T^{*}.$
(3.23)

Using Lemmas 3.6 – 3.8, we can prove 3.5:

###### Proof of 3.5..

As stated in the theorem, we set $f_{t}(E)=e^{\omega t}(E+E_{0})-E_{0}$. Note
that $f_{e^{itA}}(E),\,f_{e^{itB}}(E)\leq f_{{|t|}}(E)$.

_Step 1._ In the first step, we establish the desired bound (3.19) for $t>0$
and a maximally dissipative, i.e., contraction-generating, extension $iC$ of
the skew-symmetric operator $i(A+B)$. It follows that $-iC^{*}$ generates a
contraction semigroup as well, and, by 3.8, that $C^{*}$ is also an extension
of $A+B$. We start with 3.6, which gives

${\|(e^{itA/n}e^{itB/n})^{n}-e^{itC}\|}_{E}^{G}\leq
n{\|e^{itA/n}e^{itB/n}-e^{itC/n}\|}_{f_{2t-2t/n}(E)}^{G}.$ (3.24)

To estimate the right-hand side, we pick unit vectors and
$\phi,\psi\in{\mathcal{D}}$ with $\langle\psi,G\psi\rangle\leq E$. We have

$\displaystyle\langle\phi,(e^{itA}e^{itB}-e^{itC})\psi\rangle$
$\displaystyle=\langle
e^{-itC^{*}}\phi,\psi\rangle-\langle\phi,e^{itA}e^{itB}\psi\rangle$
$\displaystyle=\int_{0}^{t}\frac{d}{ds}\langle
e^{-isC^{*}}\phi,e^{isA}e^{isB}\psi\rangle\,ds$
$\displaystyle=\int_{0}^{t}\Big{(}\langle-
iC^{*}e^{-isC^{*}}\phi,e^{isA}e^{isB}\psi\rangle-\langle
e^{-isC^{*}}\phi,e^{isA}(A+B)e^{-isB}\psi\rangle\Big{)}ds$
$\displaystyle=\int_{0}^{t}\Big{(}\langle-
ie^{-isC^{*}}\phi,Ce^{isA}e^{isB}\psi\rangle-\langle
e^{-isC^{*}}\phi,e^{isA}(A+B)e^{-isB}\psi\rangle\Big{)}ds$
$\displaystyle=i\int_{0}^{t}\langle
e^{-isC^{*}}\phi,((A+B)e^{isA}-e^{isA}(A+B))e^{isB}\psi\rangle\,ds$
$\displaystyle=i\int_{0}^{t}\langle
e^{-isC^{*}}\phi,[B,e^{isA}]e^{isB}\psi\rangle\,ds,$

where we used that $\phi\in D(C^{*})$ and that $C$ equals $A+B$ on
$e^{isA}e^{isB}{\mathcal{D}}\subset D(A+B)$. Using that $e^{-isC^{*}}$ is a
contraction, we get

$|\langle\phi,(e^{itA}e^{itB}-e^{itC})\psi\rangle|\leq\int_{0}^{t}{\|[B,e^{isA}]e^{isB}\psi\|}\,ds.$
(3.25)

Optimizing over $\phi$ and replacing $t$ by $t/n$, gives

$\displaystyle{\|(e^{itA/n}e^{itB/n}-e^{itC/n})\psi\|}\leq\int_{0}^{t/n}{\|[B,e^{isA}]e^{isB}\psi\|}\,ds\leq\int_{0}^{t/n}{\|[B,e^{isA}]\|}_{f_{s}(E)}^{G}ds.$

Since ${\|[B,e^{itA}]\|}_{E}^{G}={\|e^{-itA}Be^{itA}-B\|}_{E}^{G}$, we can
apply 3.7 to obtain

$\displaystyle{\|(e^{itA/n}e^{itB/n}-e^{itC/n})\psi\|}$
$\displaystyle=\int_{0}^{t/n}s{\|[A,B]\|}_{f_{2s}(E)}^{G}\,ds$
$\displaystyle\leq\frac{t^{2}}{2n^{2}}{\|[A,B]\|}_{f_{2t/n}(E)}^{G}.$ (3.26)

Optimizing over $\psi$ and inserting this estimate into (3.24), yields

$\displaystyle{\|(e^{itA/n}e^{itB/n})^{n}-e^{itC}\|}_{E}^{G}\leq\frac{t^{2}}{2n}{\|[A,B]\|}_{f_{2t}(E)}^{G}.$
(3.27)

_Step 2._ Since the commutator is $G^{1/2}$-bounded by assumption, the right-
hand side of (3.27) is finite and goes to zero as $n\to\infty$. This implies
that the Trotter product converges strongly for $t>0$ to the dynamics
generated by a contraction-generating extension $i(A+B)|_{\mathcal{D}}$. Since
this holds for all contraction-generating extensions of
$i(A+B)|_{{\mathcal{D}}}$, it follows that $i(A+B)|_{\mathcal{D}}$ has a
unique contraction-generating extension. This extension generates an isometry
semigroup because strong limits of unitaries are isometries, and, hence, $C$
is a symmetric operator. Thus, one of the defect indices of
$(A+B)|_{\mathcal{D}}$ is zero. If we apply the same arguments (including step
one) with $A$ and $B$ replaced by their negatives, we learn that the same
holds for $-i(A+B)|_{\mathcal{D}}$. Thus, $(A+B)|_{\mathcal{D}}$ is
essentially self-adjoint since both defect indices are zero. We conclude that
(3.27) holds with $C=\overline{(A+B)}$ for $t\in{\mathbb{R}}$. ∎

Using the variational principle (3.8) or the estimate (3.7), we can further
bound the right-hand side of (3.19)

######  Corollary 3.9.

Under the assumptions of 3.5, the convergence rate of the Trotter product is
bounded by

$\|\big{(}e^{itA/n}e^{itB/n}\big{)}^{n}-e^{i\overline{(A+B)}}\|_{E}^{G}\leq\frac{t^{2}}{2n}{\|[A,B]\|}_{E}^{G}\cdot\sqrt{1+(e^{2\omega
t}-1)(1+\tfrac{E_{0}}{E})}$ (3.28)

Furthermore, if $M,E_{1}\geq 0$ are such that $-[A,B]^{2}\leq M(G+E_{0})$
holds in the sense

${\|[A,B]\psi\|}^{2}\leq
M^{2}(\|G^{1/2}\psi\|^{2}+E_{1}\|\psi\|^{2}),\qquad\psi\in{\mathcal{D}},$
(3.29)

then

$\|\big{(}e^{itA/n}e^{itB/n}\big{)}^{n}-e^{i\overline{(A+B)}}\|_{E}^{G}\leq\frac{t^{2}}{2n}M\sqrt{e^{2\omega
t}(E+E_{0})+E_{1}-E_{0}}.$ (3.30)

