# Aharonov-Bohm effect and superoscillations

Fabrizio Colombo  (FC) Politecnico di Milano
Dipartimento di Matematica
Via E. Bonardi 9
20133 Milano, Italy<EMAIL_ADDRESS>,  Elodie Pozzi  (EP)
Department of Mathematics and Statistics
Saint Louis University
220 N. Grand Blvd, 63103 St Louis MO, USA<EMAIL_ADDRESS>,  Irene
Sabadini  (IS) Politecnico di Milano
Dipartimento di Matematica
Via E. Bonardi 9
20133 Milano, Italy<EMAIL_ADDRESS>and  Brett D. Wick  (BDW)
Department of Mathematics, Washington University - St. Louis, One Brookings
Drive
St. Louis, MO USA 63130-4899<EMAIL_ADDRESS>
###### Abstract.

The path-integral technique in quantum mechanics provides an intuitive
framework for comprehending particle propagation and scattering. Calculating
the propagator for the Aharonov-Bohm potential fits into the range of
potentials in multiply-connected spaces, with the propagator represented
through a series expansion. In this paper, we analyze the Schrödinger
evolution of superoscillations, showing the supershift properties of the
solution to the Schrödinger equation for this potential. Our proof is based on
the continuity of particular infinite order differential operators acting on
spaces of entire functions.

FC and IS are supported by MUR grant Dipartimento di Eccellenza 2023-2027.

BDW’s research supported in part by National Science Foundation DMS awards
#2349868 and #2054863 and Australian Research Council – DP 220100285.

EP and BDW would like to thank Politecnio di Milano for hospitality during
visits in Spring 2024 when this paper was written.

AMS Classification: 35A20, 35A08.

Keywords: Superoscillations, supershift, infinite order differential
operators, spaces of entire functions.

today

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on Superoscillations and Supershifts
  3. 3 Fresnel Integral Representation of the Aharonov-Bohm Propagator
  4. 4 Aharonov-Bohm Infinite Order Differential Operators and  the Supershift Property

##  1\. Introduction

The phenomenon of superoscillations has attracted attention across
mathematics, physics and engineering, with contributions from various authors
in quantum mechanics, optics, signal processing, and wave theory. Roughly
speaking superoscillations are functions that can oscillate beyond their
highest Fourier frequency. The archetypal superoscillating function resulting
from weak values is expressed as:

$F_{n}(x,a)=\sum_{j=0}^{n}C_{j}(n,a)e^{i(1-2j/n)x},\ \ x\in\mathbb{R},$ (1)

where

$C_{j}(n,a):=\binom{n}{j}\left(\frac{1+a}{2}\right)^{n-j}\left(\frac{1-a}{2}\right)^{j}$
(2)

and $a>1$ and if we fix $x\in\mathbb{R}$ and we let $n$ go to infinity, we
obtain that

$\lim_{n\to\infty}F_{n}(x,a)=e^{iax}.$

This property holds considerable promise for advancing science and technology.
Important contributions can be found in [36]-[39]. Over the past 15 years,
significant advancements have been made in function theory, driven by insights
from quantum mechanics and the Schrödinger evolution of such functions. The
discovery of superoscillatory phenomena in quantum mechanics stemmed from weak
measurements [1, 11], pioneered by Yakir Aharonov and his collaborators.
Understanding the evolution of the quantum state is of clear interest and
using weak measurements, a type of quantum measurement that minimally disturbs
a quantum system while extracting information, allows one to do so. When
considering a general potential, the evolution of superoscillations is
encapsulated in a new concept known as the “supershift property,” which
encompasses superoscillations as a specific case within its framework. The
literature on superoscillations is nowadays quite large, and without claiming
completeness we mention the papers [13], [31]-[33] and [42, 43, 45].

