# Comments on the cosmic convergence
of nonexpansive maps

Armando W. Gutiérrez  INRIA Saclay and CMAP Ecole PolytechniqueCNRS, 91128,
Palaiseau, France<EMAIL_ADDRESS><EMAIL_ADDRESS>Anders Karlsson  Section de mathématiques,
Université de Genève2-4 Rue du Lièvre, Case Postale 64, 1211 Genève 4, Suisse
<EMAIL_ADDRESS>Matematiska institutionen, Uppsala universitetBox
256, 751 05 Uppsala, Sweden<EMAIL_ADDRESS>
(Date: August 10, 2021)

###### Abstract.

This note discusses some aspects of the asymptotic behaviour of nonexpansive
maps. Using metric functionals, we make a connection to the invariant subspace
problem and prove a new result for nonexpansive maps of $\ell^{1}$. We also
point out some inaccurate assertions appearing in the literature on this
topic.

###### Key words and phrases:

Metric fixed point theory, nonexpansive maps, invariant subspace problem,
metric functionals

###### 1991 Mathematics Subject Classification:

Primary 47H09; Secondary 47A15

##  1\. Introduction

This brief note was inspired by recent papers by Bauschke, Douglas, and Moursi
[2], and Ryu [26]. Let $(X,d)$ be a metric space and let $T$ be a nonexpansive
map of $X$ into itself, that is

$d(Tx,Ty)\leq d(x,y)$

for all $x,y\in X$. The question concerns the asymptotic behaviour of the
iterates $T^{n}x$ as $n\rightarrow\infty$. The first important case of this is
well-known: the contraction mapping principle. In this paper we are mostly
interested in the complementary case; when no fixed point exists. The case
when $X$ is a Banach space has been especially studied, see [17, 20, 1, 13,
22, 23, 18, 16, 2, 26] and references therein. It should be pointed out that a
significant special case was considered in the 1930s: the mean ergodic theorem
of von Neumann and Carleman, especially in the generality of F. Riesz [25,
Theorem 1]. Here $X$ is a Hilbert space and

$Tx=Ux+v$

where $U$ is a linear operator of norm at most one and $v\in X$. Then the
iterates $T^{n}0$ converge in the following sense

$\lim_{n\to\infty}\frac{1}{n}T^{n}0=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}U^{k}v=\pi_{U}(v)$

where $\pi_{U}$ is the projection onto the subspace of $U$-invariant vectors.

It is stated in [2, Fact 1.1] and also in [26] that for fixed-point free
nonexpansive maps of Hilbert spaces the orbit must diverge in the sense that
$\left\|T^{n}x\right\|\rightarrow\infty$ as $n\rightarrow\infty$. This may
however fail in infinite-dimensional Hilbert spaces as an isometry discovered
by Edelstein in the 1960s demonstrates, recalled in [9, p. 1453] and the nice
exposition [27, p. 6]. That is, in spaces which are not locally compact it may
happen that the orbit $(T^{n}x)_{n\geq 0}$ neither stays bounded nor leaves
every bounded set. On the other hand, this phenomenon cannot happen in spaces
where closed bounded sets are compact as it was shown by Calka in the 1980s (a
simple proof is given in [6, Lemma 2.6]).

The quantity

$\tau=\lim_{n\rightarrow\infty}\frac{1}{n}\left\|T^{n}x\right\|$ (1.1)

always exists by a well-known subadditive argument and it is independent of
$x$. Convergence in direction, _cosmic convergence_ in the terminology of [2,
26], in the case $\tau>0$ for general Banach spaces was treated in [20, 13]
among other papers. A very general result in this direction is found in [7]
which considers random products of nonexpansive maps of any metric space. In
the case $\tau=0$ one may wonder, as the authors in [2] do, if cosmic
convergence still takes place, that is, whether

$\frac{T^{n}x}{\left\|T^{n}x\right\|}$

converges as $n\rightarrow\infty$. Strong and weak limits of this expression
are called strong and weak cosmic limits respectively. This question is also
raised in [10, Problem 4.6]. The paper [2] proves several interesting cases
when this is true, and suggests a conjecture that it might always be true for
finite dimensional Hilbert spaces. A counterexample was given in [26, Section
3]. In 2005, Enrico Le Donne showed the second author another such an example
(unpublished). In other normed spaces, a counterexample was considered by
Kohlberg and Neyman in [13, p. 272], also discussed in [6, p. 1936].

It still makes sense to wonder about what the limit set can be when it is not
just one point. The theorem [11, Theorem 11] shows that it must be contained
in certain kinds of sectors. This can be compared with Corollary 5.3 to
Theorem 5.1 in [26]. These are some of the most general results presently
available on this type of convergence known to us. When $X$ is hyperbolic
there are many results starting from Wolff-Denjoy showing that cosmic
convergence holds, for example [9], see also the discussion in [2]. Let us
point out the similar question and conjecture in the case $X$ is a convex set
equipped with Hilbert’s metric, called the Karlsson-Nussbaum conjecture, see
[14] for one of the most recent significant contributions.

In Section 3 we will prove the following connection of cosmic convergence to
the problem of the existence of non-trivial closed invariant subspaces:

######  Theorem.

Suppose that $T$ is a nonexpansive map of a real Hilbert space into itself of
the form $Tx=Ux+v$ for some vector $v$ and linear operator $U$ of norm at most
one. If $0\notin\overline{Im(I-T)},$ then there is a non-zero continuous
linear functional $f$ such that

$f(Tx)\geq f(x)$

for all $x$. This inequality implies that $\ker(f)$ is a non-trivial closed
invariant subspace for $U$.

In Section 4 we will prove:

######  Theorem.

Let $T$ be a nonexpansive map of $\ell^{1}(\mathbb{Z})$ into itself. Then
there is a non-trivial metric functional $h$ so that

$h(Tx)\leq h(x)$

for every $x\in\ell^{1}(\mathbb{Z})$, and the orbit $(T^{n}0)_{n\geq 0}$ is
contained in a half-space. More precisely, there is a non-trivial continuous
linear functional $f$ so that

$f(T^{n}0)\geq 0$

for all $n\geq 0$.

Metric functionals, which constitute a main tool in this note will be recalled
below. We provide some further statements and examples, such as Proposition
2.3 about firmly nonexpansive maps. We end by giving a simple example of a
nonexpansive map whose unbounded orbits converge to a metric functional but
without having any weak cosmic limit point. This shows once more that metric
functionals are useful when linear notions fail to describe a phenomenon.

##  2\. Metric functionals

Let $(X,d)$ be any metric space, for example a Banach space. Fix $x_{0}\in X$,
in the case of a Banach space we take $x_{0}=0$. We consider the following map

$\Phi:X\rightarrow\mathbb{R}^{X}$

via

$x\mapsto h_{x}(\cdot):=d(\cdot,x)-d(x_{0},x).$

The map $\Phi$ is injective (if $h_{x}=h_{y}$, then in particular
$h_{x}(x)=h_{y}(x)$ and $h_{x}(y)=h_{y}(y)$ which together imply that
$d(x,y)=0$). As the notation indicates, we endow the target space
$\mathbb{R}^{X}$ with the topology of pointwise convergence. The map $\Phi$ is
clearly continuous. The closure $\overline{\Phi(X)}$ is compact and consists
of a subset of nonexpansive maps $h:X\to\mathbb{R}$ with $h(x_{0})=0$. Every
element of $\overline{\Phi(X)}$ is called a _metric functional_ on $X$. See
[5, 7, 8, 12] for discussions on metric functionals and the related notion of
horofunctions.

Below we will use two cases:

######  Proposition 2.1.

[12, 8] Let $H$ be a real Hilbert space with scalar product $(\cdot,\cdot)$.
Every metric functional on $H$ has precisely one of the following forms:

  1. 1.

$h(x)=\left\|x\right\|$;

  2. 2.

$h(x)=\sqrt{\left\|x\right\|^{2}-2(x,rv)+r^{2}}-r$, with $0<r<\infty$ and
$v\in H$, $\left\|v\right\|<1$;

  3. 3.

$h(x)=\left\|x-rv\right\|-r$, with $0<r<\infty$ and $v\in H$,
$\left\|v\right\|=1$;

  4. 4.

$h(x)=-(x,v)$, with $v\in H$, $\left\|v\right\|\leq 1$.

Note that only the metric functionals of the form _4._ are unbounded from
below (except $h\equiv 0$).

The first author determined all the metric functionals on
$\ell^{1}(\mathbb{Z})$:

######  Proposition 2.2.

[8] The following functions are precisely the metric functionals on
$\ell^{1}(\mathbb{Z})$:

$h(x)=\sum_{s\in I}\epsilon_{s}x_{s}+\sum_{s\in\mathbb{Z}\setminus
I}\left|x_{s}-z_{s}\right|-\left|z_{s}\right|$

where $I\subseteq\mathbb{Z}$, $\epsilon_{s}\in\left\\{-1,+1\right\\}$ for all
$s\in I$ and $z_{s}$ are arbitrary real numbers for all
$s\in\mathbb{Z}\setminus I$.

Another fact we will use below (see [7, Lemma 3.1]) is that for any metric
functional $h$ on a Banach space there is always a continuous linear
functional $f$ of norm at most 1 such that $f\leq h.$ As a first illustration
of this tool let us point out the following. Browder and Bruck introduced
_firmly nonexpansive maps_ of Banach spaces. Those are maps $T:V\rightarrow V$
of a Banach space $V$(or a convex subsets thereof) satisfying

$\left\|Tx-Ty\right\|\leq\left\|(1-t)(Tx-Ty)+t(x-y)\right\|$

for every $x,y\in V$ and $t\geq 0$. Reich and Shafrir proved in [24] that for
such maps it holds that

$\lim_{n\rightarrow\infty}\left\|T^{n+1}x-T^{n}x\right\|=\tau,$

where $\tau$ is defined above in (1.1). This is a property used in the proof
of [26, Theorem 5.1]. The following reproves the main theorem in [5, Section
3] in a very special case, but with the additional information that our metric
functional $h$ is a limit point of the orbit $(T^{n}0)_{n\geq 0}$ in the
compact space $\overline{\Phi(V)}$.

######  Proposition 2.3.

Let $T$ be a firmly nonexpansive map of a Banach space $V$ into itself with
$\tau=0$. Then there is a metric functional $h$ on $V$ which is a limit point
of $(T^{n}0)_{n\geq 0}$ in the compact space $\overline{\Phi(V)}$ and such
that

$h(Tx)\leq h(x)$

for all $x\in V.$

###### Proof.

Take a subsequence $n_{i}$ so that $h_{T^{n_{i}}0}\rightarrow h$ (in the case
that $\overline{\Phi(V)}$ is not sequentially compact we can use a similar
argument as in [7, p. 1907]). Note that for every $x\in V$,

$\displaystyle h(Tx)$
$\displaystyle=\lim_{i\rightarrow\infty}\left\|Tx-T^{n_{i}}0\right\|-\left\|T^{n_{i}}0\right\|$
$\displaystyle\leq\liminf_{i\rightarrow\infty}\left\|x-T^{n_{i}-1}0\right\|-\left\|T^{n_{i}}0\right\|$
$\displaystyle\leq\liminf_{i\rightarrow\infty}\left\|x-T^{n_{i}}0\right\|+\left\|T^{n_{i}}0-T^{n_{i}-1}0\right\|-\left\|T^{n_{i}}0\right\|$
$\displaystyle=h(x)+\tau.$

If $\tau=0$, then the proposition is proved. ∎

Note that we do not need $T$ to be defined on the whole vector space, just as
long as we can iterate the map. Firmly nonexpansive maps are of interest in
the study of certain non-linear operators as pioneered by Browder and they
often arise in optimization problems; see, for example, [21, 4, 3].

Ryu constructed an interesting firmly nonexpansive map $T$ of
$\ell^{2}(\mathbb{N})$ into itself [26, p. 11] such that
$T^{n}0/\left\|T^{n}0\right\|$ converges weakly, but not strongly, to $0$. We
note that the iterates $T^{n}0$ converge to the metric functional $h=0$.
Furthermore we note that Ryu’s map also provides a nonexpansive map of
$\ell^{1}(\mathbb{N})$ into itself, and that in this case the iterates
$T^{n}0$ converge to a non-trivial metric functional (which in fact is
linear). Indeed, the result in [26, Lemma 4.1] implies that for fixed $i$, the
$i$th coefficient of the $k$th iterate looks like $(T^{k}0)_{i}=\log k+O(1)$
as $k\rightarrow\infty$. Now if we consider $x\in\ell^{1}$ for which only a
finite number of coefficients are non-zero, say $F$ is the support of $x$,
then

$\left\|x-T^{k}0\right\|-\left\|T^{k}0\right\|=\sum_{i\in
F}\left|x_{i}-(T^{k}0)_{i}\right|-\left|(T^{k}0)_{i}\right|$

which converges to $-\sum_{i\in F}x_{i}$ as $k\rightarrow\infty$. Since such
finitely supported points are dense in $\ell^{1}$ we have that the iterates
$T^{k}0$ converge to the metric functional, indeed linear, $h(x)=-\sum_{i\geq
0}x_{i}$.

##  3\. Invariant subspaces

One of the oldest and best-known open problems in operator theory is the
invariant subspace problem. It asks whether every bounded linear operator of a
complex Hilbert space $H$ of dimension at least two must have a non-trivial
invariant closed linear subspace. For general Banach spaces the first
counterexample was found by Enflo and other examples by Read [19]. We will
here make a connection to the topic of the present paper.

First we point out that in Remark 3.1 of [13] it is stated that one can choose
a linear functional $f$ of norm $1$ such that $f(T^{n}x-x)\geq n\tau.$ But it
seems to us that this can only be guaranteed when $\tau>0$, otherwise the weak
limit of linear functionals in the proof may be $f\equiv 0.$ Compare also with
[5, p. 355].

Given any bounded linear operator $U:H\rightarrow H$, by rescaling we can
assume that its norm is at most one since this does not alter the invariant
subspaces. We will consider associated affine nonexpansive maps, that is,
$Tx=Ux+v$ for some vector $v\in H$. We will be interested in statements of the
form that there exists a linear functional $f$ such that

$f(Tx)\geq f(x)$

for all $x\in H.$

######  Theorem 3.1.

Suppose that $T$ is a nonexpansive map of a real Hilbert space into itself of
the form $Tx=Ux+v$ for some vector $v$ and linear operator $U$ of norm at most
one. If $0\notin\overline{Im(I-T)},$ then there is a non-zero continuous
linear functional $f$ such that

$f(Tx)\geq f(x)$

for all $x$. This inequality implies that $\ker(f)$ is a non-trivial closed
invariant subspace for $U$.