In particular, when we choose $E_{1}=E_{0}$ the bound (3.30) takes the simple
form

$\|\big{(}e^{itA/n}e^{itB/n}\big{)}^{n}-e^{i\overline{(A+B)}}\|_{E}^{G}\leq\frac{t^{2}}{2n}Me^{\omega
t}\sqrt{E+E_{0}}.$ (3.31)

In cases where the joint dynamics is energy-preserving, i.e., commutes with
the reference Hamiltonian, the exponential behaviour in the time parameter can
be tamed:

######  Corollary 3.10.

Let $A,B$ be self-adjoint operators satisfying the assumptions of 3.5 and let
$\omega,E_{0},f_{t}(E)$ be as in that theorem. Assume that $\overline{A+B}$
has energy-preserving dynamics with respect to $G$. Then

$\|\big{(}e^{itA/n}e^{itB/n}\big{)}^{n}-e^{i\overline{(A+B)}}\|_{E}^{G}\leq\frac{t^{2}}{2n}{\|[A,B]\|}_{f_{2t/n}(E)}^{G}$
(3.32)

In particular, if $M>0$ is such that $-[A,B]^{2}\leq M(G+E_{0})$ in the sense
of (3.29), then

$\|\big{(}e^{itA/n}e^{itB/n}\big{)}^{n}-e^{i\overline{(A+B)}}\|_{E}^{G}\leq\frac{t^{2}}{2n}Me^{\omega
t/n}\sqrt{E+E_{0}}.$ (3.33)

###### Proof.

This follows from 3.6 and (3.26):

$\displaystyle\|\big{(}e^{itA/n}e^{itB/n}\big{)}^{n}-e^{i\overline{(A+B)}}\|_{E}^{G}$
$\displaystyle\leq
n{\|{\big{(}e^{itA/n}e^{itB/n}\big{)}}^{n}-e^{it\overline{(A+B)}/n}\|}_{E}^{G}$
$\displaystyle\leq\frac{t^{2}}{2n}{\|[A,B]\|}_{f_{2t/n}(E)}^{G}.$

∎

In Section 4, we will see numerically that these bounds correctly predict the
actual convergence rates of the infinite-dimensional problem.

###  3.1. Application to Schrödinger operators

In this section, we consider Schrödinger operators $-\Delta+V(x)$ with
potentials $V$ whose second derivatives are bounded. Additionally, we need
polynomial boundedness:

We write $P$ and $Q$ for the momentum and position operators
$P\psi(x)=-i\psi^{\prime}(x)$ and $Q\psi(x)=x\psi(x)$ defined on their natural
domains. Note that the Schrödinger operator $-\Delta+V(x)$ equals
$P^{2}+V(Q)$. Throughout this section, we write $N=P^{2}+Q^{2}$ for the
harmonic oscillator.

######  Lemma 3.11.

Let $V\in C^{4}({\mathbb{R}},{\mathbb{R}})$ with $V$ and its first 4
derivatives being polynomially bounded. Then for Schwartz function
$\psi\in\mathcal{S}(\mathbb{R})$ we have that the maps

$(t,s)\mapsto e^{itP^{2}}e^{isV(Q)}\psi\quad\text{and}\quad(t,s)\mapsto
e^{itV(Q)}e^{isP^{2}}\psi$ (3.34)

are in $C^{2}({\mathbb{R}}^{2},\mathcal{H})$. In particular, the commutator
$[V(Q),P^{2}]$ is well-defined on $\mathcal{S}({\mathbb{R}})$.

###### Proof.

We shall prove the claim for $G(t,s):=e^{itP^{2}}e^{isV(Q)}\psi$, the other
ordering follows from analogous arguments. We first recall that, since
$e^{isV(Q)}\psi\in D(P^{2})=H^{2}(\mathbb{R})$, we can differentiate in $t$ to
find

$\partial_{t}G(t,s)=ie^{itP^{2}}P^{2}(e^{isV(Q)}\psi).$

Since by assumption $P^{2}(e^{isV(Q)}\psi)\in D(P^{2})$ as well, we can apply
the same argument again to find that

$\partial^{2}_{t}G(t,s)=-e^{itP^{2}}P^{4}(e^{isV(Q)}\psi).$

We can explicitly compute the derivative $(iP)^{2}(e^{isV(Q)}\psi)$ to see
that $\partial_{t}^{2}G(t,s)$ is continuous in $(t,s).$ Finally, since

$F(s):=P^{2}(e^{isV(x)}\psi(x))=e^{isV(x)}(2is\psi^{\prime}(x)V^{\prime}(x)+\psi^{\prime\prime}(x)+s\psi(x)(iV^{\prime\prime}(x)-sV^{\prime}(x)^{2}))$

is differentiable in $s$ with

$\begin{split}F^{\prime}(s)&=iV(x)e^{isV(x)}(2is\psi^{\prime}(x)V^{\prime}(x)+\psi^{\prime\prime}(x)+s\psi(x)(iV^{\prime\prime}(x)-sV^{\prime}(x)^{2}))\\\
&\quad+ie^{isV(x)}(2V^{\prime}(x)\psi^{\prime}(x)+\psi(x)(V^{\prime\prime}(x)+2isV^{\prime}(x)^{2}))\end{split}$

by Lebesgue’s differentiation theorem. This implies that $\lim_{h\rightarrow
0}\frac{\|F(s+h)-F(s)-hF^{\prime}(s)\|_{L^{2}}}{h}=0.$ Continuity of
$e^{itP^{2}}$ thus allows us to interchange limits and conclude that

$\partial_{s}\partial_{t}G(t,s)=\lim_{h\rightarrow
0}e^{itP^{2}}\frac{F(h+s)-F(s)}{h}=e^{itP^{2}}F^{\prime}(s)=-e^{itP^{2}}P^{2}(e^{isV(x)}V(x)\psi)(x)$

exists. The continuity of the derivative $\partial_{s}\partial_{t}G(t,s)$ then
just follows from the (strong) continuity of the semigroup $e^{itP^{2}}$ and
the continuity of the expression of $F^{\prime}(s)$ in $s$.

The continuity of the semigroups $(e^{itP^{2}})$, as above, and since $\psi\in
D(V(Q)^{k})$ for $k\geq 0,$ implies that

$\partial_{s}^{k}G(t,s)=e^{itP^{2}}e^{isV(x)}(iV(x))^{k}\psi.$

By strong continuity of the two groups, this also implies that
$(t,s)\mapsto\partial_{s}^{k}G(t,s)$ is continuous.