The class of superoscillations has been investigated also from the
mathematical viewpoint, focusing on its function theory, but a large part of
the results are associated with the study of the evolution of
superoscillations by quantum field equations with particular attention to the
Schrödinger equation. In the papers [2]-[9], [15]-[19], [22]-[26] and [29, 30,
46, 47], one can find an up-to-date panorama of this field. In order to have
an overview of the techniques developed in the recent years to study the
evolution of superoscillations and their function theory, we refer the reader
to the introductory papers [10, 12, 14] and [40]. Finally, we mention the
Roadmap on superoscillations, see [21], where some of the most recent advances
in superoscillations and their applications to technology are explained by
leading experts in this field.

In this paper we investigate the evolution of superoscillating functions in
the Aharonov-Bohm context. An idealized arrangement commonly used to observe
the Aharonov-Bohm effect typically involves an infinitely long solenoid, a
source emitting charged particles, for example electrons, and a screen
positioned behind the solenoid to capture interference patterns. The solenoid
is designed to be shielded to ensure that electrons are prohibited from
entering the space it occupies. The Green’s function for the Aharonov-Bohm
effect in dimension two is given by the series expansion

$K(x^{\prime\prime},x^{\prime},t)=\frac{M}{2\pi\hbar t}e^{\frac{iM}{2\hbar
t}((r^{\prime})^{2}+(r^{\prime\prime})^{2})}\sum_{n=-\infty}^{\infty}e^{in(\phi^{\prime\prime}-\phi^{\prime})}I_{|n-\xi|}\Big{(}\frac{Mr^{\prime}r^{\prime\prime}}{i\hbar
t}\Big{)}$ (3)

where $x=(r,\phi)$, $I_{|n-\xi|}$ is the Bessel function and the series over
the integer numbers takes into account the winding paths around the prohibited
region. The path-integral approach to quantum mechanics, due to Feynman, see
[34, 35], provides the most intuitive theoretical framework for describing
processes involving the propagation and scattering of particles. Behind the
computation of the exact form of the propagator are several topological
challenges that arise when particles propagate in multiply-connected spaces.
The first to analyze this fact was Schulman [48] and, in a more systematic
manner, Laidlaw and Morette-de Witt [41]. They have provided general
guidelines for computing the propagator in arbitrary multiply-connected
spaces, see also [44].

Given the propagator (3), the solution of the Schrödinger equation with the
Aharonov-Bohm potential is given by

$\psi(x^{\prime\prime},t)=\int_{\mathbb{R}^{2}}K(x^{\prime\prime},x^{\prime},t)\psi(x^{\prime})dx^{\prime},$

which can be expressed in polar coordinates $x=(r,\phi)$ as

$\psi(r,\phi,t)=\int_{0}^{2\pi}\int_{0}^{\infty}K(r,\phi;\rho,\theta,t)\
\psi_{0}(\rho,\theta)\ \rho d\rho\ d\theta.$

The Green’s function

$K(x^{\prime\prime},x^{\prime},t)=K(r,\phi;\rho,\theta,t)=\frac{M}{2\pi\hbar
t}e^{\frac{iM}{2\hbar
t}(\rho^{2}+r^{2})}\sum_{n=-\infty}^{\infty}e^{in(\phi-\theta)}I_{|n-\xi|}\Big{(}\frac{M\
r\ \rho}{i\hbar t}\Big{)}$

can be rewritten using the modified Bessel functions,
$I_{\alpha}(x)=i^{-\alpha}J_{\alpha}(ix)$, in the equivalent form

$K(x^{\prime\prime},x^{\prime},t)=K(r,\phi;\rho,\theta,t)=\frac{M}{2\pi\hbar
t}e^{\frac{iM}{2\hbar
t}(\rho^{2}+r^{2})}\sum_{n=-\infty}^{\infty}e^{in(\phi-\theta)}i^{-|n-\xi|}J_{|n-\xi|}\Big{(}\frac{M\
r\ \rho}{\hbar t}\Big{)}.$

Consider now the initial datum

$\psi_{0}(r,\phi)=e^{ig(a)x+ih(a)y}=e^{ig(a)r\cos\phi+ih(a)r\sin\phi}$

where $g$ and $h$ are entire functions, the solution to the Schrödinger
equation becomes