###### Proof.

It is known since Pazy’s work that the condition imposed implies that the
corresponding escape rate $\tau$ defined above in (1.1) is strictly positive.
This also follows from the main theorem in Gaubert-Vigeral [5] that moreover
asserts that there is a metric functional such that

$h(Tx)\leq h(x)-\tau$

for every $x$ in the Hilbert space. Since $\tau>0$, it follows from the above
inequality iterated that $h(T^{n}x)\rightarrow-\infty$ as
$n\rightarrow\infty$. This shows, in view of the identification of the metric
functionals in a Hilbert space recalled in Proposition 2.1, that $h$ must be
of the type $h(x)=-(x,q)$ for some non-zero vector $q$ which must have norm
$1$. Hence $f(x):=-h(x)$ is the linear functional as required.

Notice now that this means that $-(Ux+v,q)\leq-(x,q)$ for every $x$. In
particular, for any $x\in\ker f$, we have $(Ux,q)+(v,q)\geq 0.$ Applying the
same inequality with $tx$ with scalars $t\in\mathfrak{\mathbb{R}}$ instead of
$x,$ using linearity of the scalar product, it follows that the only
possibility is that $(Ux,q)=0,$ in other words $Ux\in\ker f$. Hence the kernel
of $f$, which clearly is a non-trivial and closed linear subspace is invariant
under $U$. ∎

There is a hope, related to cosmic convergence, that even when
$0\in\overline{Im(I-T)}\setminus Im(I-T),$ there is sometimes such a linear
functional $f.$ (The case $0\in Im(I-T)$ means that $v=x-Ux$ for a certain
$x$). It is stated in [26, Theorem 5.1] that for a non-zero cosmic weak limit
point $q$, it holds that $(Tx-x,q)\geq 0$ for all $x$ in the Hilbert space,
which is precisely what is needed in view of Theorem 3.1. Note however that in
the proof of [26, Theorem 5.1] it is assumed that
$\left\|T^{k_{j}+1}x-T^{k_{j}}x\right\|\rightarrow 0$, but for an isometry
this norm is constant and positive unless $x$ is a fixed point. As far as we
can see the proof of [26, Theorem 5.1] needs this assumption.

######  Remark 3.2.

The condition in Theorem 3.1 means that $1$ is in the compression spectrum of
the linear operator $U$. By the definition of this subset of the spectrum, the
set $\overline{Im(I-U)}$ is a non-trivial closed subspace which is clearly
invariant under $U$. In contrast, Theorem 3.1 provides a co-dimension $1$
closed invariant subspace for $U$.

##  4\. Nonexpansive maps of $\ell^{1}$

In view of Theorem 3.1 above, it is interesting to recall that for any
nonexpansive map $T$ of a Banach space, Gaubert and Vigeral in their paper [5]
(strenghtening [9] in the case of star-shaped hemi-metrics) provide a _metric_
functional $h$ so that

$h(Tx)\leq h(x)$

for every $x$. For a Hilbert space, in case $\tau=0$ there seems to be no way
in general of guaranteeing that $h$ is not the function identically $0.$ On
the other hand, Gutiérrez [8] identified explicitly all the metric functionals
of the Banach spaces $\ell^{1}$ recalled in Proposition 2.2, and none is
identically $0$. So this gives finer information than [13] in this case. It is
pointed out in [7, Lemma 3.1] that there is always a continuous linear
functional $f$ of norm at most 1 such that $f\leq h.$ Nevertheless, the
discrepancy between linear and metric functionals has as consequence that the
proof of Theorem 3.1 does not lead to an affirmative solution to the invariant
subspace problem in the case of $\ell^{1}$. Indeed, there is a celebrated
counterexample to the invariant subspace problem for $\ell^{1}$, constructed
by Read [19].

The following can be said about nonexpansive maps of $\ell^{1}$:

######  Theorem 4.1.

Let $T$ be a nonexpansive map of $\ell^{1}(\mathbb{Z})$ into itself. Then
there is a non-trivial metric functional $h$ so that

$h(Tx)\leq h(x)$

for every $x\in\ell^{1}(\mathbb{Z})$, and the orbit $(T^{n}0)_{n\geq 0}$ is
contained in a half-space. More precisely, there is a non-trivial continuous
linear functional $f$ so that

$f(T^{n}0)\geq 0$

for all $n\geq 0$.

###### Proof.

The first statement is the Gaubert-Vigeral Theorem [5, Theorem 1] with the
addition of the explicit determination of such metric functionals in [8], see
Proposition 2.2 above. Each of these metric functionals takes on negative
values somewhere, except for $h(x)=\left\|x\right\|$. In this latter
exceptional case, note that $h(Tx)\leq h(x)$ means that
$\left\|Tx\right\|\leq\left\|x\right\|$ which applied to $x=0$ gives $T(0)=0.$
In this case any linear functional $f$ would satisfy our claim since
$f(T^{n}0)=f(0)=0$. If $h$ does take negative values then take the linear
functional $g$ obtained in [7, Lemma 3.1] with $g\leq h$. This linear
functional must be non-trivial since it is forced to take on strictly negative
values at some points. Iterating the main inequality we have

$h(T^{n}0)\leq h(T^{n-1}0)\leq...\leq h(0)=0.$

Let finally $f(x)=-g(x)$, of course also not the identically 0 linear
functional, and with the property $f(T^{n}0)\geq-h(T^{n}0)\geq 0.$

∎

Let us remark that Edelstein’s example alluded to already above extends to
$\ell^{1}(\mathbb{Z})$. This isometry has unbounded orbits but they
nevertheless return infinitely often to a fixed bounded set. Let us finish by
another example:

######  Example.

Consider the nonexpansive map
$T:\ell^{1}(\mathbb{N})\rightarrow\ell^{1}(\mathbb{N})$ defined by
$T(x_{1},x_{2},...)=(1,x_{1},x_{2},...)$. This map clearly has no fixed points
in $\ell^{1}(\mathbb{N})$. Indeed the orbits tend to infinity, more precisely,
$T^{n}0=(1,1,1,...,1,0,0,...)$ and $h_{T^{n}0}$ converges to

$h(x)=\sum_{s=1}^{\infty}\left|x_{s}-1\right|-1$

as $n\rightarrow\infty$. This $h$ is the metric functional that is obtained in
[9, 12] (also the one in [5] as can be verified), and clearly
$h(T^{n}0)\rightarrow-\infty$. On the other hand, there are no weak cosmic
limit points.

### Acknowledgment

The first author acknowledges financial support from the Vilho, Yrjö and Kalle
Väisälä Foundation of the Finnish Academy of Science and Letters. The second
author acknowledges partial financial support from the Swiss NSF grant
200020_15958.

## References

  * [1] J. B. Baillon, R. E. Bruck and S. Reich. On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Houston J. Math. 4 (1978), 1–9.
  * [2] H. H. Bauschke, G. R. Douglas and W. M. Moursi. On a result of Pazy concerning the asymptotic behaviour of nonexpansive mappings. J. Fixed Point Theory Appl. 18 (2016), no. 2, 297–307.
  * [3] H. H. Bauschke, E. Matoušková and S. Reich. Projection and proximal point methods: convergence results and counterexamples. Nonlinear Anal. 56 (2004), no. 5, 715–738.
  * [4] R. E. Bruck and S. Reich. Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J. Math. 3 (1977), no. 4, 459–470.
  * [5] S. Gaubert and G. Vigeral. A maximin characterisation of the escape rate of non-expansive mappings in metrically convex spaces. Math. Proc. Cambridge Philos. Soc. 152 (2012), no. 2, 341–363.
  * [6] S. Gouëzel. Subadditive cocycles and horofunctions. Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. III. Invited lectures, 1933–1947, World Sci. Publ., Hackensack, NJ, 2018.
  * [7] S. Gouëzel and A. Karlsson. Subadditive and multiplicative ergodic theorems. J. Eur. Math. Soc. (JEMS) 22 (2020), no. 6, 1893–1915.
  * [8] A. W. Gutiérrez. On the metric compactification of infinite-dimensional $\ell_{p}$ spaces. Canad. Math. Bull. 62 (2019), no. 3, 491–507.
  * [9] A. Karlsson. Non-expanding maps and Busemann functions. Ergodic Theory Dynam. Systems 21 (2001), no. 5, 1447–1457.
  * [10] A. Karlsson. Nonexpanding maps, Busemann functions, and multiplicative ergodic theory. Rigidity in dynamics and geometry (Cambridge, 2000), 283–294, Springer, Berlin, 2002.
  * [11] A. Karlsson. On the dynamics of isometries. Geom. Topol. 9 (2005), 2359–2394.
  * [12] A. Karlsson. Elements of a metric spectral theory., https://arxiv.org/
abs/1904.01398

  * [13] E. Kohlberg and A. Neyman. Asymptotic behavior of nonexpansive mappings in normed linear spaces. Israel J. Math. 38 (1981), no. 4, 269–275.
  * [14] B. Lemmens, B. Lins, R. Nussbaum and M. Wortel. Denjoy-Wolff theorems for Hilbert’s and Thompson’s metric spaces. J. Anal. Math. 134 (2018), no. 2, 671–718.
  * [15] B. Lins. Asymptotic behavior of nonexpansive mappings in finite dimensional normed spaces. Proc. Amer. Math. Soc. 137 (2009), no. 7, 2387–2392.
  * [16] E. Matoušková and S. Reich. Reflexivity and approximate fixed points. Stud. Math. 159 (2003), no. 3, 403–415.
  * [17] A. Pazy. Asymptotic behavior of contractions in Hilbert space. Israel J. Math. 9 (1971), 235–240.
  * [18] A. T. Plant and S. Reich. The asymptotics of nonexpansive iterations. J. Funct. Anal. 54 (1983), no. 3, 308–319.
  * [19] C. J. Read. A solution to the invariant subspace problem on the space $l^{1}$. Bull. London Math. Soc. 17 (1985), no. 4, 305–317.
  * [20] S. Reich. Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 44 (1973), 57–70.
  * [21] S. Reich. Extension problems for accretive sets in Banach spaces. J. Functional Analysis. 26 (1977), no. 4, 378–395.
  * [22] S. Reich. On the asymptotic behavior of nonlinear semigroups and the range of accretive operators. J. Math. Anal. Appl. 79 (1981), 113–126.
  * [23] S. Reich. On the asymptotic behavior of nonlinear semigroups and the range of accretive operators. II. J. Math. Anal. Appl. 87 (1982), 134–146.
  * [24] S. Reich and I. Shafrir. The asymptotic behavior of firmly nonexpansive mappings. Proc. Amer. Math. Soc. 101 (1987), no. 2, 246–250.
  * [25] F. Riesz. Some Mean Ergodic Theorems. J. London Math. Soc. 13 (1938), no. 4, 274–278.
  * [26] E. K. Ryu. Cosmic divergence, weak cosmic convergence, and fixed points at infinity. J. Fixed Point Theory Appl. 20 (2018), no. 3, Paper No. 109
  * [27] A. Valette. Affine isometric actions on Hilbert spaces and amenability. Lectures at ESI, Vienna in 2007, https://www.unine.ch/files/live/sites/
math/files/shared/documents/articles/ESIValette.pdf


# A uniform framework for quantum diffraction at a planar pattern, including
double-slit diffraction, via Schrödinger equations with generalized potentials

Günther Hörmann  ,  Ljubica Oparnica  and  Christian Spreitzer  Fakultät für
Mathematik
Universität Wien, Austria<EMAIL_ADDRESS>Faculty of Education
University of Novi Sad<EMAIL_ADDRESS><EMAIL_ADDRESS>Fakultät für Mathematik
Universität Wien
& Pädagogische Hochschule Niederösterreich, Austria
<EMAIL_ADDRESS>
(Date: August 30, 2024)

###### Abstract.

We discuss a direct Schrödinger equation set-up for the diffraction of a
quantum particle at (almost) planar patterns with slit-type configurations.
Physically meaningful initial values and potentials are modeled via
regularizations and the solutions can be interpreted as generalized functions
in this approach. We discuss fundamental spectral and scattering theoretical
properties of the regularizing solution families and provide also some
comparison with the more direct approximations and simplifications used in
physics.

###### Key words and phrases:

Generalized functions, regularization methods, Schrödinger equation

###### 2020 Mathematics Subject Classification:

Primary: 81Q05; Secondary: 46F30

##  1\. Introduction

Discussions of double-slit experiments are very often included in the
introductory parts of quantum physics text books. They perfectly serve as an
illustration of the fundamental so-called quantum logic and are still a
prominent subject of research, not only as a _Gedankenexperiment_ but also in
contemporary experimental physics (see, e.g., [Fein:19] and [Brand:21]).
Somewhat surprisingly however, it seems difficult to find any discussion of
these experiments in the context of quantum mechanics with a Schrödinger
equation set-up, although it is the standard foundation of text book models.
The desire to have such a theory in early quantum physics is nicely
illustrated by the following quote from Weinberg’s text book ([Weinberg:15,
pages 14-15]):“There is a story that in his oral thesis examination, de
Broglie was asked what other evidence might be found for a wave theory of the
electron, and he suggested that perhaps diffraction phenomena might be
observed in the scattering of electrons by crystals. …. What was needed was
some way of extending the wave idea from free particles, described by waves …,
to particles moving in a potential, …”.

The standard physics argument justifying the occurrence of the interference
pattern observed in a double-slit experiment is, in fact, a pure classical
approximation based on point sources for waves at the narrow slits. The two
discussions of the double-slit in [Manoukian:89] and [Beau:12] are based on
the Feynman path technique, which means that mathematically speaking they make
use of the typical distribution theoretic fundamental solution to the free
Schrödinger equation and produce a somewhat refined justification of the usual
approximation via classical wave theory and diffraction. A very recent
numerical and qualitative analysis based on the Bohmian point of view with
particle trajectories is presented in [DDS:22], which employs a type of
modeling of the initial value and of the potential very similar to our
approach in Subsections 2.1 and 2.2. In fact, as noted already in
[Hoermann:17, Remark 2.4], the regularization methods from theories of
generalized functions would apply also in studies of the Bohmian flow related
to double-slit diffraction and related problems. However, there seems to be a
certain lack of standard quantum mechanical and analytical treatments of the
double-slit configuration, at least with a plain Schrödinger equation model.

A main criticism we have is that the double slit problem is often treated as a
“boundary value” problem with a free wave function “approaching the double-
slit”. However, the Schrödinger equation does not have the property of finite
propagation speed like a wave equation—or any strictly hyperbolic equation,
more generally speaking. Thus on a fundamental level it is not justified a
priori why an approximation with classical wave propagation starting from the
slits works so well in explaining the measured diffraction pattern on a screen
at some distance from the slits. Moreover, at least in principle, one would
expect or at least cannot rule out that something is also reflected from the
blocking objects and therefore a stationary state would already be a
superposition of what came in from one side of the slits and what was
reflected.