Finally, since $e^{isV(x)}(iV(x))\psi\in D(P^{2})$, we have

$\partial_{t}\partial_{s}G(t,s)=-e^{itP^{2}}P^{2}e^{isV(x)}V(x)\psi(x)$

and by working out the derivative $P^{2}e^{isV(x)}{V(x)}\psi(x)$ and using the
strong continuity of the semigroups, we find that
$(t,s)\mapsto\partial_{t}\partial_{s}G(t,s)$ is continuous which implies the
claim. ∎

Since we assume $V^{\prime\prime}$ is uniformly bounded, we can bound111111
Let $f$ be Lipschitz continuous (e.g. $V^{\prime}$ above). Then it has a weak
derivative $f^{\prime}\in L^{\infty}$. Let $a=|f(0)|$ and
$M=\|f^{\prime}\|_{L^{\infty}}$ be the optimal Lipschitz constant. Then noting
that $|x|\leq(1+x^{2})/2$ we have  $\displaystyle
f(x)^{2}=(f(x)-f(0)+f(0))^{2}\leq M^{2}x^{2}+2aM|x|+a^{2}\leq
M(M+a)x^{2}+a(M+a)\leq w^{2}(1+x^{2})$ where $w=M+a$.

$|V^{\prime}(x)|\leq(|V^{\prime}(0)|+{\|V^{\prime\prime}\|}_{L^{\infty}})(1+x^{2})^{1/2}$
(3.35)

and, hence, $|V(x)|\lesssim 1+x^{2}$. In particular, this implies that $V(Q)$
is relatively bounded with respect to $N=P^{2}+Q^{2}$.

######  Lemma 3.12.

Let $V$ satisfy the assumption of 3.11. Then the commutator $[V(Q),P^{2}]$ is
relatively bounded with respect to $N$. The energy-constrained operator norm
with respect to the reference Hamiltonian $G=N^{2}-1$ is bounded by

$\|[V(Q),P^{2}]\|_{E}^{G}\leq\nu\sqrt{E+1}$ (3.36)

where $\nu=5\|V^{\prime\prime}\|_{L^{\infty}}+4|V^{\prime}(0)|$.

###### Proof.

First note that

$-i[V(Q),P^{2}]=PV^{\prime}(Q)+V^{\prime}(Q)P=2V^{\prime}(Q)P-iV^{\prime\prime}(Q).$

Since $V^{\prime\prime}\in L^{\infty}$, we only have to show that
$V^{\prime}(Q)P$ is $N$-bounded. Let
$w={\|V^{\prime\prime}\|}_{L^{\infty}}+|V^{\prime}(0)|$ such that
$|V^{\prime}(x)|^{2}\leq w^{2}(1+x^{2})$, cf. (3.35). Then
$V^{\prime}(Q)^{2}\leq w^{2}(1+Q^{2})$ and, hence,

$\|V^{\prime}(Q)P\psi\|^{2}\leq
w^{2}\|(1+Q^{2})^{1/2}P\psi\|^{2}=w^{2}(\|QP\psi\|^{2}+\|P\psi\|^{2})$

for $\psi\in D(N)$. This reduces the problem to showing that $QP$ is
$N$-bounded, which follows from the canonical commutation relations
($[Q,P]=i$):

$\displaystyle 2PQ^{2}P$ $\displaystyle=PQPQ+PQ[Q,P]+QPQP+[P,Q]QP$
$\displaystyle=PQPQ+QPQP+i(PQ-QP)$
$\displaystyle=P^{2}Q^{2}+P[Q,P]Q+Q^{2}P^{2}+Q[P,Q]P+i(PQ-QP)$
$\displaystyle=P^{2}Q^{2}+Q^{2}P^{2}+2i(PQ-QP)$
$\displaystyle\leq(P^{2}+Q^{2})^{2}+2=N^{2}+2\leq 3N^{2},$

where we used that $N\geq 1$. Putting everything together, we get

$\displaystyle\|[V(Q),P^{2}]\psi\|$
$\displaystyle\leq\|V^{\prime\prime}\|_{L^{\infty}}{\|\psi\|}+2\|V^{\prime}(Q)P\psi\|$
$\displaystyle\leq\|V^{\prime\prime}\|_{L^{\infty}}{\|\psi\|}+2w\big{(}{\|QP\psi\|}^{2}+{\|P\psi\|}^{2}\big{)}^{1/2}$
$\displaystyle\leq\|V^{\prime\prime}\|_{L^{\infty}}{\|\psi\|}+2w\big{(}\tfrac{3}{2}{\|N\psi\|}^{2}+{\|N\psi\|}^{2}\big{)}^{1/2}$
$\displaystyle=\|V^{\prime\prime}\|_{L^{\infty}}{\|\psi\|}+\sqrt{10}w\|N\psi\|=\nu{\|N\psi\|}$
$\displaystyle=\nu{\|(G_{0}+1)\psi\|}=\nu{\|(G+1)^{1/2}\psi\|}$ (3.37)

where (with $\sqrt{10}<4$)

$\nu=4w+\|V^{\prime\prime}\|_{L^{\infty}}=4|V^{\prime}(0)|+5\|V^{\prime\prime}\|_{L^{\infty}}.$

Finally, eq. (3.36) follows from eq. (3.8). ∎

######  Lemma 3.13.

Let $V$ satisfy the assumption of 3.11. Take $G=N^{2}-1$ as the reference
Hamiltonian.

  1. (1)

The unitary one-parameter group $e^{itP^{2}}$ is energy-limited with stability
constants $20,1$

  2. (2)

The unitary one-parameter group $e^{itV(Q)}$ is energy-limited with stability
constants $2\nu,1$ where $\nu$ is as in 3.12.

In particular, $\gamma,1$ are joint stability constants with
$\gamma=\max\\{2\nu,20\\}$.

###### Proof.

We prove both items by applying the second part of 3.4 with reference
Hamiltonian $G_{0}=N-1$. We start with the second statement. Clearly both
$V(Q)$ and $P^{2}$ are $N$-bounded. By eq. (3.37), the inequality

$-[V(Q),N]^{2}=-[V(Q),P^{2}]^{2}\leq\nu(G+1)=\nu(G_{0}+1)^{2}$ (3.38)

holds in the sense of quadratic forms. Thus, 3.4 implies that $e^{itV(Q)}$ is
energy-limited with stability constants $2\nu,1$ with respect to
$(G_{0}+1)^{2}-1=G=N^{2}+1$.