$\psi_{a,b}(r,\phi,t)=\int_{0}^{2\pi}\int_{0}^{\infty}K(r,\phi;\rho,\theta,t)e^{ig(a)\rho\cos\theta+ih(a)\rho\sin\theta}\
\rho d\rho\ d\theta.$

The aim of this paper is to establish the supershift property for the Cauchy
problem for the Schrödinger equation with the Aharonov-Bohm potential with a
very general initial superoscillatory datum

$Y_{n}(x,y,a)=\sum_{j=0}^{n}C_{j}(n,a)e^{ig(1-2j/n)x}e^{ih(1-2j/n)y},$ (4)

where $x,y\in\mathbb{R}$, the coefficients $C_{j}(n,a)$ in (2) are related to
the weak measurements, and $g$ and $h$, are given entire functions, monotone
increasing in $a$, of the following form:

$g(\lambda)=\sum_{u=0}^{\infty}g_{u}\lambda^{u},\ \ \
h(\lambda)=\sum_{v=0}^{\infty}h_{v}\lambda^{v}.$

One can prove that

$\lim_{n\to\infty}Y_{n}(x,y,a)=e^{ig(a)x}e^{ih(a)y}.$

Precisely, we study the supershift property of the solution given by

$\Psi_{n}(r,\phi,t)=\sum_{j=0}^{n}C_{j}(n,a)\psi_{g\left(1-\frac{2j}{n}\right),h\left(1-\frac{2j}{n}\right)}(r,\phi,t)$

for all $r\in(0,\infty)$ $\phi\in[0,2\pi]$, $t>0$, where

$\psi_{g(1-2j/n),h(1-2j/n)}(r,\phi,t):=\int_{0}^{2\pi}\int_{0}^{\infty}K(r,\phi;\rho,\theta,t)e^{ig(1-2j/n)\rho\cos\theta+ih(1-2j/n)\rho\sin\theta}\
\rho d\rho\ d\theta.$

The plan of the paper is as follows. In Section 2 we summarize the main
concepts of superoscillations and the notion of supershift property of the
solution of the Schrödinger equation. In Section 3 we represent the
oscillating integral in the Aharonov-Bohm propagator in terms of Fresnel
integrals and we give the main estimates in order to guarantee the limit
exchanges in the proof of the main results. In Section 4 we state and prove
our main results based on the so-called Aharonov-Bohm infinite order
differential operator. In particular the continuity of this operator on the
space of holomorphic functions with suitable growth conditions is a crucial
tool to prove the supershift property of the solution of the Schrödinger
equation in the Aharonov-Bohm field.

##  2\. Preliminaries on Superoscillations and Supershifts

We begin by collecting all the necessary background for the paper regarding
the precise definition of superoscillatory functions and the crucial notion of
supershift which encompasses superoscillations as a particular case.

######  Definition 2.1.

A generalized Fourier sequence is a sequence of the form

$f_{n}(x):=\sum_{j=0}^{n}Z_{j}(n,a)e^{ih_{j}(n)x},\ \ \ n\in\mathbb{N},\ \ \
x\in\mathbb{R},$ (5)

where $a\in\mathbb{R}$, $Z_{j}(n,a)$ and $h_{j}(n)$ are complex and real
valued functions of the variables $n,a$ and $n$, respectively. The sequence
(5) is said to be a superoscillating sequence if $\sup_{j,n}|h_{j}(n)|\leq 1$
and there exists an open subset of $\mathbb{R}$, which will be called a
superoscillation set, on which $f_{n}(x)$ converges locally uniformly to
$e^{ig(a)x}$, where $g$ is a continuous real valued function in an open subset
of $\mathbb{R}$ such that $|g(a)|>1$.