A further difficulty is that modeling the potential for a realistic double-
slit configuration in a Schrödinger operator is mathematically delicate and
even distributional potentials, e.g. producing Dirac-type “sources for passing
waves” at the slits, seem not truly appropriate due to its idealizations from
the outset. Instead one might rather strive for an implementation via
regularizations that can more accurately capture the nature of an essentially
unsurmountable high barrier away from the slits that is at the same time
infinitesimally narrow in the transversal direction, thus almost located in a
plane perpendicular to the “main propagation direction” as seen from the
source. Thus we will attempt to accurately model such potentials by
corresponding generalized functions which can be conveniently represented via
families of regularizations. These do not have to be convergent in
distributions, but instead obey certain asymptotic estimates.

Well-posedness of Cauchy problems for Schrödinger equations allowing for
discontinuous or distributional coefficients, initial data, and right-hand
sides was shown in [Hoermann:11]. Previously, several Colombeau-generalized
solutions to special types of linear and nonlinear Schrödinger equations have
been constructed in [Bu:96, Sto:06b, Sto:06a]. The special case of Schrödinger
operators with $\delta$-potential could be treated in a non-standard analysis
setting (see [Albeverio:88]), but also with quadratic forms in terms of a
Friedrichs extension (see [Thirring:02, Example 2.5.19]). Further applications
with a mixed setting involving distribution theory, Hilbert space techniques
or measures and invariant means can be found in the context of seismic wave
propagation (see [dHHO:08]) or generalizations of the usual quantum mechanical
initial values (see [Hoermann:17]).

Recall that in quantum mechanics one is often interested in allowing for the
zero-order term $V$ in the Schrödinger equation
$\partial_{t}u=i\Delta_{x}u+iVu$ to model non-smooth potentials, as, e.g.,
already with Coulomb-type potentials. In the classical $L^{2}$ theory we have
initial data $u\\!\mid_{t=0}=u_{0}$ and $|u_{0}|^{2}$ gives an initial
probability density while $|u(.,t)|^{2}$ represents the result of the
evolution of this probability density at time $t>0$.

In the following subsection we briefly review concepts from [Hoermann:11] in
the regularization approach to generalized functions in the sense of
Colombeau. We also describe the main results on unique existence of solutions
to the Schrödinger equation Cauchy problem stated in (1-2) and their
compatibility with classical and distributional solution concepts.

Section 2 then discusses the details of the mathematical regularization
modeling for the potential and the initial values in Subsections 2.1 and 2.2,
while Subsections 2.3 and 2.4 establish the key spectral properties of these
regularizations and ensure the applicability of basic methods from scattering
theory. Section 3 makes an effort to connect the regularization and
generalized function set-up more directly with various approximations or
calculational simplifications employed in physics texts on such problem.

###  1.1. Regularizations, generalized function solutions, and coherence
properties

In this section, we review the main results of [Hoermann:11]. Before going
into details, we recall a few basics from the theory of Colombeau generalized
functions.

The fundamental idea of Colombeau-type regularization methods is to model non-
smooth objects by approximating nets of smooth functions, convergent or not,
but with _moderate_ asymptotics and to identify regularizing nets whose
differences compared to the moderateness scale are _negligible_. For a modern
introduction to Colombeau algebras we refer to [GKOS:01]. Here we will also
make use of constructions and notations from [Garetto:05b], where generalized
functions based on a locally convex topological vector space $E$ are defined:
Let $E$ be a locally convex topological vector space whose topology is given
by the family of seminorms $\\{p_{j}\\}_{j\in J}$. The elements of

$\mathcal{M}_{E}:=\\{(u_{\varepsilon})_{\varepsilon}\in E^{(0,1]}:\,\forall
j\in J\,\,\exists N\in\mathbb{N}\quad
p_{j}(u_{\varepsilon})=O(\varepsilon^{-N})\,\text{as}\,\varepsilon\to 0\\}$

and

$\mathcal{N}_{E}:=\\{(u_{\varepsilon})_{\varepsilon}\in E^{(0,1]}:\,\forall
j\in J\,\,\forall q\in\mathbb{N}\quad
p_{j}(u_{\varepsilon})=O(\varepsilon^{q})\,\text{as}\,\varepsilon\to 0\\},$

are called _$E$ -moderate_ and _$E$ -negligible_, respectively. With
operations defined componentwise, e.g.,
$(u_{\varepsilon})+(v_{\varepsilon}):=(u_{\varepsilon}+v_{\varepsilon})$ etc.,
$\mathcal{N}_{E}$ becomes a vector subspace of $\mathcal{M}_{E}$. We define
the _generalized functions based on $E$_ as the factor space
$\mathcal{G}_{E}:=\mathcal{M}_{E}/\mathcal{N}_{E}$. If $E$ is a differential
algebra then $\mathcal{N}_{E}$ is an ideal in $\mathcal{M}_{E}$ and
$\mathcal{G}_{E}$ is a differential algebra as well.

Particular choices of $E$ reproduce the standard Colombeau algebras of
generalized functions. For example, $E=\mathbb{C}$ with the absolute value as
norm yields the generalized complex numbers
$\mathcal{G}_{E}=\widetilde{\mathbb{C}}$; for $\Omega\subseteq\mathbb{R}^{d}$
open, $E=\mathcal{C}^{\infty}(\Omega)$ with the topology of compact uniform
convergence of all derivatives provides the so-called special Colombeau
algebra $\mathcal{G}_{E}=\mathcal{G}(\Omega)$. Recall that
$\Omega\mapsto\mathcal{G}(\Omega)$ is a fine sheaf, thus, in particular, the
restriction $u|_{B}$ of $u\in\mathcal{G}(\Omega)$ to an arbitrary open subset
$B\subseteq\Omega$ is well-defined and yields $u|_{B}\in\mathcal{G}(B)$.
Moreover, we may embed $\mathcal{D}^{\prime}(\Omega)$ into
$\mathcal{G}(\Omega)$ by appropriate localization and convolution
regularization.

If $E\subseteq\mathcal{D}^{\prime}(\Omega)$, then certain generalized
functions can be projected into the space of distributions by taking weak
limits: We say that $u\in\mathcal{G}_{E}$ is _associated_ with
$w\in\mathcal{D}^{\prime}(\Omega)$, if $u_{\varepsilon}\to w$ in
$\mathcal{D}^{\prime}(\Omega)$ as $\varepsilon\to 0$ holds for any (hence
every) representative $(u_{\varepsilon})$ of $u$. This fact is also denoted by
$u\approx w$.

Consider open strips of the form
$\Omega_{T}=\mathbb{R}^{n}\times\,]0,T[\subseteq\mathbb{R}^{n+1}$ (with $T>0$
arbitrary) and the spaces
$E=H^{\infty}({\Omega_{T}})=\\{h\in\mathcal{C}^{\infty}(\Omega_{T}):\partial^{\alpha}h\in
L^{2}(\Omega_{T})\;\forall\alpha\in\mathbb{N}^{n+1}\\}$ with the family of
(semi-)norms

${\left\|h\right\|}_{H^{k}}=\Big{(}\sum_{|\alpha|\leq
k}{\left\|\partial^{\alpha}h\right\|}_{L^{2}}^{2}\Big{)}^{1/2}\quad(k\in\mathbb{N}),$

as well as
$E=W^{\infty,\infty}({\Omega_{T}})=\\{h\in\mathcal{C}^{\infty}(\Omega_{T}):\partial^{\alpha}h\in
L^{\infty}(\Omega_{T})\;\forall\alpha\in\mathbb{N}^{n+1}\\}$ with the family
of (semi-)norms

${\left\|h\right\|}_{W^{k,\infty}}=\max_{|\alpha|\leq
k}{\left\|\partial^{\alpha}h\right\|}_{L^{\infty}}\quad(k\in\mathbb{N}).$

Clearly, $\Omega_{T}$ satisfies the strong local Lipschitz property [Adams:75,
Chapter IV, 4.6, p. 66], hence every element of $H^{\infty}(\Omega_{T})$ and
$W^{\infty,\infty}(\Omega_{T})$ belongs to
$\mathcal{C}^{\infty}(\overline{\Omega_{T}})$ by the Sobolev embedding theorem
[Adams:75, Chapter V, Theorem 5.4, Part II, p. 98].

In the sequel, we will employ the following notation

$\mathcal{G}_{L^{2}}(\mathbb{R}^{n}\times[0,T]):=\mathcal{G}_{H^{\infty}({\Omega_{T}})}\quad\text{
and
}\quad\mathcal{G}_{L^{\infty}}(\mathbb{R}^{n}\times[0,T]):=\mathcal{G}_{W^{\infty,\infty}({\Omega_{T}})}.$

Thus, we will represent a generalized function
$u\in\mathcal{G}_{L^{2}}(\mathbb{R}^{n}\times[0,T])$ by a net
$(u_{\varepsilon})$ with the moderateness property

$\forall k\,\exists
m:\quad{\left\|u_{\varepsilon}\right\|}_{H^{k}}=O(\varepsilon^{-m})\quad(\varepsilon\to
0).$

If $(\widetilde{u_{\varepsilon}})$ is another representative of $u$, then

$\forall k\,\forall
p:\quad{\left\|u_{\varepsilon}-\widetilde{u_{\varepsilon}}\right\|}_{H^{k}}=O(\varepsilon^{p})\quad(\varepsilon\to
0).$

Similar constructions and notations are used in case of
$E=H^{\infty}(\mathbb{R}^{n})$ and $E=W^{\infty,\infty}(\mathbb{R}^{n})$. Note
that by Young’s inequality ([Folland:99, Proposition 8.9.(a)]) any standard
convolution regularization with a scaled mollifier of Schwartz class provides
embeddings $L^{2}\hookrightarrow\mathcal{G}_{L^{2}}$ and
$L^{p}\hookrightarrow\mathcal{G}_{L^{\infty}}$ ($1\leq p\leq\infty$).

We recall below the main existence and uniqueness result for the following
Cauchy problem for the Schrödinger equation: Find a (unique) generalized
function $u$ on $\mathbb{R}^{n}\times[0,T]$ solving

(1)
$\displaystyle\partial_{t}u-i\,\sum_{k=1}^{n}\partial_{x_{k}}(c_{k}\partial_{x_{k}}u)+iVu$
$\displaystyle=f$ (2) $\displaystyle u\mid_{t=0}$ $\displaystyle=g,$

where $c_{1},\ldots,c_{n}$, $V$, and $f$ are generalized functions on
$\mathbb{R}^{n}\times[0,T]$ and $g$ is a generalized function on
$\mathbb{R}^{n}$.

######  Theorem 1.1.

Let $c_{k}$ ($k=1,\ldots,n$) and $V$ be generalized functions in
$\mathcal{G}_{L^{\infty}}(\mathbb{R}^{n}\times[0,T])$ possessing representing
nets of real-valued functions, $f$ in
$\mathcal{G}_{L^{2}}(\mathbb{R}^{n}\times[0,T])$, and $g$ be in
$\mathcal{G}_{L^{2}}(\mathbb{R}^{n})$. Suppose
(a) $c_{k}$ ($k=1\ldots,n$) and $V$ are of log-type, that is, for some (hence
every) representative $(c_{k\varepsilon})$ of $c_{k}$ and $(V_{\varepsilon})$
of $V$ we have
${\left\|\partial_{t}{c_{k\varepsilon}}\right\|}_{L^{\infty}}=O(\log({1}/{\varepsilon}))$
and
${\left\|\partial_{t}{V_{\varepsilon}}\right\|}_{L^{\infty}}=O(\log({1}/{\varepsilon}))$
as $\varepsilon\to 0$
and
(b) that the positivity conditions $c_{k\varepsilon}(x,t)\geq c_{0}$ for all
$(x,t)\in\mathbb{R}^{n}\times[0,T]$, $\varepsilon\in\,]0,1]$, $k=1,\ldots,n$
with some constant $c_{0}>0$ hold (hence with $c_{0}/2$ for any other
representative and small $\varepsilon$).
Then the Cauchy problem (1-2) has a unique solution
$u\in\mathcal{G}_{L^{2}}(\mathbb{R}^{n}\times[0,T])$.

We note that a regularization of an arbitrary finite-order distribution which
meets the log-type conditions on the coefficients $c_{k}$ and $V$ in the above
statement is easily achieved by employing a re-scaled mollification process as
described in [O:89].

In case of smooth coefficients a simple integration by parts argument shows
that any solution to the Cauchy problem obtained from the variational method
as in [DL:V5, Chapter XVIII, $\mathchar 632\relax$7, Section 1]) is a solution
in the sense of distributions as well. In addition, the following result from
[Hoermann:11] shows further coherence with the generalized function solution.

######  Corollary 1.2.

Let $V$ and $c_{k}$ ($k=1,\ldots,n$) belong to $C^{\infty}(\Omega_{T})\cap
L^{\infty}(\Omega_{T})$ with bounded time derivatives of first-order,
$g_{0}\in H^{1}(\mathbb{R}^{n})$, and $f_{0}\in
C^{1}([0,T],L^{2}(\mathbb{R}^{n}))$. Let $u$ denote the unique Colombeau
generalized solution to the Cauchy problem (1-2), where $g$, $f$ denote
standard embeddings of $g_{0}$, $f_{0}$, respectively. Then $u\approx w$,
where $w\in C([0,T],H^{1}(\mathbb{R}^{n}))$ is the unique distributional
solution obtained from the variational method.

##  2\. Modeling of the Cauchy problem and the Hamiltonian

###  2.1. Regularizations representing the potential

We consider a model potential heuristically of the form
$V(x,y,z)=\delta_{0}(x)h(y)$ for diffraction at a pattern in the plane $x=0$,
which is invariant with respect to height $z$ and has a horizontal structure
described by $h$ as a “function” of $y$. The value $h(y)$ should be
“essentially zero” where slits are located, say if $y$ belongs to a subset
$S\subseteq\mathbb{R}$ being the disjoint union of intervals (possibly
infinite in case of grating), and $h(y)$ should be “essentially infinite” at
locations that block classical objects, i.e., if $y\in\mathbb{R}\setminus S$.
Since the whole problem is invariant with respect to $z$-translations, we
reduce it immediately to a problem in the $(x,y)$-plane.

As a representation of the potential we consider the family of regularizations
given by

(3)
$V_{\varepsilon}(x,y):=\delta_{0}^{\varepsilon}(x)h_{\varepsilon}(y)\quad(\varepsilon>0,(x,y)\in\mathbb{R}^{2}),$

where $\delta_{0}^{\varepsilon},h_{\varepsilon}\in L^{\infty}(\mathbb{R})$ for
every $\varepsilon\in\,]0,1]$, $\delta_{0}^{\varepsilon}$ and
$h_{\varepsilon}$ are real-valued,

$\delta_{0}^{\varepsilon}\to\delta_{0}$ as $\varepsilon\to 0$, and
$h_{\varepsilon}\to 0$ pointwise on $S$, but $h_{\varepsilon}\to\infty$ on
$\mathbb{R}\setminus S$.