The statement for $e^{itP^{2}}$ follows from this. Since
$[P^{2},N]=[P^{2},Q^{2}]$ is of above form with $V(x)=-x^{2}$, we have
$|V^{\prime}(0)|=0$, ${\|V^{\prime\prime}\|}_{L^{\infty}}=2$ and thus, cf.
(3.36),

$-[P^{2},N]^{2}\leq((4+1)\cdot 2)N^{2}=10(G+1).$ (3.39)

Therefore, $P^{2}$ is energy-limited with stability constants $20,1$ with
respect to $G$, and the claim follows. ∎

######  Theorem 3.14.

Let $V\in C^{4}({\mathbb{R}},{\mathbb{R}})$ have bounded second derivative and
assume that $V^{(3)}$ and $V^{(4)}$ are polynomially bounded. Then
$H=P^{2}+V(Q)$ is essentially self-adjoint on $\mathcal{S}({\mathbb{R}})$. Set
$\nu=5{\|V^{\prime\prime}\|}_{L^{\infty}}+4|V^{\prime}(0)|$ and
$\gamma=\max\\{2\nu,20\\}$.

Then

${\|\big{(}e^{-itP^{2}/n}e^{tV/n}\big{)}^{n}\psi-e^{-itH}\psi\|}\leq\frac{t^{2}}{2n}\nu
e^{\gamma t}{\|N\psi\|}$ (3.40)

for all $\psi\in D(N)$.

###### Proof.

We apply 3.9 with $A=P^{2}$, $B=V(Q)$ and
${\mathcal{D}}=\mathcal{S}({\mathbb{R}})$. Energy-limitedness of $e^{itP^{2}}$
and $e^{itV(Q)}$ with the claimed stability constants is proved in 3.13.
$G^{1/2}$-boundedness of the commutator $[A,B]=[V(Q),P^{2}]$ is proved in
3.12, and the $C^{2}$ condition is proved in 3.11. Then, (3.29) holds with
$M=\nu$ and $E_{0}=1$. Therefore, we get (cp. (3.31))

${\|\big{(}e^{-itP^{2}/n}e^{tV/n}\big{)}^{n}-e^{-itH}\|}_{E}^{G}\leq\frac{t^{2}}{2n}\nu
e^{\gamma t}\sqrt{E+1}.$ (3.41)

If $\psi\in D(N)$ is a unit vector, the left-hand side of (3.40) is bounded by
the operator $E$-norm
${\|\big{(}e^{-itP^{2}/n}e^{tV/n}\big{)}^{n}-e^{-itH}\|}_{E}^{G}$ with
$E=\mathbf{E}(\psi)={\|N\psi\|}^{2}-1$. Thus, (3.40) follows from (3.41). ∎

###  3.2. Explicit example: Trotter splitting for the Harmonic oscillator

Let us specialize to the Harmonic oscillator. In this case, the joint dynamics
is energy-preserving relative to the reference Hamiltonian $G=N^{2}-1$. Making
use of 3.10, we get:

######  Theorem 3.15.

Let $\psi\in D(N)$. Then

${\|\big{(}e^{itP^{2}/n}e^{tQ^{2}/n}\big{)}^{n}\psi-e^{itN}\psi\|}\leq
6\,\frac{t^{2}{\|N\psi\|}}{n}$ (3.42)

for all $n\geq 55t$.

We will see numerically in Section 4 that this bound correctly predicts the
actual convergence rates.

###### Proof.

By 3.13, the dynamics of $P^{2}$ and $Q^{2}$ have joint stability constants
$20,1$. 3.10 gives

${\|\big{(}e^{itP^{2}/n}e^{itQ^{2}/n}\big{)}^{n}-e^{itN}\|}_{E}^{G}\leq\frac{t^{2}}{2n}{\|[P^{2},Q^{2}]\|}^{N^{2}-1}_{e^{20t/n}(E+1)-1}.$

By 3.12, we have

${\|[P^{2},Q^{2}]\|}_{E}^{N^{2}-1}\leq 10\sqrt{E+1}.$

Taken together, these imply

${\|\big{(}e^{itP^{2}/n}e^{itQ^{2}/n}\big{)}^{n}-e^{itN}\|}_{E}^{G}\leq\frac{t^{2}}{2n}10\,e^{10t/n}\sqrt{E+1}$

Since $10t/n\leq\log(6/5)$ guarantees $5e^{10t/n}\leq 6$ and since and
$10/\log(6/5)\leq 55$ the claim follows. ∎

###  3.3. Application to magnetic Dirac operators

We consider the free Dirac operator in 2 and 3 dimensions, which takes the
abstract form, for $D_{x_{i}}:=\frac{1}{i}\frac{\partial}{\partial x_{i}},$

$H_{\text{free}}=\begin{pmatrix}mc^{2}&cD^{*}\\\
cD&-mc^{2}\end{pmatrix}\in\mathbb{C}^{2(n-1)\times 2(n-1)},$

with Pauli matrices $\sigma_{i}$

$D=\begin{cases}\sum_{i=1}^{3}\sigma_{i}D_{x_{i}}&\text{ if }n=3,\\\
D_{x_{1}}+iD_{x_{2}}&\text{ if }n=2,\end{cases}\text{ with
}D\in\mathbb{C}^{(n-1)\times(n-1)},$

see [Th92] for a general introduction to Dirac operators. One then has:

######  Lemma 3.16.

Let $V\in C^{2}({\mathbb{R}}^{n},{\mathbb{C}}^{2(n-1)\times 2(n-1)})$
hermitian for $n\in\\{2,3\\}$ with $V$ and its first $2$ derivatives being
polynomially bounded. If
$\psi\in\mathcal{S}(\mathbb{R}^{n};{\mathbb{C}}^{2(n-1)}),$ then the maps

$(t,s)\mapsto
e^{itH_{\operatorname{free}}}e^{isV(Q)}\psi\quad\text{and}\quad(t,s)\mapsto
e^{itV(Q)}e^{isH_{\operatorname{free}}}\psi$ (3.43)

are in $C^{2}({\mathbb{R}}^{2},\mathcal{H})$, where
$\mathcal{H}=L^{2}(\mathbb{R}^{n};\mathbb{C}^{2(n-1)\times 2(n-1)}).$

We omit its proof as it is very similar to Lemma 3.11. It follows from [Th92,
Thm. 4.3] that the Dirac operator $H_{\operatorname{free}}+V$ with $V$ as in
Lemma 3.16 is essentially self-adjoint on
$\mathcal{S}(\mathbb{R}^{n};{\mathbb{C}}^{2(n-1)}).$

Magnetic Dirac operators.

Unlike potential perturbations that we considered for Schrödinger operators,
we may also consider quantum systems incorporating a magnetic field.

For Dirac operators, magnetic fields are incorporated as follows: In three
dimensions, the magnetic vector potential $A\in
C^{\infty}({\mathbb{R}}^{3};{\mathbb{R}}^{3})$ gives a magnetic field
$\mathbf{B}=\operatorname{curl}(A),$ whereas in two dimensions, the magnetic
field strength just reduces to a scalar field
$B=\partial_{x_{1}}A_{2}-\partial_{x_{2}}A_{1}$.