######  Definition 2.2 (Supershift).

Let $\mathcal{I}\subseteq\mathbb{R}$ be an interval with
$[-1,1]\subset\mathcal{I}$ and let
$\varphi:\,\mathcal{I}\times\mathbb{R}\to\mathbb{R}$ be a continuous function
on $\mathcal{I}$. Set

$\varphi_{h}(x):=\varphi(h,x),\ \ h\in\mathcal{I},\ \ x\in\mathbb{R}$

and consider a sequence of points $\\{h_{j}(n)\\}$ such that
$h_{j}(n)\in[-1,1]\ \ {\rm for}\ \ j=0,\ldots,n\ \ {\rm and}\ \
n\in\mathbb{N}_{0}$. Define the functions

$\psi_{n}(x):=\sum_{j=0}^{n}c_{j}(n)\varphi_{h_{j}(n)}(x),$ (6)

where $\\{c_{j}(n)\\}$ is a sequence of complex numbers for $j=0,\ldots,n$ and
$n\in\mathbb{N}_{0}$. If

$\lim_{n\to\infty}\psi_{n}(x)=\varphi_{a}(x)$

for some $a\in\mathcal{I}$ with $|a|>1$, we say that the functions
$\psi_{n}(x)$, for $x\in\mathbb{R}$, admits a supershift.

######  Remark 2.3.

The term supershift comes from the fact that the interval $\mathcal{I}$ can be
arbitrarily large (it could be $\mathbb{R}$) and that the constant $a$ can be
arbitrarily far away from the interval $[-1,1]$ where the functions
$\varphi_{h_{j(n)}}(\cdot)$ are indexed, see (6).

The definition of superoscillations can be extended to several variables.

######  Definition 2.4 (Generalized Fourier sequence in several variables).

Let $d\in\mathbb{N}$ be such that $d\geq 2$, and assume that
$(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}$. Let $\\{h_{j,\ell}(n)\\}$,
$j=0,\ldots,n$ for $n\in\mathbb{N}_{0}$, be real-valued sequences for
$\ell=1,\ldots,d$. A generalized Fourier sequence in several variables is a
sequence of functions of the form

$F_{n}(x_{1},\ldots,x_{d})=\sum_{j=0}^{n}c_{j}(n)e^{ix_{1}h_{j,1}(n)}e^{ix_{2}h_{j,2}(n)}\cdots
e^{ix_{d}h_{j,d}(n)},$ (7)

where $\\{c_{j}(n)\\}$, for $j=0,\ldots,n$ and $n\in\mathbb{N}_{0}$, is a
complex-valued sequence.

######  Definition 2.5 (Superoscillating sequence).

Let $d\in\mathbb{N}$ be such that $d\geq 2$. A generalized Fourier sequence in
several variables $F_{n}(x_{1},\ldots,x_{d})$, is said to be a
superoscillating sequence if

$\sup_{j=0,\ldots,n,\ n\in\mathbb{N}}\ |h_{j,\ell}(n)|\leq 1,\ \ {\rm for}\
\ell=1,\ldots,d,$

and there exists an open subset of $\mathbb{R}^{d}$, which will be called a
superoscillation set, on which $F_{n}(x_{1},\ldots,x_{d})$ converges locally
uniformly to $e^{ix_{1}g_{1}(a)}e^{ix_{2}g_{2}(a)}\cdots e^{ix_{d}g_{d}(a)}$,
where $a$ belongs to an open subset $U$ of $\mathbb{R}$, the $g_{\ell}$ are
continuous functions of real variables whose domain contains $U$ and
$|g_{\ell}(a)|>1$ for $\ell=1,\ldots,d$ and $a\in U$.

In our discussion, the concept of supershift, which extends the idea of
superoscillation, illustrates how sampling a function within an interval
enables the computation of its values far beyond that interval. In the paper
[27], the relationship between supershift and real analyticity was explored.
Leveraging Serge Bernstein’s classical theorem, it was demonstrated that real
analyticity in a complex-valued function implies a robust form of supershift.
Conversely, employing a parametric version of a result by Leonid Kantorovitch,
it was established that the reverse implication does not hold in general.
Another work, see [28], delves into the notion of irregular sampling and the
correlation between supershift and real analyticity across the entire real
line. These findings reveal that any entire function, when restricted to
$\mathbb{R}$, exhibits a supershift, whereas the converse is generally false.