We assume that $W^{\infty,\infty}$-moderate regularizing families
$(\delta_{0}^{\varepsilon})$ and $(h_{\varepsilon})$ of nonnegative functions
with the above properties and such that
$\mathop{\mathrm{supp}}(\delta_{0}^{\varepsilon})$ and
$\mathop{\mathrm{supp}}(h_{\varepsilon})$ are compact for every
$\varepsilon>0$, can be achieved. We may then consider $V$ as the element in
$\mathcal{G}_{L^{\infty}}(\mathbb{R}^{2})$ represented by the family
$(V_{\varepsilon})$ and apply Theorem 1.1 for arbitrary $T>0$ to obtain the
following statement. (Note that the conditions (a) and (b) in Theorem 1.1 are
automatically satisfied, since we have $c_{k}=1$ and a potential $V$ without
$t$-dependence.)

######  Remark 2.1 (Relation with the indicator function of the slit
configuration).

The standard approximations in theoretical physics as described in
Subsubsection 3.1.1 circumvent the introduction of an explicit potential for
the Schrödinger operator. Instead they encode the slit configuration,
corresponding to the subset $S\subseteq\mathbb{R}$ above, into a specific
initial condition (“boundary value”) involving the characteristic function
$1_{S}$ of $S$, i.e., $1_{S}(y)=1$ for $y\in S$ and $0$ otherwise.

Note that by our specifications,
${\left\|h_{\varepsilon}\right\|}_{L^{\infty}}\to\infty$ as $\varepsilon\to
0$, in particular, ${\left\|h_{\varepsilon}\right\|}_{L^{\infty}}>0$ for small
$\varepsilon>0$. We clearly have
$h_{\varepsilon}(y)/{\left\|h_{\varepsilon}\right\|}_{L^{\infty}}\to 0$ for
any $y\in S$ and in addition, we would like to guarantee that
$h_{\varepsilon}(y)/{\left\|h_{\varepsilon}\right\|}_{L^{\infty}}\to 1$, if
$y\in\mathbb{R}\setminus S$, so that we obtain the pointwise convergence

(4)
$b_{\varepsilon}(y):=1-\frac{h_{\varepsilon}(y)}{{\left\|h_{\varepsilon}\right\|}_{L^{\infty}}}\to
1_{S}(y)\quad\varepsilon\to 0.$

The required property
$h_{\varepsilon}(y)/{\left\|h_{\varepsilon}\right\|}_{L^{\infty}}\to 1$ for
any $y\in\mathbb{R}\setminus S$ can easily be achieved by a stricter
formulation of the rather vague statement $h_{\varepsilon}\to\infty$ on
$\mathbb{R}\setminus S$ in the above specifications. For example, in addition
to $h_{\varepsilon}\to 0$ on $S$, we may require that

(5)
${\left\|h_{\varepsilon}\right\|}_{L^{\infty}}\stackrel{{\scriptstyle\varepsilon\to
0}}{{\longrightarrow}}\infty\quad\mathrm{and}\quad\forall
y\in\mathbb{R}\setminus S\,\exists c>0\,\exists\varepsilon_{0}>0\colon
h_{\varepsilon}(y)\geq{\left\|h_{\varepsilon}\right\|}_{L^{\infty}}-c>0\quad(0<\varepsilon\leq\varepsilon_{0}).$

In particular, this condition is still compatible with choosing
$h_{\varepsilon}$ of compact support (which is certainly growing as
$\varepsilon$ becomes smaller) and implies (4).

######  Remark 2.2 (Alternative modeling).

The above specifications of the potential are led by certain idealizations
which, however, are not implied by the physics of the problem. Instead of
modeling the impenetrable potential barrier at $x=0$ by a delta distribution
(in the limit $\varepsilon\to 0$) and the slits by $h_{\varepsilon}$ via
condition (4), we may work with some fixed potential $V\in
W^{\infty,\infty}(\mathbb{R}^{2})$ with
$\mathrm{supp}(V)=[-c,c]\times\overline{\mathbb{R}\setminus S}$. In this case,
the properties discussed in Subsections 2.1, 2.2, and 2.3 are still valid.
However, the proof of Proposition 2.4 has to be adapted similarly to that of
Proposition 3.2.

######  Theorem 2.3.

Let $V\in\mathcal{G}_{L^{\infty}}(\mathbb{R}^{2})$ denote the potential
defined via the regularizations (3). For every
$g\in\mathcal{G}_{L^{2}}(\mathbb{R}^{2})$, there is a unique solution
$u\in\mathcal{G}_{L^{2}}(\mathbb{R}^{2}\times[0,T])$ to the Cauchy problem

(6) $\partial_{t}u=i\,\Delta u-iVu,\quad u|_{t=0}=g.$

In the description of a scattering experiment, one is interested in the
limiting behavior as $\varepsilon\to 0$ of solution representatives
$(u_{\varepsilon})$ in case of an initial value $g$, modeled by a regularizing
family $(g_{\varepsilon})$, corresponding to a quantum particle that is
“spread out” considerably in the $y$-direction and “approaches” the plane
$x=0$ from the side of the half plane $x<0$. In particular, the aim would be
to study the properties of the _intensity distribution_

(7) $y\mapsto|u_{\varepsilon}(t_{1},x_{1},y)|^{2}$

at some time $t_{1}>0$ and as $\varepsilon\to 0$, where $x_{1}>0$ represents
the location of some screen parallel to the diffraction plane.

For the purpose of asymptotics with $\varepsilon\to 0$ one does not rely on
the full framework of generalized functions and Theorem 1.1, but may instead
look more directly at the family of regularized problems

(8) $\partial_{t}u_{\varepsilon}=i\,\Delta
u_{\varepsilon}-iV_{\varepsilon}u_{\varepsilon},\quad
u_{\varepsilon}|_{t=0}=g_{\varepsilon}.$

For every $\varepsilon>0$, $V_{\varepsilon}$ is bounded and real-valued, thus
the _Hamiltonians_

(9) $H_{\varepsilon}:=-\Delta+V_{\varepsilon}$

are an $\varepsilon$-parametrized family of self-adjoint operators with domain
$H^{2}(\mathbb{R}^{2})$ (independent of $\varepsilon$). We obtain the solution
$u_{\varepsilon}$ via the unitary group generated by $H_{\varepsilon}$, i.e.,

$\forall\varepsilon\in\,]0,1],\forall t\in\mathbb{R}\colon\quad
u_{\varepsilon}(t)=e^{-itH_{\varepsilon}}g_{\varepsilon}.$

###  2.2. Regularizations representing the initial configuration

Intuitively, during the time while the “source quantum particle approaches the
scattering obstacle”, the idealization of “least possible localization in $y$”
would be a plane wave of the form $(x,y,t)\mapsto Ae^{i(kx-\omega t)}$
($A\in\mathbb{C}$), following de Broglie’s correspondence with a free particle
of momentum $\hbar(k,0)\in\mathbb{R}^{2}$ and energy
$E=\hbar\omega\in[0,\infty[$. However, it is a more realistic model to
implement instead the initial value $g$, or rather its regularizations
$g_{\varepsilon}$, as a “wave packet” with an average momentum $p_{0}=\hbar k$
in the $x$-direction. Dropping again the explicit reference to $\hbar$, we may
make the ansatz for the initial value representatives as the family of
functions

(10)
$g_{\varepsilon}(x,y)=\rho_{\varepsilon}(y)\varphi_{\varepsilon}(x)e^{ip_{0}x},$

where $(\rho_{\varepsilon})_{\varepsilon\in(0,1]}$ and
$(\varphi_{\varepsilon})_{\varepsilon\in(0,1]}$ are $H^{\infty}$-moderate
families of nonnegative functions on $\mathbb{R}$ such that
$\mathop{\mathrm{supp}}(\varphi_{\varepsilon})\subseteq(-\infty,-1]$, and
${\left\|\rho_{\varepsilon}\right\|}_{L^{2}}=1={\left\|\varphi_{\varepsilon}\right\|}_{L^{2}}$
for every $\varepsilon>0$. Note that, by construction, $g_{\varepsilon}$ has
the following momentum expectation values

$\langle
g_{\varepsilon}|-i\partial_{y}g_{\varepsilon}\rangle=0\quad\text{and}\quad\langle
g_{\varepsilon}|-i\partial_{x}g_{\varepsilon}\rangle=p_{0}.$

###  2.3. Basic properties of the Hamiltonian $\bm{H_{\varepsilon}}$

For every $\varepsilon>0$, the operator $H_{\varepsilon}$ is positive, since
$\langle
f|H_{\varepsilon}f\rangle=\langle\partial_{x}f|\partial_{x}f\rangle+\langle\partial_{y}f|\partial_{y}f\rangle+\langle
f|V_{\varepsilon}f\rangle\geq 0$ holds for every $f\in H^{2}(\mathbb{R}^{2})$,
hence the spectrum of $H_{\varepsilon}$ is contained in $[0,\infty)$. We can
easily derive more details about the spectrum.

######  Proposition 2.4.

For the spectrum of $H_{\varepsilon}=-\Delta+V_{\varepsilon}$ with
$V_{\varepsilon}(x,y)=\delta_{0}^{\varepsilon}(x)h_{\varepsilon}(y)$, we
obtain $\sigma(H_{\varepsilon})=[0,\infty[$, while the point spectrum
$\sigma_{p}(H_{\varepsilon})$, i.e. the set of eigenvalues of
$H_{\varepsilon}$, is empty.

###### Proof.

Since $V_{\varepsilon}$ is a bounded real-valued function of compact support,
the corresponding multiplication operator is relatively ($-\Delta$)-compact
and hence $H_{\varepsilon}$ has the same essential spectrum111Recall: The
discrete spectrum consists of the isolated spectral points that are
eigenvalues of finite multiplicity and the essential spectrum is its
complement within the spectrum. The essential spectrum may include eigenvalues
which are non-isolated or have infinite dimensional eigenspace. as $-\Delta$
(cf. [HS:96, Section 14.3] or [RS:V4, Theorem XIII.15]). Therefore, the
essential spectrum of $H_{\varepsilon}$ is $[0,\infty)$, which equals also the
entire spectrum of $H_{\varepsilon}$ and the discrete spectrum is empty.

This leaves us with the question whether there can be eigenvalues embedded in
$[0,\infty[$: Since the (virial) function $(x,y)\mapsto
x\partial_{x}V_{\varepsilon}(x,y)+y\partial_{y}V_{\varepsilon}(x,y)$ has
compact support, we obtain from [HS:96, Theorem 16.1] that $H_{\varepsilon}$
cannot possess any strictly positive eigenvalues (alternatively, this follows
also from [RS:V4, Corollary to Theorem XIII.58]). Finally, we see that $0$
cannot be an eigenvalue either: Otherwise, there would be a nonvanishing $f\in
H^{2}(\mathbb{R}^{2})$ with $-\Delta f+V_{\varepsilon}f=0$ on
$\mathbb{R}^{2}$; upon multiplication by $\overline{f}$, integration over
$\mathbb{R}^{2}$, an integration by parts implies
$0=\langle\partial_{x}f|\partial_{x}f\rangle+\langle\partial_{y}f|\partial_{y}f\rangle+\langle
V_{\varepsilon}f|f\rangle\geq\langle\partial_{x}f|\partial_{x}f\rangle+\langle\partial_{y}f|\partial_{y}f\rangle$
due to the nonnegativity of $V_{\varepsilon}$; hence $f$ would have to be a
constant function. ∎

In summary, $H_{\varepsilon}$ has no eigenvalues and its continuous spectrum
is $[0,\infty)$ (and equals the essential spectrum). Therefore, the dynamics
given by $\exp(itH_{\varepsilon})$ does not have any bound states.

By some abuse of notation the symbol of the second-order differential operator
$H_{\varepsilon}$ is the function on $\mathbb{R}^{2}\times\mathbb{R}^{2}\cong
T^{*}(\mathbb{R}^{2})$ given by
$H_{\varepsilon}(q,\theta):=|\theta|^{2}+V_{\varepsilon}(q)\geq|\theta|^{2}$,
hence it is obviously uniformly elliptic.

In passing, let us show in addition that $H_{\varepsilon}$ is an operator of
constant strength on $\mathbb{R}^{2}$ in the sense of [Hoermander:V2,
Definition 13.1.1].

######  Proposition 2.5.

$H_{\varepsilon}$ is a differential operator of constant strength.

###### Proof.

With the weight function as defined in [Hoermander:V2, Example 10.1.3],

$\widetilde{H_{\varepsilon}}(q,\theta)^{2}:=\sum_{|\alpha|\leq
2}|\partial_{\theta}^{\alpha}H_{\varepsilon}(q,\theta)|^{2}=\left||\theta|^{2}+|V_{\varepsilon}(q)|\right|^{2}+4|\theta|^{2}+8,$

we obtain for arbitrary $q,r,\theta\in\mathbb{R}^{2}$ the estimate

$\left(\frac{\widetilde{H_{\varepsilon}}(q,\theta)}{\widetilde{H_{\varepsilon}}(r,\theta)}\right)^{2}\leq\frac{|\theta|^{4}+4|\theta|^{2}+8+(2|\theta|^{2}+{\left\|V_{\varepsilon}\right\|}_{L^{\infty}}){\left\|V_{\varepsilon}\right\|}_{L^{\infty}}}{|\theta|^{4}+4|\theta|^{2}+8}\leq
1+(1+{\left\|V_{\varepsilon}\right\|}_{L^{\infty}}){\left\|V_{\varepsilon}\right\|}_{L^{\infty}}.$

∎

###  2.4. Application of scattering theory to $\bm{H_{\varepsilon}}$

######  Lemma 2.6.

As a multiplication operator on $L^{2}(\mathbb{R}^{2})$, $V_{\varepsilon}$
defines a _short range perturbation_ of $P_{0}:=-\Delta$ in the sense of
[Hoermander:V2, Definition 14.4.1].