The magnetic Dirac operator then takes the abstract form

$H=\begin{pmatrix}mc^{2}&cD^{*}\\\ cD&-mc^{2}\end{pmatrix},$

with

$D=\begin{cases}\sum_{i=1}^{3}\sigma_{i}(D_{x_{i}}-A_{i})&\text{ if }n=3,\\\
(D_{x_{1}}-A_{1})+i(D_{x_{2}}-A_{2})&\text{ if }n=2\end{cases}$

and $D_{x_{1}}=-i\partial_{x_{1}}.$ The Pauli operator is then, for
$\mathbf{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$, defined by

$H_{P}=D^{*}D=(-i\nabla-A)^{2}-\begin{cases}\mathbf{\sigma}\cdot\mathbf{B}&\text{
if }n=3,\\\ B&\text{ if }n=2.\end{cases}$ (3.44)

Homogeneous magnetic fields. For illustration purposes, we shall now consider
a homogeneous magnetic field in two dimensions, i.e., $\mathbf{B}=(0,0,B_{0})$
and $B_{0}>0.$ We can then choose the unbounded(!) magnetic potential
$A(x)=\frac{B_{0}}{2}(-x_{2},x_{1},0)$ that yields the constant magnetic field
and observe that

$[D,D^{*}]=2B_{0}.$ (3.45)

From this, we can deduce that

$\operatorname{Spec}(H_{P})=2\mathbb{N}_{0}B_{0}$

such that $G:=H_{P}$ is a positive self-adjoint operator with
$\inf\operatorname{Spec}G=0.$ We also notice that

$D=2D_{\bar{z}}-\frac{iB_{0}}{2}z\text{ with }z=x_{1}+ix_{2}$

where $\partial_{\bar{z}}=\partial_{x_{1}}+i\partial_{x_{2}}$ with
$D_{\bar{z}}=-i\partial_{\bar{z}}$ and
$\partial_{z}=\partial_{x_{1}}-i\partial_{x_{2}}$ with $D_{z}=-i\partial_{z}$
such that

$\begin{split}H_{P}&=\left(D_{x_{1}}-\frac{B_{0}x_{2}}{2}\right)^{2}+\left(D_{x_{2}}+\frac{B_{0}x_{1}}{2}\right)^{2}-B_{0}\\\
&=-\Delta+B_{0}(\bar{z}\partial_{\bar{z}}-z\partial_{z})+\frac{B_{0}^{2}}{4}|z|^{2}-2B_{0}.\end{split}$

We therefore consider a Trotter-splitting of the Hamiltonian with

$K=\begin{pmatrix}0&2D_{z}\\\ 2D_{\bar{z}}&0\end{pmatrix}\text{ and
}P=\begin{pmatrix}0&\frac{iB_{0}}{2}\bar{z}\\\
-\frac{iB_{0}}{2}z&0\end{pmatrix}.$ (3.46)

######  Theorem 3.17.

For kinetic energy $K$ and magnetic potential $P$ as in (3.46) and
$G=(-\Delta+|z|^{2}-2)^{2}$ as in Eq. 3.44, the Trotter splitting for the
magnetic Dirac operator satisfies for $\omega:=2\varepsilon\max\\{1,B_{0}\\}$,
$E_{0}:=\frac{4}{\varepsilon\omega}(2+\tfrac{1}{\varepsilon^{2}})\max\left\\{B_{0},1\right\\}$
with $\varepsilon>0$ arbitrary

${\|\big{(}e^{-itK/n}e^{-itP/n}\big{)}^{n}-e^{-it(K+P)}\|}_{E}^{G}\leq\frac{t^{2}B_{0}(\sqrt{6}+1)}{2n}\sqrt{e^{2\omega
t}(E+E_{0})+2-E_{0}}.$ (3.47)

In particular, for $\varepsilon=1/t$, we obtain a global-in time bound of the
form

${\|\big{(}e^{-itK/n}e^{-itP/n}\big{)}^{n}-e^{-it(K+P)}\|}_{E}^{G}=\mathcal{O}_{B_{0}}(t^{2}(1+t)(1+\sqrt{E})/n)$
(3.48)

###### Proof.

We apply Eq. 3.31 with $A=K$ and $B=P$. The $C^{2}$ condition follows from
3.16. We write $N=-\Delta+|z|^{2}$ and $G_{0}=N-2$ such that $G_{0}^{2}=G.$

It then suffices, by 3.4, to compute and bound the relevant commutators: We
compute

$\displaystyle[K,N]$ $\displaystyle=\begin{pmatrix}0&-2i\bar{z}\\\
-2iz&0\end{pmatrix}$ and $\displaystyle[P,N]$
$\displaystyle=2B_{0}\begin{pmatrix}0&-D_{\bar{z}}\\\ D_{z}&0\end{pmatrix},$

so in particular $|[K,N]|^{2},B_{0}^{-2}|[P,N]|^{2}\leq 4N=4(G_{0}+2)$.
Moreover, we recall the identity

$[K,G]=[K,N]G_{0}+G_{0}[K,N].$

By this, the Cauchy-Schwarz inequality, Young’s inequality for products121212
It holds for $x,y\in{\mathbb{R}}$, $\lambda>0$ and $p,q>1$ with $1/p+1/q=1$
that  $|xy|\leq\tfrac{\lambda}{p}|x|^{p}+\tfrac{\lambda^{1-q}}{q}|y|^{q}$
which we apply here for $p=q=2$ and once for $\lambda=\varepsilon$, resp.
$\lambda=B_{0}\varepsilon$, and once for $\lambda=\delta$.  and by above
operator inequalities for the commutators we have for $\varepsilon>0$
arbitrary and $\delta=\varepsilon^{2}/2$ that

$\begin{split}|\langle\psi,[K,G]\psi\rangle|&\leq 2|\langle
G_{0}\psi,[K,N]\psi\rangle|\\\
&\leq\varepsilon{\|G_{0}\psi\|}^{2}+\tfrac{1}{\varepsilon}{\|[K,N]\psi\|}^{2}\\\
&\leq\varepsilon{\|G_{0}\psi\|}^{2}+\tfrac{4}{\varepsilon}\langle(G_{0}+2)\psi,\psi\rangle\\\
&=\varepsilon{\|G_{0}\psi\|}^{2}+\tfrac{4}{\varepsilon}\langle
G_{0}\psi,\psi\rangle+\frac{8}{\varepsilon}\|\psi\|^{2}\\\
&\leq\varepsilon{\|G_{0}\psi\|}^{2}+\tfrac{2\delta}{\varepsilon}\|G_{0}\psi\|^{2}+\frac{2}{\varepsilon\delta}\|\psi\|^{2}+\frac{8}{\varepsilon}\|\psi\|^{2}\\\
&=2\varepsilon{\|G_{0}\psi\|}^{2}+\frac{4}{\varepsilon^{3}}\|\psi\|^{2}+\frac{8}{\varepsilon}\|\psi\|^{2}\\\
&=2\varepsilon{\|G_{0}\psi\|}^{2}+4\left(\frac{1}{\varepsilon^{3}}+\frac{2}{\varepsilon}\right)\|\psi\|^{2}\end{split}$