We next introduce the space of functions that plays an important role in our
analysis.

######  Definition 2.6.

The space $A_{1}$ is the complex algebra of entire functions such that there
exists $B>0$ with

$\sup\limits_{w\in\mathbb{C}}\left(|f(w)|\,\exp(-B|w|)\right)<+\infty.$ (8)

The space $A_{1}$ has a rather complicated topology, see e.g. [20], since it
is a linear space obtained via an inductive limit. For our purposes, it is
enough to consider, for any fixed $B>0$, the set $A_{1,B}$ of functions $f$
satisfying (8), and to observe that

$\|f\|_{B}:=\sup\limits_{w\in\mathbb{C}}\big{(}|f(w)|\,\exp(-B|w|)\big{)}$

defines a norm on $A_{1,B}$, called the $B$-norm. One can prove that $A_{1,B}$
is a Banach space with respect to this norm.

Moreover, for $f$ and a sequence $\\{f_{n}\\}$ belonging to $A_{1}$, $f_{n}$
converges to $f$ in $A_{1}$ if and only if there exists $B$ such that
$f,f_{n}\in A_{1,B}$ and

$\lim\limits_{n\rightarrow\infty}\sup\limits_{w\in\mathbb{C}}\big{|}f_{n}(w)-f(w)\big{|}e^{-B|w|}=0.$

To prove our main results we need an important lemma that characterizes the
coefficients of entire functions with growth conditions.

######  Lemma 2.7.

The function

$f(z)=\sum_{j=0}^{\infty}f_{j}z^{j}$

belongs to $A_{1}$ if and only if there exists $C_{f}>0$ and $b>0$ such that

$|f_{j}|\leq C_{f}\frac{b^{j}}{\Gamma(j+1)}=C_{f}\frac{b^{j}}{j!}.$

Lemma 2.7 has been proved in [15] and is a crucial fact in what follows. With
this notation and these definitions we can make the notion of continuity on
the space $A_{1}$ explicit, see [15]. A linear operator $\mathcal{U}:A_{1}\to
A_{1}$ is continuous if and only if for any $B>0$ there exists $B^{\prime}>0$
and $C>0$ such that

$\mathcal{U}(A_{1,B})\subset A_{1,B^{\prime}}\ {\rm
and}\qquad\|\mathcal{U}(f)\|_{B^{\prime}}\leq C\|f\|_{B},\qquad\forall f\in
A_{1,B}.$ (9)

We will also be in need of the following estimate for the Gamma function.

######  Lemma 2.8.

Let $q\in[1,\infty)$. Then we have

$\Gamma\left(\frac{n}{q}+1\right)\leq(n!)^{1/q}.$

###### Proof.

This is a direct consequence of Hölder’s inequality. Let $p$ and $q$ satisfy
$\frac{1}{p}+\frac{1}{q}=1$, we observe that

$\begin{split}\Gamma\left(\frac{n}{q}+1\right)&=\int_{0}^{\infty}e^{-t}t^{\frac{n}{q}}\,dt=\int_{0}^{\infty}e^{-t\left(\frac{1}{p}+\frac{1}{q}\right)}t^{\frac{n}{q}}\,dt\\\
&\leq\left(\int_{0}^{\infty}e^{-t}t^{n}\
\,dt\right)^{\frac{1}{q}}\left(\int_{0}^{\infty}e^{-t}\,dt\right)^{1/p}=\left(\int_{0}^{\infty}e^{-t}t^{n}\
\,dt\right)^{\frac{1}{q}}=(n!)^{\frac{1}{q}}.\end{split}$

∎

##  3\. Fresnel Integral Representation of the Aharonov-Bohm Propagator

We now want to represent via a power series the solution of the Schrödinger
equation with the exponential function as initial datum, which is a crucial
step in order to study superoscillations. The Green’s function for the
Aharonov-Bohm effect in two dimensions is