###### Proof.

In fact, the two conditions in [Hoermander:V2, Theorem 14.4.2] are easily seen
to hold: Let $\Omega$ denote the open unit ball in $\mathbb{R}^{2}$. i) For
any $y\in\mathbb{R}^{2}$, the set $E_{y}:=\\{V_{\varepsilon}(.+y)u\mid
u\in\mathcal{D}(\Omega),{\left\|\Delta u\right\|}_{L^{2}}\leq 1\\}$ is bounded
in $H^{2}_{0}(\Omega)$ thanks to Poincaré’s inequality, thus precompact in
$L^{2}(\Omega)$ by Rellich’s embedding theorem, and the continuous inclusion
$L^{2}(\Omega)\subseteq L^{2}(\mathbb{R}^{2})$ gives precompactness in
$L^{2}(\mathbb{R}^{2})$. ii) There is some $R>0$ such that
$\mathop{\mathrm{supp}}(V_{\varepsilon}(.+y))\cap\Omega=\emptyset$, if
$y\in\mathbb{R}^{2}$ with $|y|\geq R$. Thus for large $j\in\mathbb{N}$ we have
$V_{\varepsilon}(.+y)u=0$ for every $y\in\mathbb{R}^{2}$ with
$R_{j-1}<|y|<R_{j}$ and may put $M_{j}=0$ (notation of [Hoermander:V2, Theorem
14.4.2] and recall $R_{j}=2^{j-1}$ from [Hoermander:V2, Equation (14.1.2)]),
while for finitely many $j\in\mathbb{N}$ we may put
$M_{j}=C{\left\|V_{\varepsilon}\right\|}_{L^{\infty}}$, where $C$ is a
constant taken from Poincare’s inequality to obtain
${\left\|V_{\varepsilon}(.+y)u\right\|}_{L^{2}}\leq{\left\|V_{\varepsilon}\right\|}_{L^{\infty}}{\left\|u\right\|}_{L^{2}}\leq{\left\|V_{\varepsilon}\right\|}_{L^{\infty}}C{\left\|\Delta
u\right\|}_{L^{2}}$; then clearly, $\sum_{j=1}^{\infty}R_{j}M_{j}<\infty$. ∎

We may combine the property of $V_{\varepsilon}$ being a short range
perturbation with the information from 2.3 about the spectrum of
$H_{\varepsilon}$ being purely continuous and apply [Hoermander:V2, Theorems
14.4.6, 14.6.4, and 14.6.5]. We obtain that the _wave operators_
$W_{\pm}^{\varepsilon}$, defined by

$W_{\pm}^{\varepsilon}\varphi=\lim_{t\to\pm\infty}e^{itH_{\varepsilon}}e^{it\Delta}\varphi\qquad(\varphi\in
L^{2}(\mathbb{R}))$

are unitary intertwiners for $H_{\varepsilon}$ and $-\Delta$, in particular,
the unitary group providing the solution to (8) can be described in the form

(11)
$e^{-itH_{\varepsilon}}=W_{+}^{\varepsilon}e^{it\Delta}(W_{+}^{\varepsilon})^{-1}.$

Moreover, in our situation, the _distorted Fourier transforms_
([Hoermander:V2, Definition 14.6.3]) yield unitary operators
$F_{\pm}^{\varepsilon}$ on $L^{2}(\mathbb{R}^{2})$ such that both compositions
$F_{+}^{\varepsilon}\circ W_{+}^{\varepsilon}$ and $F_{-}^{\varepsilon}\circ
W_{-}^{\varepsilon}$ are equal to the Fourier transform ${\mathcal{F}}$ on
$L^{2}(\mathbb{R}^{2})$. We therefore have as an alternative to (11),

(12)
$F_{+}^{\varepsilon}(e^{-itH_{\varepsilon}}\varphi)(\theta)=e^{-it|\theta|^{2}}(F_{+}^{\varepsilon}\varphi)(\theta)\qquad(\varphi\in
L^{2}(\mathbb{R}),\theta\in\mathbb{R}^{2}).$

We recall how the action of $F_{+}^{\varepsilon}$ can be described more
explicitly on the subspace $B$ of $L^{2}(\mathbb{R}^{2})$ (defined in
[Hoermander:V2, Section 14.1]) consisting of those $L^{2}$-functions $\varphi$
such that
${\left\|\varphi\right\|}_{B}:=\sqrt{a_{1}(\varphi)}+\sum_{j=2}^{\infty}2^{(j-1)/2}\sqrt{a_{j}(\varphi)}<\infty$,
where $a_{1}(\varphi):=\int_{|x|<1}|\varphi|^{2}$ and
$a_{j}(\varphi):=\int_{2^{j-2}<|x|<2^{j-1}}|\varphi|^{2}$ ($j\geq 2$);
equipped with ${\left\|\ \right\|}_{B}$, $B$ becomes a Banach space with
$\mathcal{D}(\mathbb{R}^{2})$ as a dense subspace.

Let $R_{0}(z)$ denote the resolvent of $-\Delta$ for
$z\in\mathbb{C}\setminus[0,\infty)$. For $\varphi\in B$ we have by
[Hoermander:V2, Theorem 14.5.4] that
$z\mapsto(I+V_{\varepsilon}R_{0}(z))^{-1}\varphi$ gives continuous maps from
both $\\{z\in\mathbb{C}\mid\mathop{\mathrm{Im}}z\geq 0,z\neq 0\\}$ and
$\\{z\in\mathbb{C}\mid\mathop{\mathrm{Im}}z\leq 0,z\neq 0\\}$ into $B$, hence
for every $\varphi\in B$ and $\lambda>0$, the limits

$\varphi_{\lambda\pm i0}^{\varepsilon}:=\lim_{\alpha\downarrow
0}(I+V_{\varepsilon}R_{0}(\lambda\pm i\alpha))^{-1}\varphi$

exist in $B$, in particular, $\varphi_{\lambda\pm i0}^{\varepsilon}\in
L^{2}(\mathbb{R}^{2})$. Now $F_{+}^{\varepsilon}\varphi$ can be described for
any $\varphi\in B$ as follows ([Hoermander:V2, Lemma 14.6.2 and Definition
14.6.3]): For every $\lambda>0$, we have for $\theta\in\mathbb{R}^{2}$ with
$|\theta|^{2}=\lambda$, almost everywhere in the sense of the line measure
along the circle of radius $\sqrt{\lambda}$,

(13)
$(F_{+}^{\varepsilon}\varphi)(\theta)={\mathcal{F}}\varphi_{\lambda+i0}^{\varepsilon}(\theta).$

We may apply the distorted Fourier transform in a more concrete distributional
description of the solution $u_{\varepsilon}$ of (8). There is no substantial
loss for the physics of the problem to consider only initial values from $B$.

######  Proposition 2.7.

Suppose that the initial value regularizations $g_{\varepsilon}$, in addition
to (10), also satisfy

(14) $\forall\varepsilon>0\colon\quad g_{\varepsilon}\in B.$

Then we have for the distributional action of
$F_{+}^{\varepsilon}u_{\varepsilon}(t)$ (at fixed time $t$) on a test function
$\psi$ on $\mathbb{R}^{2}$,

(15) $\langle
F_{+}^{\varepsilon}u_{\varepsilon}(t),\psi\rangle=\int_{0}^{\infty}re^{-itr^{2}}\int_{S^{1}}{\mathcal{F}}(g_{r^{2}+i0}^{\varepsilon})(r\omega)\psi(r\omega)\,d\omega
dr,$

where $d\omega$ denotes the line measure on $S^{1}$ and we define, for every
$\lambda>0$,

$g_{\lambda+i0}^{\varepsilon}:=(I+V_{\varepsilon}R_{0}(\lambda+i0))^{-1}g_{\varepsilon}.$

###### Proof.

Upon applying (12) and (13), we obtain

$\langle F_{+}^{\varepsilon}u_{\varepsilon}(t),\psi\rangle=\langle
e^{-it|.|^{2}}F_{+}^{\varepsilon}g_{\varepsilon},\psi\rangle=\int_{\mathbb{R}^{2}}e^{-it|\theta|^{2}}F_{+}^{\varepsilon}g_{\varepsilon}(\theta)\psi(\theta)\,d\theta\\\
=\int_{S^{1}}\int_{0}^{\infty}e^{-itr^{2}}F_{+}^{\varepsilon}g_{\varepsilon}(r\omega)\psi(r\omega)\,r\,drd\omega=\int_{S^{1}}\int_{0}^{\infty}e^{-itr^{2}}{\mathcal{F}}(g_{r^{2}+i0}^{\varepsilon})(r\omega)\psi(r\omega)\,r\,drd\omega\\\
=\int_{0}^{\infty}re^{-itr^{2}}\int_{S^{1}}{\mathcal{F}}(g_{r^{2}+i0}^{\varepsilon})(r\omega)\psi(r\omega)\,d\omega
dr.$

∎

######  Remark 2.8 (Attempt at a formula for the “boundary value”
$R_{0}(\lambda+i0)$ of the resolvent of $-\Delta$).

The resolvent $R_{0}(z)=(-\Delta-z)^{-1}$ is defined (and holomorphic) for
$z\in\mathbb{C}\setminus[0,\infty)$. Let $f$ be a test function on
$\mathbb{R}^{2}$, then we have

${\mathcal{F}}(R_{0}(z)f)(\theta)=\frac{f(\theta)}{|\theta|^{2}-z}\qquad(\theta\in\mathbb{R}^{2}),$

where we note that $\theta\mapsto 1/(|\theta|^{2}-z)$ is a smooth bounded
function, hence belongs to $\mathscr{S}^{\prime}(\mathbb{R}^{2})$. Denote by
$r_{0}(z)\in\mathscr{S}^{\prime}(\mathbb{R}^{2})$ its inverse Fourier
transform, so that we have (with convolution
$\mathscr{S}^{\prime}*\mathscr{S}$)

$R_{0}(z)f=r_{0}(z)*f.$

We should have

$R_{0}(\lambda+i0)f=r_{0}(\lambda+i0)*f\quad\text{with}\quad
r_{0}(\lambda+i0)=\mathscr{S}^{\prime}\text{-}\lim_{\mu\to
0+}r_{0}(\lambda+i\mu).$

A formula for ${\mathcal{F}}r_{0}(\lambda+i0)$ could be obtained by
determining $\lim_{\mu\to 0+}v_{\mu}$ in
$\mathscr{S}^{\prime}(\mathbb{R}^{2})$, where

$v_{\mu}(\theta):=\frac{1}{|\theta|^{2}-\lambda-i\mu}\qquad(\theta\in\mathbb{R}^{2},\lambda\in\mathbb{R},\mu>0).$

Let $g\in\mathcal{D}(\mathbb{R}^{2})$, use polar coordinates $\theta=r\omega$
with $r\geq 0$, $\omega\in S^{1}$, and introduce
$Mg(r):=\int_{S^{1}}g(r\omega)\,d\omega$ to deduce in a first step (with the
change of coordinates $r^{2}=s$ in the last equality)

$\langle
v_{\mu},g\rangle=\int_{\mathbb{R}^{2}}\frac{g(\theta)}{|\theta|^{2}-\lambda-i\mu}\,d\theta=\int_{0}^{\infty}\frac{r}{r^{2}-\lambda-i\mu}\;\int_{S^{1}}g(r\omega)\,d\omega\,dr=\frac{1}{2}\int_{0}^{\infty}\frac{Mg(\sqrt{s})}{s-\lambda-i\mu}\,ds.$

The interesting case is $\lambda\geq 0$, for otherwise
$1/(s-\lambda)=\lim_{\mu\to 0}1/(s-\lambda-i\mu)$ is a locally integrable
function on $[0,\infty)$ and we directly obtain
$\langle{\mathcal{F}}r_{0}(\lambda+i0),g\rangle=\int_{0}^{\infty}\int_{S^{1}}g(\sqrt{s}\omega)/(2(s-\lambda))\,d\omega\,ds$.
If $\lambda\geq 0$, then $1/(s-\lambda)$ has a non-integrable singularity at
$s=\lambda$. Upon a shift by $\lambda$, we would like to interpret the above
integral with the help of the one-dimensional distribution $(s-i0)^{-1}$,
which can be defined as distributional boundary value of the holomorphic
function $z\mapsto 1/z$ in the lower complex half plane ([Hoermander:V1,
Theorem 3.1.11]). The explicit action of $(s-\lambda-i0)^{-1}$ on a test
function $\phi$ can be given by (cf. [Hoermander:V1, Equations (3.2.10),
(3.2.10’), and bottom of page 72])

$\langle(s-\lambda-i0)^{-1},\phi\rangle=-\int_{\mathbb{R}}(\log(|s-\lambda|)\phi^{\prime}(s)\,ds+i\pi\phi(\lambda),$

hence can be extended to functions $\phi\in C^{1}_{c}(\mathbb{R})$. The
remaining difficulty now is that $\phi(s):=Mg(\sqrt{s})$ is only defined for
$s\geq 0$ and will in general not be $C^{1}$ up to $s=0$. However, if $g$
vanishes in a neighborhood of $0$, then $Mg(\sqrt{s})=0$ for small $s\geq 0$
and we may take the $C^{1}$ extension of $s\mapsto Mg(\sqrt{s})$ to the
function $m_{g}\in C^{1}_{c}(\mathbb{R})$ with $m_{g}(s)=0$ for $s<0$ and
write

$\langle{\mathcal{F}}r_{0}(\lambda+i0),g\rangle=\lim_{\mu\to 0+}\langle
v_{\mu},g\rangle=\frac{1}{2}\langle(s-\lambda-i0)^{-1},m_{g}(s)\rangle.$

##  3\. Alternative approximations and regularizations

We now leave the detailed specifications of the previous sections, vary
certain aspects of the modeling and allow for simplifications and
approximations. Interpreting the term involving the potential,
$V_{\varepsilon}u_{\varepsilon}$, as a source term (right-hand side) for the
equation, we investigate connections to approximate solution methods used in
physics. In particular, we will employ iterative procedures and a concrete
regularization for diffraction at a single-slit.

Suppose that we would have convergence $u_{\varepsilon}\to u$ in
$C(\mathbb{R},\mathscr{S}^{\prime}(\mathbb{R}^{2}))$ as $\varepsilon\to 0$,
then necessarily

$g_{\varepsilon}=u_{\varepsilon}(0)\to u(0),$

and, in the sense of distributions,

(16) $iV_{\varepsilon}u_{\varepsilon}=i\Delta
u_{\varepsilon}-\partial_{t}u_{\varepsilon}\to i\Delta u-\partial_{t}u.$

We would then obtain that both $u_{\varepsilon}$ and
$V_{\varepsilon}u_{\varepsilon}$ converge as distributions, suggesting that
$u_{\varepsilon}\to 0$ near $\\{0\\}\times(\mathbb{R}\setminus S)$ since
$V_{\varepsilon}\to\infty$ on that same set, where classical particles should
be blocked. Thus the approximative idea to consider the set $S$ as sources of
waves propagating into the region $x>0$, as it is often described in the
physics literature, can be given some mathematical support.