and similarly

$\begin{split}|\langle\psi,[P,G]\psi\rangle|&\leq\varepsilon
B_{0}{\|G_{0}\psi\|}^{2}+\tfrac{1}{\varepsilon B_{0}}{\|[P,N]\psi\|}^{2}\\\
&\leq\varepsilon
B_{0}{\|G_{0}\psi\|}^{2}+\tfrac{4B_{0}}{\varepsilon}\langle(G_{0}+2)\psi,\psi\rangle\\\
&\leq\varepsilon B_{0}{\|G_{0}\psi\|}^{2}+\tfrac{2\delta
B_{0}}{\varepsilon}\|G_{0}\psi\|^{2}+\frac{2B_{0}}{\varepsilon\delta}\|\psi\|^{2}+\frac{8B_{0}}{\varepsilon}\|\psi\|^{2}\\\
&=2\varepsilon
B_{0}\|G_{0}\psi\|^{2}+4B_{0}\left(\frac{1}{\varepsilon^{3}}+\frac{2}{\varepsilon}\right)\|\psi\|^{2}.\end{split}$

Thus, we find that $e^{itP}$ and $e^{itK}$ are energy-limited with stability
constants $\omega:=2\varepsilon\max\\{1,B_{0}\\}$,
$E_{0}:=\frac{4}{\varepsilon\omega}(2+\tfrac{1}{\varepsilon^{2}})\max\left\\{B_{0},1\right\\}$
with respect to $G.$ We also compute

$\begin{split}[K,P]&=iB_{0}\operatorname{diag}(-(D_{z}z+\bar{z}D_{\bar{z}}),D_{\bar{z}}\bar{z}+zD_{z})\\\
&=B_{0}\operatorname{diag}(-1-(z\partial_{z}+\bar{z}\partial_{\bar{z}}),1+(\bar{z}\partial_{\bar{z}}+z\partial_{z}))\\\
&=B_{0}\operatorname{diag}(-1-(x_{1}\partial_{x_{1}}+x_{2}\partial_{x_{2}}),1+x_{1}\partial_{x_{1}}+x_{2}\partial_{x_{2}}).\end{split}.$

From (3.37) we conclude that

$\begin{split}\|[K,P]\psi\|&\leq
B_{0}(\|\psi\|+\|x_{1}\partial_{x_{1}}\psi\|+\|x_{2}\partial_{x_{2}}\psi\|)\\\
&\leq B_{0}(\|\psi\|+\sqrt{6}\|N\psi\|)\\\ &\leq
B_{0}(\sqrt{6}+1)\|(G+2)^{1/2}\psi\|.\end{split}$ (3.49)

This shows that in the notation of (3.29)

$\|[K,P]\psi\|^{2}\leq
B_{0}^{2}(\sqrt{6}+1)^{2}(\|G^{1/2}\psi\|^{2}+2\|\psi\|^{2}),$

we have $M=B_{0}(\sqrt{6}+1)$ and $E_{1}=2$. Thus, the result follows from
(3.30). ∎

######  Remark 3.18.

To close the estimate, it seemed necessary to consider again the harmonic
oscillator as a reference Hamiltonian in the previous theorem. We were
initially considering the Pauli operator too but were unable to bound $[K,P]$
in (3.49) with respect to this reference operator.

##  4\. Numerical examples

In this section, we compare our convergence rates on suitably chosen vectors
to the true convergence rates of the Trotter splitting in large finite-
dimensional approximations of the full infinite-dimensional Hilbert space. We
verified numerically that the dimension of the approximation is sufficiently
large in the sense that the Trotter error remains constant when increasing it
further. We interpret this as a heuristic confirmation that our numerics
capture the convergence rate of the full infinite-dimensional Trotter
splitting. A more comprehensive discussion of this latter point can be found
in [B+23b].

###  4.1. Singular potentials

Here, we numerically evaluate the scaling of the Trotter error for the case
$A=i\Delta$ on $X=L^{2}([0,1])$ and $L=iV$ for some potential $V(x)$. Our
focus lies on bounded but non-smooth potentials in which case we expect a slow
decay of the Trotter error from our results Theorem 2.7 and Corollary 2.21 as
the condition $VD((-\Delta))\subseteq D((-\Delta)^{\beta})$ can only hold true
for small values of $\beta.$ As shown for the considered examples in Figures 1
and 2, this expectation is met supporting the corresponding assumption in
Theorem 2.7 and Corollary 2.21.

Figure 1.  Left: Trotter error for the ground state of respective Schrödinger
operator (numerically computed with finite matrix truncation in
$\mathbb{C}^{2M+1}$ of size $M=400$) after time $t=1$. On the right, we see
the potential $V$ with its Fourier approximations.

Figure 2.  Trotter error for ground state of respective Schrödinger operator
(numerically computed with finite matrix truncation in $\mathbb{C}^{2M+1}$ of
size $M=400$) after time $t=1$ (very singular potential on the left, more
regular one on the right). $N$ indicates the number of iterations in the
Trotter product.

In particular, we consider the square wave potential over the interval
$V(x)=2(H(2x)-H(2x-1))-1$ where $H$ is the Heaviside function
$H(x)=\operatorname{1\hskip-2.75ptl}_{[0,\infty)}(x).$ When periodically
continued, this function has a Fourier series representation

$V(x)=\frac{4}{\pi}\sum_{n\in 2\mathbb{N}_{0}+1}\frac{e^{2\pi inx}-e^{-2\pi
inx}}{2in}.$

This implies that with respect to the Fourier basis $V$ can be written as

$V=\frac{2}{\pi}\sum_{n\in 2\mathbb{N}_{0}+1}\frac{J^{n}-(J^{*})^{n}}{in},$

where $J$ is the right shift. The Fourier series of the Laplacian with respect
to the basis $(e^{2\pi inx})_{n\in{\mathbb{Z}}}$ is

$-\Delta=\operatorname{diag}(4\pi^{2}n^{2})_{n\in{\mathbb{Z}}}.$

The Trotter error $\|(e^{i\Delta/n}e^{-iV/n})^{n}\psi-e^{-i(-\Delta+V)}\psi\|$
for $V$ above and $V=\sin(2\pi x)$ on the interval $[0,1]$ with periodic
boundary conditions is shown in Figure 1. Since we use matrix approximations
for our numerics, the Trotter error always scales as $\mathcal{O}(n^{-1}),$
but the error decays much slower for the singular potential.

We can generalize this by defining the family of potentials

$V_{\alpha}(x)=\frac{4}{\pi}\sum_{n\in 2\mathbb{N}_{0}+1}\frac{e^{2\pi
inx}-e^{-2\pi inx}}{2in^{\alpha}}.$

This way $V_{\alpha}\in L^{2}(\mathbb{R}/\mathbb{Z})$ once $\alpha>1/2.$ Since
a slow decay of Fourier coefficients is directly linked to low regularity of
$V_{\alpha}$, the parameter $\alpha$ directly tunes the regularity. We see in
Figure 2 that this directly influences the Trotter error.

###  4.2. Confining potentials

In Figures 3 and 4, we consider the one-dimensional quantum harmonic
oscillator $H=P^{2}+Q^{2}$ with the Trotter splitting into $P^{2}$ and
$Q^{2}$, which we implement using the usual creation and annihilation operator
representation in the Fock basis, where

Figure 3.  Trotter error for eigenstates (Fock states) $\\{1,4,9,14,21,29\\}$
of quantum harmonic oscillator (numerically computed with finite matrix
truncation in $\mathbb{C}^{400}$) after time $t=1$. (log-log plot (left) and
regular plot (right)).

Figure 4.  On the left, we see the Trotter error at $t=1$ for all first 81
quantum harmonic oscillator eigenstates (Fock states) after $N=1000$ Trotter
iterations (linear scaling) with matrix truncation in $\mathbb{C}^{400}$. On
the right, we see the time dependence of the Trotter error for eigenstates
(Fock states) $\\{1,4,9,14,21,29\\}$.

$a=\begin{pmatrix}0&\sqrt{1}&0&0&\dots&0&\dots\\\
0&0&\sqrt{2}&0&\dots&0&\dots\\\ 0&0&0&\sqrt{3}&\dots&0&\dots\\\
0&0&0&0&\ddots&\vdots&\dots\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\sqrt{n}&\dots\\\ 0&0&0&0&\dots&0&\ddots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}$

with $Q=\frac{a+a^{*}}{\sqrt{2}}$ and $P=\frac{a-a^{*}}{\sqrt{2}i}$. The
$\mathcal{O}(n^{-1})$ convergence rate, the linear error in the eigenvalues of
the Fock states, and the quadratic time dependence for are correctly predicted
by our estimate in 3.15.

##  Appendix A Proofs of 2.2 and 2.17

In this section, we prove 2.2 and 2.17.

###### Proof of Theorem 2.2.

First of all, note that by (2.8), since the $A$-bound of $L$ is strictly
smaller 1, and $(L,D(L))$ is dissipative [EN00, Proposition II.3.23], we have
that $(A+L,D(A))$ generates a strongly continuous contraction semigroup [EN00,
Thm. III.2.7]. Hence, we can apply Lemma 2.1 which gives for $x\in D(A^{2})$

$\displaystyle\left\|\left(\left(e^{tL/n}e^{tA/n}\right)^{n}-e^{t(A+L)}\right)x\right\|\leq
t\sup_{s,\tau\in[0,t]}\left\|[L,e^{sA/n}]x_{\tau}\right\|,$ (A.1)

where we denoted $x_{\tau}=e^{\tau(A+L)}x\in D(A^{2})$ and used that since
$D(A^{2})=D((A+L)^{2})$ [P18, Lem. IV.3], the operator $e^{\tau(A+L)}$ leaves
$D(A^{2})$ invariant. Then, using again that $L$ is relatively $A$-bounded, we
have

$\displaystyle\left\|[L,e^{sA/n}]x_{\tau}\right\|$
$\displaystyle=\left\|[L,e^{sA/n}-I]x_{\tau}\right\|$
$\displaystyle\leq\left\|L(e^{sA/n}-I)x_{\tau}\right\|+\left\|(e^{sA/n}-I)Lx_{\tau}\right\|$
$\displaystyle\leq
a\left\|A(e^{sA/n}-I)x_{\tau}\right\|+b\left\|(e^{sA/n}-I)x_{\tau}\right\|+\left\|(e^{sA/n}-I)Lx_{\tau}\right\|.$

Furthermore, since for $y\in D(A)$ we have

$\displaystyle\left\|(e^{sA/n}-I)y\right\|=\frac{s}{n}\left\|\int_{0}^{1}e^{rsA/n}Ay\,dr\right\|\leq\frac{s}{n}\left\|Ay\right\|,$

we find, as $A$ and $e^{sA}-I$ commute, that

$\displaystyle\left\|[L,e^{sA/n}]x_{\tau}\right\|$
$\displaystyle\leq\frac{s}{n}\left(a\left\|A^{2}x_{\tau}\right\|+b\left\|Ax_{\tau}\right\|+\left\|ALx_{\tau}\right\|\right),$
(A.2)

where we have used that by assumption $Ax_{\tau},Lx_{\tau}\in D(A).$ For the
term in the middle, we use

$\begin{split}\|Ax_{\tau}\|&\leq\|(A+L)x_{\tau}\|+\|Lx_{\tau}\|\\\
&\leq\|(A+L)x\|+\|Lx_{\tau}\|\\\
&\leq\|(A+L)x\|+b\|x\|+a\|Ax_{\tau}\|.\end{split}$

Using now $a<1$, we can bring $a\|A\psi_{\tau}\|$ to the other side and get a
bound of the form

$\|Ax_{\tau}\|\leq\frac{1}{1-a}((1+a)\|Ax\|+2b\|x\|).$

By the same argument, we get

$\displaystyle{\|A(A+L)x_{\tau}\|}$ $\displaystyle={\|A[(A+L)x]_{\tau}\|}$
(A.3) $\displaystyle\leq\frac{1}{1-a}((1+a){\|A(A+L)x\|}+2b\|x\|).$ (A.4)

Since $A^{2}=A(A+L)-AL$ on $D(A^{2})$ and by assumption of $AL$ being
relatively $A^{2}$-bounded we find

$\displaystyle{\|A^{2}x_{\tau}\|}$
$\displaystyle\leq{\|A(A+L)x_{\tau}\|}+{\|ALx_{\tau}\|}$
$\displaystyle\leq\frac{1}{1-a}((1+a){\|A(A+L)x\|}+2b\|x\|)+{\|ALx_{\tau}\|}$
$\displaystyle\leq\frac{1}{1-a}((1+a+a^{\prime}){\|A^{2}x\|}+(2b+b^{\prime})\|x\|)+a^{\prime}{\|A^{2}x_{\tau}\|}+b^{\prime}\|x\|.$
(A.5)