###  3.1. Cauchy problems with source term or initial value replacing the
potential

Suppose that instead of employing truly a (regularized) potential function in
the Schrödinger operator we look at a simplified Cauchy problem, where the
influence of the “interaction product” $Vu$ of the potential with the wave
function is somehow replaced by a source term $F$ and an initial value $f$ so
that we have the following inhomogeneous Cauchy problem without potential

(17) $\partial_{t}w=i\,\Delta w-iF,\quad w|_{t=0}=f.$

The solution can be written in terms of the fundamental solution $E\in
C(\mathbb{R},\mathscr{S}^{\prime}(\mathbb{R}^{2}))$,
$E(t;x,y)=\exp(i(x^{2}+y^{2})/4t)/(4\pi it)$, satisfying

$\partial_{t}E-i\Delta E=0,\quad E(0)=\delta,$

with Duhamel’s principle in the form

$w(t)=E(t)*f-i\int_{0}^{t}E(t-\tau)*F(\tau)\,d\tau.$

Equivalently, upon applying a spatial Fourier transform to (17) and using a
notation like $\widehat{w}(t,\xi,\eta)$ etc., we obtain an ordinary
differential equation with initial condition

$\partial_{t}\widehat{w}(t,\xi,\eta)=-i(|\xi|^{2}+|\eta|^{2})\widehat{w}(t,\xi,\eta)-i\widehat{F}(t,\xi,\eta),\quad\widehat{w}(0,\xi,\eta)=\widehat{f}(\xi,\eta).$

Employing the abbreviation $\theta=(\xi,\eta)$, the spatially Fourier
transformed solution is given by

(18)
$\widehat{w}(t,\theta)=e^{-it|\theta|^{2}}\widehat{f}(\theta)-i\int_{0}^{t}e^{-i(t-\tau)|\theta|^{2}}\,\widehat{F}(\tau,\theta)\,d\tau.$

(Checking with [Hoermander:V1, Theorem 7.6.1] also confirms that
$\widehat{E}(t;\xi,\eta)=\exp(-it(\xi^{2}+\eta^{2}))$.)

####  3.1.1. The approximate solutions from theoretical physics

An inspection of the discussions in [Beau:12] and [Manoukian:89] shows that
the basic solution formulae obtained in theoretical physics can be put into
the context of (17) as follows: In a first step, let $w_{0}$ denote the
solution to (17) with initial value $f=\delta_{(-x_{0},0)}$ and $F=0$, where
$x_{0}>0$; thus, $w_{0}$ corresponds to the wave function of a free particle
emitted at time $t=0$ at the location $(-x_{0},0)$ and is given by
$w_{0}(t,x,y)=(E(t)*\delta_{(-x_{0},0)})(x,y)=E(t,x+x_{0},y)$. Suppose that
$y\mapsto b_{0}(y)$ describes the pattern and shapes of slits in the plane
$x=0$, but contrary to the potential function above now with value $1$ for
passing through and $0$ for blocking, e.g., [Beau:12] uses the characteristic
function of one or two bounded intervals, and that the particle passes through
$x=0$ at time $t_{0}>0$.

In a second step, we consider now the solution $w_{1}$ to a Cauchy problem for
the free Schrödinger equation with initial value corresponding to the particle
represented by $w_{0}$ passing through the slits at time $t_{0}$, namely

(19) $\partial_{t}w_{1}=i\,\Delta w_{1},\quad
w_{1}(t_{0},x,y)=\delta_{0}(x)w_{0}(t_{0},0,y)b_{0}(y)=\delta_{0}(x)E(t_{0},x_{0},y)b_{0}(y)=:f_{0}(x,y).$

The qualitative properties of the intensity distribution
$y\mapsto|w_{1}(t_{0}+T,x_{1},y)|^{2}$ found on a screen located at distance
$x_{1}>0$ from the slits and at time $t_{0}+T$, $T>0$, are then studied in
detail in [Beau:12, Manoukian:89] for appropriate asymptotic relations between
the relevant parameters from physics (de Broglie wavelength of the particle,
$t_{0}$, and $T$) and geometry ($x_{0}$, $x_{1}$ and shape of the slits) and
seem to give reasonable approximations to the interference features seen in
actual experiments.

We can easily obtain an explicit expressions from (19) upon applying the above
solution formulae to $w_{1}(t+t_{0})$, i.e., $w_{1}(t)=E(t-t_{0})*f_{0}$,
noting that the $x$-convolution is trivial due to the factor $\delta_{0}(x)$
in $f_{0}$, i.e.,
$w_{1}(t,x,y)=\Big{(}E(t-t_{0},x,.)*\big{(}E(t_{0},x_{0},.)b_{0}(.)\big{)}\Big{)}(y)$,
and writing out the remaining $y$-convolution as an integral:

$w_{1}(t,x,y)=\int_{-\infty}^{\infty}E(t-t_{0},x,y-s)E(t_{0},x_{0},s)b_{0}(s)\,ds\\\
=\frac{1}{(4\pi
i)^{2}(t-t_{0})t_{0}}\int_{-\infty}^{\infty}e^{i(x^{2}+(y-s)^{2})/4(t-t_{0})}e^{i(x_{0}^{2}+s^{2})/4t_{0}}b_{0}(s)\,ds\\\
=\frac{-e^{\frac{i}{4}(\frac{x^{2}+y^{2}}{t-t_{0}}+\frac{x_{0}^{2}}{t_{0}})}}{16\pi^{2}(t-t_{0})t_{0}}\int_{-\infty}^{\infty}e^{i(s^{2}-2ys)/4(t-t_{0})}e^{is^{2}/4t_{0}}b_{0}(s)\,ds\\\
=\frac{-e^{\frac{i}{4}(\frac{x^{2}+y^{2}}{t-t_{0}}+\frac{x_{0}^{2}}{t_{0}})}}{16\pi^{2}(t-t_{0})t_{0}}\int_{-\infty}^{\infty}e^{\frac{-isy}{2(t-t_{0})}}\underbrace{e^{\frac{its^{2}}{t_{0}(t-t_{0})}}}_{\phi(t,t_{0},s)}b_{0}(s)\,ds=\frac{-e^{\frac{i}{4}(\frac{x^{2}+y^{2}}{t-t_{0}}+\frac{x_{0}^{2}}{t_{0}})}}{16\pi^{2}(t-t_{0})t_{0}}\,{\mathcal{F}}\Big{(}\phi(t,t_{0},.)\,b_{0}(.)\Big{)}\big{(}y/2(t-t_{0})\big{)}.$

Therefore, the corresponding intensity distribution as a function of $y$ is
proportional to

$16^{2}\pi^{4}T^{2}t_{0}^{2}\cdot|w_{1}(t_{0}+T,x_{1},y)|^{2}=\Big{|}{\mathcal{F}}\Big{(}\phi_{0}b_{0}\Big{)}\big{(}y/2T\big{)}\Big{|}^{2},$

where $\phi_{0}(s):=\phi(t_{0}+T,t_{0},s)=\exp(i(t_{0}+T)s^{2}/t_{0}T)$ and we
recall that $b_{0}$ is the characteristic function of the slits.

####  3.1.2. A plausibility check

Let $u_{0,\varepsilon}$ denote the solution to the free Schrödinger equation
with initial value $g_{\varepsilon}$, i.e.,

$\partial_{t}u_{0,\varepsilon}=i\Delta u_{0,\varepsilon},\quad
u_{0,\varepsilon}|_{t=0}=g_{\varepsilon}.$

Comparing with $w_{0}$ in Subsubsection 3.1.1 we have: If
$g_{\varepsilon}\to\delta_{(-x_{0},0)}$ as $\varepsilon\to 0$, then
$u_{0,\varepsilon}\to w_{0}$.

Consider the difference between the scattered and the free solution
$w_{\varepsilon}:=u_{\varepsilon}-u_{0,\varepsilon}$, which is characterized
by

$\partial_{t}w_{\varepsilon}=i\Delta
w_{\varepsilon}-iV_{\varepsilon}u_{\varepsilon},\quad
w_{\varepsilon}|_{t=0}=0.$

Interpreting this via Equation (17) we have the “source term”
$V_{\varepsilon}u_{\varepsilon}$ and the corresponding solution formula
implies

$w_{\varepsilon}(t)=-i\int_{0}^{t}E(t-\tau)*(V_{\varepsilon}u_{\varepsilon}(\tau))\,d\tau.$

Now from the set-up of $V_{\varepsilon}$ we could argue for an immediate
plausible approximation in the integrand by using

(20)
$F_{\varepsilon}(t,x,y):=V_{\varepsilon}(x,y)u_{\varepsilon}(t,x,y)\approx\delta_{0}(x)h_{\varepsilon}(y)u_{\varepsilon}(t,0,y).$

In view of the observations following (16), arguing that the “free solution
from the region $x<0$ enters through the slits” (represented by the set $S$)
at an instance of time $t_{0}$ according to a “typical travel time from the
source to the plane $x=0$”, we might do a further, rather bold, step and try
with the following additional “approximate replacement”

(21)
$\beta_{\varepsilon}(t,x,y):=\delta_{0}(x)h_{\varepsilon}(y)u_{\varepsilon}(t,0,y)\approx\delta_{0}(x)1_{S}(y)u_{0,\varepsilon}(t,0,y)\delta_{t_{0}}(t).$

Combining (20) and (21) we would be using (see also Proposition 3.1)

(22)
$F_{\varepsilon}(t,x,y)\approx\beta_{\varepsilon}(t,x,y)\approx\delta_{t_{0}}(t)\delta_{0}(x)1_{S}(y)u_{0,\varepsilon}(t_{0},0,y),$

which yields the following simplified approximate solution formula

(23)
$w_{\varepsilon}(t,x,y)\approx-i\Big{(}E(t-t_{0},x,.)*\big{(}u_{0,\varepsilon}(t_{0},0,.)1_{S}(.)\big{)}\Big{)}(y).$

On the other hand, if we consider $w_{1}$ given by (19) in Subsubsection 3.1.1
and put $\widetilde{w_{1}}(t):=-iH(t-t_{0})w_{1}(t)$, then an elementary
computation shows that

$\partial_{t}\widetilde{w_{1}}=i\Delta\widetilde{w_{1}}-i\delta_{t_{0}}(t)\delta_{0}(x)b_{0}(y)w_{0}(t_{0},0,y),\quad\mathop{\mathrm{supp}}\widetilde{w_{1}}\subseteq[t_{0},\infty[\times\mathbb{R}^{2}.$

Thus, $\widetilde{w_{1}}$ satisfies the Cauchy problem (17) with initial value
$0$ (at time $t=0$) and source term

$F(t,x,y)=\delta_{t_{0}}(t)\delta_{0}(x)b_{0}(y)w_{0}(t_{0},0,y),$

which nicely matches (22) in the typical case where $b_{0}=1_{S}$ and implies

$\widetilde{w_{1}}(t,x,y)=-i\Big{(}E(t-t_{0},x,.)*\big{(}w_{0}(t_{0},0,.)1_{S}(.)\big{)}\Big{)}(y),$

which happens to agree with the distributional limit, as $\varepsilon\to 0$,
of the right-hand side in (23), if $g_{\varepsilon}\to\delta_{(-x_{0},0)}$.

####  3.1.3. Improving on the plausibility of (22)

The coherence result in Corollary 1.2 tells us that in case of an $H^{1}$
initial value and a smooth bounded potential the generalized solution is
associated with the solution in $C([0,T],H^{1}(\mathbb{R}^{2}))$ obtained from
the variational method. In fact, as the proof of Corollary 1.2 (to be found in
[Hoermann:11, Corollary 3.2]) shows, we even have convergence in this latter
function space. Using this special case as a motivation, we may consider the
convergence property

(24) $u_{\varepsilon}\to u\quad(\varepsilon\to 0)\quad\text{ in
}C([0,T],H^{1}(\mathbb{R}^{2}))$

as a basis of a more general analysis. In such circumstance we have support
for the approximation (20) under the following technical conditions for the
potential regularization:

(25) $\delta_{0}^{\varepsilon}\geq
0,\quad\int_{\mathbb{R}}\delta_{0}^{\varepsilon}(x)\,dx=1,\quad\mathop{\mathrm{supp}}(\delta_{0}^{\varepsilon})\subseteq[-c_{\varepsilon},c_{\varepsilon}],\quad{\left\|h_{\varepsilon}\right\|}_{L^{\infty}(\mathbb{R})}\sqrt{c_{\varepsilon}}\to
0\;(\varepsilon\to 0).$

######  Proposition 3.1.

Suppose (24) and (25) hold and let $\beta_{\varepsilon}(t)$ denote the
distribution

$\langle\beta_{\varepsilon}(t),\varphi\rangle:=\int_{\mathbb{R}}h_{\varepsilon}(y)u_{\varepsilon}(t,0,y)\varphi(0,y)\,dy\qquad(\varphi\in\mathcal{D}(\mathbb{R}^{2})),$

then we have $\lim\limits_{\varepsilon\to
0}\big{(}V_{\varepsilon}u_{\varepsilon}(t)-\beta_{\varepsilon}(t)\big{)}=0$ in
$\mathcal{D}^{\prime}(\mathbb{R}^{2})$ for every $t\in[0,T]$.

###### Proof.

For any $\varphi\in\mathcal{D}(\mathbb{R}^{2})$, we have (since
$\int\delta_{0}^{\varepsilon}=1$ and
$h_{\varepsilon},\delta_{0}^{\varepsilon}\geq 0$)

$|\langle
V_{\varepsilon}u_{\varepsilon}(t)-\beta_{\varepsilon}(t),\varphi\rangle|=|\int_{\mathbb{R}}h_{\varepsilon}(y)\int_{\mathbb{R}}\delta_{0}^{\varepsilon}(x)\big{(}u_{\varepsilon}(t,x,y)\varphi(x,y)-u_{\varepsilon}(t,0,y)\varphi(0,y)\big{)}\,dx\,dy|\\\
\leq\int_{\mathbb{R}}h_{\varepsilon}(y)\int_{\mathbb{R}}\delta_{0}^{\varepsilon}(x)\,|\underbrace{u_{\varepsilon}(t,x,y)\varphi(x,y)-u_{\varepsilon}(t,0,y)\varphi(0,y)}_{=:\gamma_{\varepsilon}(t,x,y)}|\,dx\,dy,$

where we may write

$\gamma_{\varepsilon}(t,x,y)=\int_{0}^{x}\frac{d}{ds}(u_{\varepsilon}(t,s,y)\varphi(s,y))\,ds=\int_{0}^{x}\big{(}\partial_{x}u_{\varepsilon}(t,s,y)\varphi(s,y)+u_{\varepsilon}(t,s,y)\partial_{x}\varphi(s,y)\big{)}\,ds.$

Inserting this above and using the fact that $|x|\leq c_{\varepsilon}$ in
$\mathop{\mathrm{supp}}(\delta_{0}^{\varepsilon})$ we obtain