Using $a^{\prime}<1$ we get

${\|A^{2}x_{\tau}\|}\leq\frac{(1+a+a^{\prime}){\|A^{2}x\|}+(2b+(2-a)b^{\prime})\|x\|}{(1-a)(1-a^{\prime})}.$
(A.6)

Thus, using relative $A^{2}$-boundedness of $AL$ again in (A.2), we have

$\displaystyle{\|[L,e^{sA/n}]x_{\tau}\|}\leq\frac{s}{n}\left((a+a^{\prime})\left\|A^{2}x_{\tau}\right\|+b\left\|Ax_{\tau}\right\|+b^{\prime}\|x\|\right)$
$\displaystyle\leq\frac{s}{n}\left((a+a^{\prime})\frac{(1+a+a^{\prime}){\|A^{2}x\|}+2b+(2-a)b^{\prime}}{(1-a)(1-a^{\prime})}+\frac{b}{1-a}((1+a)\|Ax\|+2b\|x\|)+b^{\prime}\|x\|\right)$
$\displaystyle=\frac{s}{n}\left(c_{2}{\|A^{2}x\|}+c_{1}\|Ax\|+c_{0}\|x\|\right).$

where

$\displaystyle c_{0}$
$\displaystyle=\frac{(a+a^{\prime})[2b+(2-a)b^{\prime}]}{(1-a)(1-a^{\prime})}+\frac{2b^{2}}{1-a}+b^{\prime},\quad
c_{1}=\frac{1+a}{1-a}b$ $\displaystyle c_{2}$
$\displaystyle=\frac{(a+a^{\prime})(1+a+a^{\prime})}{(1-a)(1-a^{\prime})}.$

Plugging this into (A.1) finishes the proof of (2.10).

∎

###### Proof of Prop. 2.17.

From the elementary inequality
$|z|^{\alpha}=(|\operatorname{Re}(z)|+|\operatorname{Im}(z)|)^{\alpha}\leq|\operatorname{Re}(z)|^{\alpha}+|\operatorname{Im}(z)|^{\alpha},$
we conclude upon integrating against the spectral measure that by using Lemma
2.15

$(F_{\alpha}(\operatorname{Re}(N))=D(|\operatorname{Re}(N)|^{\alpha}))\cap(F_{\alpha}(\operatorname{Im}(N))=D(|\operatorname{Im}(N)|^{\alpha}))\subset
D(|N|^{\alpha}).$

Similarly,

$|\operatorname{Re}(z)|^{\alpha}\leq|z|^{\alpha}\text{ and
}|\operatorname{Im}(z)|^{\alpha}\leq|z|^{\alpha},$

implies that $D(|N|^{\alpha})\subset D(|\operatorname{Re}(N)|^{\alpha})$ and
$D(|N|^{\alpha})\subset D(|\operatorname{Im}|^{\alpha})$ which shows that

$D(|N|^{\alpha})=(F_{\alpha}(\operatorname{Re}(N))=D(|\operatorname{Re}(N)|^{\alpha}))\cap(F_{\alpha}(\operatorname{Im}(N))=D(|\operatorname{Im}(N)|^{\alpha})).$

We conclude that if $x\in D(|N|^{\alpha})$, then

$\begin{split}\|(T_{t}-I)x\|&\leq\|e^{it\operatorname{Im}(N)}(e^{t\operatorname{Re}(N)}-I)x\|+\|(e^{it\operatorname{Im}(N)}-I)x\|\\\
&=\|(e^{t\operatorname{Re}(N)}-I)x\|+\|(e^{it\operatorname{Im}(N)}-I)x\|=\mathcal{O}(t^{\alpha}).\end{split}$

Thus, $x\in F_{\alpha}(N).$ On the other hand,
$F_{\alpha}(N)=F_{\alpha}(N^{*})$ by normality and
$e^{t\operatorname{Re}(N)}=e^{tN/2}e^{tN^{*}/2}.$ Thus, we conclude, since
$\|e^{tN/2}\|\leq 1,$ that

$\begin{split}\|(e^{t\operatorname{Re}(N)}-I)x\|&\leq\|e^{tN/2}(e^{tN^{*}/2}-I)x\|+\|(e^{tN/2}-I)x\|\\\
&\leq\|(e^{tN^{*}/2}-I)x\|+\|(e^{tN/2}-I)x\|=\mathcal{O}(t^{\alpha}),\end{split}$

which shows that $x\in F_{\alpha}(\operatorname{Re}(N)).$ All the previous
inclusions follow very directly, yet the inclusion $F_{\alpha}(N)\subset
F_{\alpha}(\operatorname{Im}(N))$ seems less obvious. Thus, for the final
inclusion, we proceed as in Lemma 2.15. Writing $t=t_{1}+it_{2}$ for
$t_{i}\in\mathbb{R}$ we find

$\begin{split}\|\lambda^{r}N(\lambda-
iN)^{-1}x\|^{2}&=\lambda^{2r}\int_{\\{\operatorname{Re}(t)\leq
0\\}}\frac{t_{1}^{2}+t_{2}^{2}}{(\lambda-t_{1})^{2}+t_{2}^{2}}d\langle
E_{N}(t)x,x\rangle.\end{split}$ (A.7)

Optimizing over $\lambda$ we find

$\lambda_{\text{opt}}=\frac{t_{1}(1-2r)+\sqrt{t_{1}^{2}+4t_{2}^{2}(1-r)r}}{2(1-r)},$

such that we find the elementary upper bound

$\lambda_{\text{opt}}\leq\frac{-2rt_{1}+2|t_{2}|\sqrt{(1-r)r}}{2(1-r)}\leq\max\\{\tfrac{r}{1-r},\sqrt{\tfrac{r}{1-r}}\\}\\}|t|.$

By establishing a similar lower bound

$\lambda_{\text{opt}}\geq\sqrt{2}(r-r^{2})|t|,$

we find that $\lambda_{\text{opt}}^{2r}$ in front of the integral in (A.7)
behaves like $|t|^{2r}.$

Similar arguments show that the integrand

$\frac{|t|^{2}}{(\lambda_{\text{opt}}-t_{1})^{2}+t_{2}^{2}}=\frac{4(1-r)^{2}|t|^{2}}{4(1-r)^{2}t_{2}^{2}+(|t_{1}|+\sqrt{t_{1}^{2}+4t_{2}^{2}(1-r)r})^{2}}$

is uniformly bounded from above and below, i.e., there are $K_{r},k_{r}>0$
such that for all admissible $t\in(-\infty,0)\times i\mathbb{R}$

$k_{r}\leq\frac{|t|^{2}}{(\lambda_{\text{opt}}-t_{1})^{2}+t_{2}^{2}}\leq
K_{r}.$

Thus, arguing as in the proof of the previous Lemma, we conclude that

$F_{\alpha}(N)=D(|N|^{\alpha}).$

∎

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