$|\langle
V_{\varepsilon}u_{\varepsilon}(t)-\beta_{\varepsilon}(t),\varphi\rangle|\leq\int_{\mathbb{R}}h_{\varepsilon}(y)\int_{\mathbb{R}}\delta_{0}^{\varepsilon}(x)\int_{-c_{\varepsilon}}^{c_{\varepsilon}}|\partial_{x}u_{\varepsilon}(t,s,y)\varphi(s,y)+u_{\varepsilon}(t,s,y)\partial_{x}\varphi(s,y)|\,ds\,dx\,dy\\\
=\int_{\mathbb{R}}h_{\varepsilon}(y)\int_{-c_{\varepsilon}}^{c_{\varepsilon}}|\partial_{x}u_{\varepsilon}(t,s,y)\varphi(s,y)+u_{\varepsilon}(t,s,y)\partial_{x}\varphi(s,y)|\,ds\,dy\\\
\leq{\left\|h_{\varepsilon}\right\|}_{L^{\infty}(\mathbb{R})}\int_{[-c_{\varepsilon},c_{\varepsilon}]\times\mathbb{R}}\big{(}|\partial_{x}u_{\varepsilon}(t,s,y)\varphi(s,y)|+|u_{\varepsilon}(t,s,y)\partial_{x}\varphi(s,y)|\big{)}\,d(s,y)\\\
\leq{\left\|h_{\varepsilon}\right\|}_{L^{\infty}(\mathbb{R})}\Big{(}{\left\|\partial_{x}u_{\varepsilon}(t)\right\|}_{L^{2}(M_{\varepsilon})}{\left\|\varphi\right\|}_{L^{2}(M_{\varepsilon})}+{\left\|u_{\varepsilon}(t)\right\|}_{L^{2}(M_{\varepsilon})}{\left\|\partial_{x}\varphi\right\|}_{L^{2}(M_{\varepsilon})}\Big{)}\\\
\leq
C_{0}{\left\|h_{\varepsilon}\right\|}_{L^{\infty}(\mathbb{R})}{\left\|u_{\varepsilon}(t)\right\|}_{H^{1}(M_{\varepsilon})}{\left\|\varphi\right\|}_{H^{1}(M_{\varepsilon})},$

where we have put
$M_{\varepsilon}:=[-c_{\varepsilon},c_{\varepsilon}]\times\mathbb{R}$ and
$C_{0}$ is some positive constant. If
$\mathop{\mathrm{supp}}(\varphi)\subseteq[-l_{\varphi},l_{\varphi}]^{2}$, then
we clearly have

${\left\|\varphi\right\|}_{H^{1}(M_{\varepsilon})}^{2}\leq 3\cdot
2l_{\varphi}\cdot
2c_{\varepsilon}\cdot{\left\|\varphi\right\|}_{W^{1,\infty}(\mathbb{R}^{2})}^{2}$

and this implies (with some constant $C_{1}$ depending on $\varphi$)

(26) $|\langle
V_{\varepsilon}u_{\varepsilon}(t)-\beta_{\varepsilon}(t),\varphi\rangle|\leq
C_{1}\,\sqrt{c_{\varepsilon}}\,{\left\|h_{\varepsilon}\right\|}_{L^{\infty}(\mathbb{R})}{\left\|u_{\varepsilon}(t)\right\|}_{H^{1}(M_{\varepsilon})}.$

Due to (24) we have
${\left\|u_{\varepsilon}(t)\right\|}_{H^{1}(\mathbb{R}^{2})}\to{\left\|u(t)\right\|}_{H^{1}(\mathbb{R}^{2})}$
as $\varepsilon\to 0$, hence there is some $1>\varepsilon_{0}>0$ such that
${\left\|u(t)\right\|}_{H^{1}(M_{\varepsilon})}\leq{\left\|u_{\varepsilon}(t)\right\|}_{H^{1}(\mathbb{R}^{2})}\leq
2{\left\|u(t)\right\|}_{H^{1}(\mathbb{R}^{2})}$ for all
$0<\varepsilon<\varepsilon_{0}$, inserted into (26) we finally obtain from
(25) that

$|\langle
V_{\varepsilon}u_{\varepsilon}(t)-\beta_{\varepsilon}(t),\varphi\rangle|\leq
2C_{1}{\left\|u(t)\right\|}_{H^{1}(\mathbb{R}^{2})}\,\sqrt{c_{\varepsilon}}\,{\left\|h_{\varepsilon}\right\|}_{L^{\infty}(\mathbb{R})}\to
0\qquad(\varepsilon_{0}>\varepsilon\to 0).$

∎

####  3.1.4. Basic idea of an iterated regularized approximation scheme

Let us apply (17) to a situation, where we still take the same initial value
as in the regularized problem (8) but shift the effect of the regularized
potential into a source term $F_{\varepsilon}$, i.e.,

(27) $\partial_{t}w_{\varepsilon}=i\,\Delta
w_{\varepsilon}-iF_{\varepsilon},\quad
w_{\varepsilon}|_{t=0}=g_{\varepsilon}.$

Suppose now that $w_{0,\varepsilon}$ is the solution to the free Schrödinger
equation with initial value $g_{\varepsilon}$ and that $w_{1,\varepsilon}$ is
the solution to (27) with $F_{\varepsilon}:=V_{\varepsilon}w_{0,\varepsilon}$.
Then (18) implies

$\widehat{w_{1,\varepsilon}}(t,\theta)=\widehat{w_{0,\varepsilon}}(t,\theta)-i\int_{0}^{t}e^{-i(t-\tau)|\theta|^{2}}\,{\mathcal{F}}(V_{\varepsilon}w_{0,\varepsilon}(\tau,.))(\theta)\,d\tau,$

where the second term on the right-hand side might be considered a rough first
approximation to the scattering contribution. Proceeding inductively we
obtain, for every $\varepsilon>0$, a sequence
$(w_{n,\varepsilon})_{n\in\mathbb{N}}$ of functions satisfying

$\partial_{t}w_{n+1,\varepsilon}=i\Delta
w_{n+1,\varepsilon}-iV_{\varepsilon}w_{n,\varepsilon},\quad
w_{n+1,\varepsilon}|_{t=0}=g_{\varepsilon}.$

Let us write the solution to (18) in the more compact form
$w_{\varepsilon}=w_{0,\varepsilon}-iLF_{\varepsilon}$, where the linear
operator $f\mapsto Lf$ is given by
${\mathcal{F}}(Lf)(t,\theta):=\int_{0}^{t}\exp(-i(t-\tau)|\theta|^{2})\widehat{f}(\tau,\theta)\,d\tau$.
Then the iterative definition of $(w_{n,\varepsilon})_{n\in\mathbb{N}}$ gives

$w_{n+1,\varepsilon}=w_{0,\varepsilon}+\sum_{k=1}^{n}(-iLV_{\varepsilon})^{k}w_{0,\varepsilon},$

where $V_{\varepsilon}$ has to be understood as the linear operator of
multiplication by $V_{\varepsilon}$.

We observe that formally, as $n\to\infty$, one would expect that

$w_{n+1,\varepsilon}\to
w_{\varepsilon}:=\sum_{k=0}^{\infty}(-iLV_{\varepsilon})^{k}w_{0,\varepsilon}=(I-iLV_{\varepsilon})^{-1}w_{0,\varepsilon},$

i.e., $w_{\varepsilon}=w_{0,\varepsilon}+iLV_{\varepsilon}w_{\varepsilon}$,
and therefore
$w_{\varepsilon}|_{t=0}=w_{0,\varepsilon}|_{t=0}+i(LV_{\varepsilon}w_{\varepsilon})|_{t=0}=g_{\varepsilon}+0=g_{\varepsilon}$
as well as

$(\partial_{t}-i\Delta)w_{\varepsilon}=(\partial_{t}-i\Delta)w_{0,\varepsilon}+(\partial_{t}-i\Delta)iLV_{\varepsilon}w_{\varepsilon}=0+i(-V_{\varepsilon}w_{\varepsilon})=-iV_{\varepsilon}w_{\varepsilon},$

because $v:=-iLF$ solves $\partial_{t}v=i\Delta v-iF$ by construction of $L$.
In other words, the formal limit $w_{\varepsilon}$ of the sequence
$(w_{n,\varepsilon})_{n\in\mathbb{N}}$ satisfies

(28) $\partial_{t}w_{\varepsilon}=i\Delta
w_{\varepsilon}-iV_{\varepsilon}w_{\varepsilon},\quad
w_{\varepsilon}|_{t=0}=g_{\varepsilon},$

which is precisely (8) and thus suggests that $w_{n,\varepsilon}\to
u_{\varepsilon}$ as $n\to\infty$.

###  3.2. An explicit “non-smooth regularization”

For somewhat more explicit representations we might consider specific non-
smooth approximations $g_{\varepsilon}$ and $V_{\varepsilon}$ for the initial
value and potential that offer some convenience in calculations. For example,
let us look at

(29)
$g_{\varepsilon}(x,y):=\varphi_{\varepsilon}(x)e^{ip_{0}x}\cdot\sqrt{\frac{\varepsilon}{2}}\chi(\varepsilon
y),$

where $\chi:=\chi_{[-1,1]}$ is the characteristic function of the interval
$[-1,1]$, and

(30)
$V_{\varepsilon}(x,y):=\frac{1}{2\varepsilon}\chi\Big{(}\frac{x}{\varepsilon}\Big{)}\cdot\frac{1}{\varepsilon}\Big{(}1-\chi\Big{(}\frac{y}{d}\Big{)}\Big{)}$

with $d>0$ denoting half the width of a single slit centered at $y=0$ in the
$x=0$ plane, so that $S=[-d,d]$ in this case.

We have
$\mathop{\mathrm{supp}}(V_{\varepsilon})=[-\varepsilon,\varepsilon]\times\big{(}\mathbb{R}\setminus]-d,d[\big{)}$
and the Fourier transforms of (29) and (30) can be written conveniently in
terms of the sinus cardinalis $\mathrm{sinc}(z)=\sin(z)/z=\widehat{\chi}(z)/2$
in the form

(31)
$\widehat{g_{\varepsilon}}(\xi,\eta)=\sqrt{\frac{2}{\varepsilon}}\,\widehat{\varphi_{\varepsilon}}(\xi-
p_{0})\,\mathrm{sinc}\Big{(}\frac{\eta}{\varepsilon}\Big{)}\quad\text{and}\quad\widehat{V_{\varepsilon}}(\xi,\eta)=\frac{1}{\varepsilon}\,\mathrm{sinc}(\varepsilon\xi)\big{(}\delta_{0}(\eta)-2d\,\mathrm{sinc}(d\eta)\big{)}.$

Note that $V_{\varepsilon}$ does not have compact support (and does not belong
to any $L^{p}$ with $1\leq p<\infty$). Therefore, we cannot apply the same
reasoning as in Subsections 2.3 and 2.4 assessing spectral properties and
scattering theory for the Hamiltonian
$H_{\varepsilon}=-\Delta+V_{\varepsilon}$. In particular, $V_{\varepsilon}$ is
not a short-range potential (as can be checked with test functions of tensor
product form) and not a relatively ($-\Delta$)-compact perturbation (e.g., the
image of a bounded sequence of $L^{2}$-orthonormal test functions with
supports contained in the interior of
$\mathop{\mathrm{supp}}(V_{\varepsilon})$ is not relatively compact in
$L^{2}$). Nevertheless, we can prove that the spectral properties are
analogous to those in Proposition 2.4.

######  Proposition 3.2.

If $H_{\varepsilon}$ is the Hamiltonian corresponding to the potential (30),
then we have $\sigma(H_{\varepsilon})=[0,\infty)$, while the point spectrum
$\sigma_{p}(H_{\varepsilon})$ is empty.

###### Proof.

_Step 1, determining the spectrum:_ At fixed $\varepsilon>0$ let us simplify
the notation temporarily and write
$V_{\varepsilon}(x,y)=\chi_{1}(x)(1-\chi_{2}(y))$ with
$\chi_{1}(x):=\chi(x/\varepsilon)/(2\varepsilon^{2})$ and
$\chi_{2}(y):=\chi(\frac{y}{d})$. Thus we have

$H_{\varepsilon}=-\Delta+\chi_{1}\otimes(1-\chi_{2})=\underbrace{-\partial_{x}^{2}+\chi_{1}\otimes
1-\partial_{y}^{2}}_{A}{-\underbrace{\chi_{1}\otimes\chi_{2}}_{B}},$

where $B$ is a compact perturbation of the self-adjoint operator $A$ (which
itself is a bounded perturbation of $-\Delta$; [RS:V2, Theorem X.12]) and
therefore does not change the essential spectrum (by the classical form of
Weyl’s theorem [RS:V4, Example 3 in Section XIII.4]), i.e.,

$\sigma_{\text{ess}}(H_{\varepsilon})=\sigma_{\text{ess}}(A).$

In addition, we observe that the operator $A$ has the form

$A=S_{1}\otimes I+I\otimes S_{2}\quad\text{with}\quad
S_{1}:=-\partial_{x}^{2}+\chi_{1},\;{S_{2}:=-\partial_{y}^{2}}.$

By [RS:V1, Theorem VIII.33] we therefore have the following relation between
the spectra

$\sigma(A)=\overline{\sigma(S_{1})+\sigma(S_{2})}.$

Each $S_{j}$ ($j=1,2$) is self-adjoint and positive with domain
$H^{2}(\mathbb{R})\subseteq L^{2}(\mathbb{R})$, thus
$\sigma(S_{j})\subseteq[0,\infty[$. Clearly,
$\sigma_{\text{ess}}(S_{2})=\sigma_{\text{ess}}(-\partial_{y}^{2})=[0,\infty[$.
As above, $\chi_{1}$ being a compact perturbation of the one-dimensional
Laplacian $-\partial_{x}^{2}$ also implies
$\sigma_{\text{ess}}(S_{1})=\sigma_{\text{ess}}(-\partial_{x}^{2})=[0,\infty[$.
Hence we have

$\sigma(S_{j})=\sigma_{\text{ess}}(S_{j})=[0,\infty[\quad(j=1,2),$

which then implies that

$\sigma(A)=[0,\infty[.$

Since there are no isolated points in the spectrum of $A$, we also obtain

$\sigma_{\text{ess}}(H_{\varepsilon})=\sigma_{\text{ess}}(A)=\sigma(A)=[0,\infty[$

and positivity of $H_{\varepsilon}$ yields
$\sigma(H_{\varepsilon})\subseteq[0,\infty[$, hence

$\sigma(H_{\varepsilon})=\sigma_{\text{ess}}(H_{\varepsilon})=[0,\infty[.$

_Step 2, $0$ cannot be an eigenvalue of $H_{\varepsilon}$:_ This follows by
the same reasoning as in Subsection 2.3.

_Step 3, $H_{\varepsilon}$ has no positive eigenvalues:_ We will show this by
suitable adaptations of the statements and proofs in [RS:V4, Corollary to and
Theorem XIII.58] and noting that [RS:V4, Theorem XIII.57] is applicable to the
(bounded) potential $V_{\varepsilon}$ given in (30) (so that the operator of
multiplication by $V_{\varepsilon}$ is $-\Delta$-bounded with relative bound
less than $1$).

_Substep 3(a), preparatory remarks:_ A basic observation is that the so-called
_radial vector field_ $r\partial_{r}:=x\partial_{x}+y\partial_{y}$ can be
applied as a differential operator in the sense of distributions to
$V_{\varepsilon}$ and yields, upon recalling
$\chi^{\prime}=\delta_{-1}-\delta_{1}$,
$\delta_{a}(\frac{x}{b})=b\,\delta_{ba}(x)$ for $b>0$, $a\in\mathbb{R}$, and
that $x\delta_{a}(x)=a\delta_{a}(x)$,

$r\partial_{r}V_{\varepsilon}(x,y)=\frac{1}{2\varepsilon^{2}}\left(x\partial_{x}+y\partial_{y}\right)\left(\chi\left(\frac{x}{\varepsilon}\right)\left(1-\chi\left(\frac{y}{d}\right)\right)\right)\\\
=-\frac{1}{2\varepsilon}\left(\delta_{-\varepsilon}(x)+\delta_{\varepsilon}(x)\right)\left(1-\chi\left(\frac{y}{d}\right)\right)+\frac{d}{2\varepsilon^{2}}\chi\left(\frac{x}{\varepsilon}\right)\left(\delta_{-d}(y)+\delta_{d}(y)\right).$

Thus we have

(32) $r\partial_{r}V_{\varepsilon}=-\mu_{1}+\mu_{2},$

where $\mu_{1}$ and $\mu_{2}$ are positive Borel measures on $\mathbb{R}^{2}$,
$\mu_{2}$ being finite with compact support
$[-\varepsilon,\varepsilon]\times\\{-d,d\\}$ ,and $\mu_{1}$ is concentrated on
the four vertical half-lines
$\\{-\varepsilon,\varepsilon\\}\times\left(\mathbb{R}\setminus[-d,d]\right)$.
Noting that $H^{2}(\mathbb{R}^{2})={\mathcal{F}}^{-1}\\{f\in
L^{2}(\mathbb{R}^{2})\mid z\mapsto(1+|z|^{2})f(z)\in
L^{2}(\mathbb{R}^{2})\\}\subseteq{\mathcal{F}}^{-1}L^{1}(\mathbb{R}^{2})$ (by
the Cauchy-Schwarz inequality; this is also a special case of [Hoermander:V1,
Lemma 7.9.2 and Theorem 7.9.3]), we deduce that for any $\psi\in
H^{2}(\mathbb{R}^{2})$, the non-negative function $|\psi|^{2}$ on
$\mathbb{R}^{2}$ is continuous, integrable, and
$\lim_{|z|\to\infty}|\psi(z)|^{2}=0$. In particular, we may apply the
distribution of order $0$ given in (32) to it and obtain a distributional
interpretation of a would-be $L^{2}$ inner product in case of smooth
$V_{\varepsilon}$, namely of
$\langle\psi|r\partial_{r}V_{\varepsilon}\psi\rangle$ in terms of
$\langle\mu_{2},|\psi|^{2}\rangle-\langle\mu_{1},|\psi|^{2}\rangle$. Moreover,
upon introducing the scaling $V_{\varepsilon}^{a}(z):=V_{\varepsilon}(az)$
($z\in\mathbb{R}^{2}$, $a>0$) and noting that $\lim_{a\to
1}\langle\psi|\frac{V_{\varepsilon}^{a}-V_{\varepsilon}}{a-1}\psi\rangle=-\langle\mu_{1},|\psi|^{2}\rangle+\langle\mu_{2},|\psi|^{2}\rangle$,
we may even state the following variation of the _virial theorem_ (cf. [RS:V4,
Theorem XIII.59]): If $\psi\in H^{2}(\mathbb{R}^{2})$ is a solution to the
eigenvalue equation $-\Delta\psi+V_{\varepsilon}\psi=\lambda\psi$, then

(33) $2\langle\psi|-\Delta\psi\rangle=2\langle\psi|(\lambda-
V_{\varepsilon})\psi\rangle=\left(\langle\mu_{2},|\psi|^{2}\rangle-\langle\mu_{1},|\psi|^{2}\rangle\right)=:\langle\psi|r\partial_{r}V_{\varepsilon}\psi\rangle.$

Recalling now from (32) that outside the compact support of $\mu_{2}$, the
distribution $r\partial_{r}V_{\varepsilon}$ equals $-\mu_{1}$, one might hope
also for an appropriate extension of the classical assertion in [RS:V4,
Corollary to Theorem XIII.58], saying that for a bounded and differentiable
potential $V$, the condition of repulsiveness ($\partial_{r}V\leq 0$) near
infinity implies non-existence of positive eigenvalues. We argue for this to
be true in the final paragraph of this subsubsection by sketching the few
changes required in the proof of [RS:V4, Theorem XIII.58], where compared to
the statement in [RS:V4, Theorem XIII.58] we consider $V_{1}=0$ and
$V_{2}=V_{\varepsilon}$, having to replace the conditions (ii) and (iii) in
the hypothesis of that theorem by appropriate alternative properties of
$V_{\varepsilon}$ along the way.

_Substep 3(b):_ Following the strategy in the proof of [RS:V4, Theorem
XIII.58], entering there at the third paragraph on page 227 (the one including
Equation (89)), let us suppose that $\lambda>0$ and $\psi\in
H^{2}(\mathbb{R}^{2})$ satisfy

$-\Delta\psi+V_{\varepsilon}\psi=\lambda\psi,$

i.e., $\psi$ is an eigenfunction for the positive eigenvalue $\lambda>0$, and
define

$w(r,\omega):=\sqrt{r}\,\psi(r\cos\omega,r\sin\omega)\quad(r>0,\omega\in[0,2\pi[).$

We may consider $w$ as a function $]0,\infty[\to L^{2}(S^{1})$, where we will
denote the inner product in the latter space by $(.\mid.)$. With the notation
$\tilde{f}(r,\omega):=f(r\cos\omega,r\sin\omega)$ we may write
$w(r)=\sqrt{r}\tilde{\psi}(r,.)$. Recalling
$\widetilde{(\Delta\psi)}=\partial_{r}^{2}\tilde{\psi}+\frac{1}{r}\partial_{r}\tilde{\psi}+\frac{1}{r^{2}}\partial_{\omega}^{2}\tilde{\psi}$,
an elementary calculation shows that the eigenvalue equation implies (denoting
$w^{\prime}=\partial_{r}w$)

$w^{\prime\prime}(r)+\frac{1}{r^{2}}\partial_{\omega}^{2}w(r)-\tilde{V_{\varepsilon}}(r)w(r)=-\frac{1}{4r^{2}}w(r)-\lambda
w(r).$

_Substep 3(c):_ In analogy to the role of Equations (90) and (91) on page 227
in [RS:V4]), one can show from the above relation (and the positivity of the
self-adjoint operator $-\partial_{\omega}^{2}$ in $L^{2}(S^{1})$) that the
function

$F(r):=(w^{\prime}(r)\mid
w^{\prime}(r))+\frac{1}{r^{2}}(w(r)\mid\partial_{\omega}^{2}w(r))+(w(r)\mid(\lambda-\tilde{V_{\varepsilon}}(r))w(r))$

satisfies for sufficiently large $r$ the inequality

(34)
$\frac{d}{dr}\left(rF(r)\right)\geq(1-\frac{1}{4r}){\left\|w^{\prime}(r)\right\|}_{L^{2}(S^{1})}^{2}+(\lambda-\frac{1}{4r}){\left\|w(r)\right\|}_{L^{2}(S^{1})}^{2}\\\
-(w(r)\mid\tilde{V_{\varepsilon}}(r)w(r))-(w(r)\mid
r\tilde{V_{\varepsilon}}^{\prime}(r)w(r)),$

where the final term still has to be interpreted in the sense of an
$r$-parametrized distribution acting on $S^{1}$, namely as $(w(r)\mid
r\tilde{V_{\varepsilon}}^{\prime}(r)w(r)):=\langle\widetilde{r\partial_{r}V_{\varepsilon}}(r),|w(r)|^{2}\rangle=-\langle\tilde{\mu_{1}}(r),|w(r)|^{2}\rangle$,
if $r$ is also larger than the diameter of the support of $\mu_{2}$; in fact,
upon defining $\tilde{\mu_{1}}$ by distributional pull-back to polar
coordinates one can check that we may write
$\langle\tilde{\mu_{1}},\tilde{f}\rangle=\int_{\sqrt{\varepsilon^{2}+d^{2}}}^{\infty}\langle\tilde{\mu_{1}}(r),\tilde{f}(r,.)\rangle\,dr$,
where $\tilde{\mu_{1}}(r)\in\mathcal{D}^{\prime}(S^{1})$ is given by a sum of
four terms of the form
$\delta_{\theta(r)}/(\varepsilon\sqrt{r^{2}-\varepsilon^{2}})$, where
$\theta(r)$ is one of the four angular values in $[0,2\pi[$, where the circle
of radius $r$ intersects the vertical lines constituting
$\mathop{\mathrm{supp}}(\mu_{1})=\\{-\varepsilon,\varepsilon\\}\times\left(\mathbb{R}\setminus[-d,d]\right)$.
Having settled for its meaning, we see immediatly that the fourth term on the
right-hand of inequality (34) gives the non-negative contribution $-(w(r)\mid
r\tilde{V_{\varepsilon}}^{\prime}(r)w(r))=\langle\tilde{\mu_{1}}(r),|w(r)|^{2}\rangle$
to the lower bound. Since the scalar factors of the first and second term are
clearly positive for sufficiently large $r>0$, it remains to investigate the
third term
$-(w(r)\mid\tilde{V_{\varepsilon}}(r)w(r))=-\int_{S^{1}}r|\tilde{\psi}(r,\omega)|^{2}\tilde{V_{\varepsilon}}(r,\omega)\,d\omega$,
which is clearly non-positive. However, since
$\mathop{\mathrm{supp}}(\tilde{V_{\varepsilon}}(r,.))$ is contained in two
small angular intervals of size proportional to $1/r$ and
$\lim_{|z|\to\infty}|\psi(z)|^{2}=0$, as noted in the discussion following
(32) above, we conclude that
$\lim_{r\to\infty}(w(r)\mid\tilde{V_{\varepsilon}}(r)w(r))=0$. Therefore, (34)
implies that one can find some $R>0$ guaranteeing

$\frac{d}{dr}\left(rF(r)\right)\geq\quad(r\geq R)$

and we may directly deduce

$rF(r)\geq RF(R)\quad(r\geq R).$

_Substep 3(d):_ Now from the above the reasoning at the bottom of page 228 in
[RS:V4] applies to yield that $F(r)\leq 0$ for all $r\geq R$ and that it
suffices to show $w(r)=0$ for large $r$ to finish the proof. The arguments on
pages 229-230 in [RS:V4] for the remaining part of the proof need adaptation
only at one point, namely on lines 11-14 near the middle of page 229, because
we do not have an analogue of the inequality $-(r^{2}V_{2})^{\prime}\geq 0$.
Instead, in estimating $\frac{d}{dr}(r^{2}G(m,r))$ for the analogue of the
function $G(m,r)$ introduced in [RS:V4] on page 229 (with the notation
$w_{m}=r^{m}w$, $m\geq 0$, as in [RS:V4] and $R$, $\sqrt{\lambda}$ in place of
$R_{1}$, $k$ used there, respectively), namely,

$G(m,r):={\left\|w_{m}^{\prime}(r)\right\|}_{L^{2}(S^{1})}^{2}+(\lambda-\frac{\lambda
R}{r}+\frac{m(m+1)}{r^{2}}){\left\|w_{m}(r)\right\|}_{L^{2}(S^{1})}^{2}\\\
+\frac{1}{r^{2}}(w_{m}(r)\mid\partial_{\omega}^{2}w_{m}(r)-(w_{m}(r)\mid\tilde{V_{\varepsilon}}(r)w_{m}(r),$

we inspect a combination of two specific terms occurring upon differentiating
$r^{2}G(m,r)$:

$2r\lambda(1-\frac{R}{2r}){\left\|w_{m}(r)\right\|}_{L^{2}(S^{1})}^{2}-(w_{m}(r)\mid(r^{2}\tilde{V_{\varepsilon}}(r))^{\prime}w_{m}(r))\\\
=r^{2m+1}2\lambda(1-\frac{R}{2r}){\left\|w(r)\right\|}_{L^{2}(S^{1})}^{2}-r^{2m+1}\Big{(}2(w(r)\mid\tilde{V_{\varepsilon}}(r)w(r))+(w(r)\mid
r\tilde{V_{\varepsilon}}^{\prime}(r)w(r))\Big{)}\\\
=r^{2m+1}\Big{(}2\lambda(1-\frac{R}{2r}){\left\|w(r)\right\|}_{L^{2}(S^{1})}^{2}-2(w(r)\mid\tilde{V_{\varepsilon}}(r)w(r))-(w(r)\mid
r\tilde{V_{\varepsilon}}^{\prime}(r)w(r))\Big{)}=:r^{2m+1}h(r)$

and recognize that by the reasoning detailed above we have $h(r)\geq 0$ for
sufficiently large $r>0$. ∎

Calculations for the approximation scheme described in Subsubsection 3.1.4
with initial data $g_{\varepsilon}$ and potential $V_{\varepsilon}$ as in
(29-30) would start out with $w_{0,\varepsilon}$ and $w_{1,\varepsilon}$ as
follows: With the solution formula (18) we immediately obtain, based on (31),

$\widehat{w_{0,\varepsilon}}(t,\xi,\eta)=\widehat{g_{\varepsilon}}(\xi,\eta)e^{-it(\xi^{2}+\eta^{2})}=\sqrt{\frac{2}{\varepsilon}}\,\widehat{\varphi_{\varepsilon}}(\xi-
p_{0})\,\mathrm{sinc}\Big{(}\frac{\eta}{\varepsilon}\Big{)}e^{-it(\xi^{2}+\eta^{2})}.$

Equation (31) also provides an explicit expression for
$\widehat{V_{\varepsilon}}$, which could be used to evaluate the following
formula determining $w_{1,\varepsilon}$,

$\widehat{w_{1,\varepsilon}}(t,\xi,\eta)=\widehat{w_{0,\varepsilon}}(t,\xi,\eta)-i\int_{0}^{t}e^{-i(t-\tau)(\xi^{2}+\eta^{2})}\,{\mathcal{F}}(V_{\varepsilon}w_{0,\varepsilon}(\tau,.))(\xi,\eta)\,d\tau\\\
=\sqrt{\frac{2}{\varepsilon}}\,\widehat{\varphi_{\varepsilon}}(\xi-
p_{0})\,\mathrm{sinc}\Big{(}\frac{\eta}{\varepsilon}\Big{)}e^{-it(\xi^{2}+\eta^{2})}-i\int_{0}^{t}e^{-i(t-\tau)(\xi^{2}+\eta^{2})}\,\widehat{V_{\varepsilon}}\ast{\mathcal{F}}(w_{0,\varepsilon}(\tau,.))(\xi,\eta)\,d\tau.$

## References