$K(x^{\prime\prime},x^{\prime},t)=\frac{M}{2\pi\hbar t}e^{\frac{iM}{2\hbar
t}((r^{\prime})^{2}+(r^{\prime\prime})^{2})}\sum_{n=-\infty}^{\infty}e^{in(\phi^{\prime\prime}-\phi^{\prime}))}I_{|n-\xi|}\Big{(}\frac{Mr^{\prime}r^{\prime\prime}}{i\hbar
t}\Big{)}$

where $x=(r,\phi)$ and $\xi$ is a real number that is proportional to the
magnetic flux of the infinite solenoid centered at the origin of the reference
system. The solution to the Schrödinger equation with initial data $\psi$ is
given by

$\psi(x^{\prime\prime},t)=\int_{\mathbb{R}^{2}}K(x^{\prime\prime},x^{\prime},t)\psi(x^{\prime})dx^{\prime}$

which can be written in polar coordinates as

$\psi(r,\phi,t)=\int_{0}^{2\pi}\int_{0}^{\infty}K(r,\phi;\rho,\theta,t)\
\psi_{0}(\rho,\theta)\ \rho d\rho\ d\theta.$

Here the kernel $K(x^{\prime\prime},x^{\prime},t)$ has the following
equivalent representations:

$\displaystyle K(x^{\prime\prime},x^{\prime},t)$
$\displaystyle=K(r,\phi;\rho,\theta,t)$ $\displaystyle=\frac{M}{2\pi\hbar
t}e^{\frac{iM}{2\hbar
t}(\rho^{2}+r^{2})}\sum_{n=-\infty}^{+\infty}e^{in(\phi-\theta)}I_{|n-\xi|}\Big{(}\frac{M\
r\ \rho}{i\hbar t}\Big{)}$ $\displaystyle=\frac{M}{2\pi\hbar
t}e^{\frac{iM}{2\hbar
t}(\rho^{2}+r^{2})}\sum_{n=-\infty}^{\infty}e^{in(\phi-\theta)}i^{-|n-\xi|}J_{|n-\xi|}\Big{(}\frac{M\
r\ \rho}{\hbar t}\Big{)}$ $\displaystyle=\frac{M}{2\pi\hbar
t}e^{\frac{iM}{2\hbar t}(\rho^{2}+r^{2})}F_{\xi}(r,\phi,\theta,t,\rho)$

where we have defined:

$F_{\xi}(r,\phi,\theta,t,\rho):=\sum_{n=-\infty}^{\infty}e^{in(\phi-\theta)}i^{-|n-\xi|}J_{|n-\xi|}\Big{(}\frac{M\
r\ \rho}{\hbar
t}\Big{)}=\sum_{n=-\infty}^{\infty}e^{in(\phi-\theta)}I_{|n-\xi|}\Big{(}\frac{M\
r\ \rho}{i\hbar t}\Big{)}$ (10)

and used the modified Bessel functions
$I_{\alpha}(x)=i^{-\alpha}J_{\alpha}(ix)$.

Important for the remaining portions of the argument, we will need to obtain
some estimates on the function $F_{\xi}$ and its growth as a function of $z$.
We turn to this now.

######  Lemma 3.1 (Estimate on the function $F_{\xi}$).

Let $F_{\xi}(r,\phi,\theta,t,\rho)$ be the function defined in (10). For
$\rho$ real and non-negative there exists a positive constant $c$ such that

$\left|F_{\xi}(r,\phi,\theta,t,\rho)\right|\leq q(\rho)e^{{c}\rho}\ \ {\rm
and}\ \ \left|F_{\xi}(r,\phi,\theta,t,\rho e^{i\pi/4})\right|\leq
q(\rho)e^{{c}\rho}$ (11)

where $q(\rho)$ is a function that has at most polynomial growth in $\rho$ and
the coefficients in the function and in $c$ depend upon $r,M,t$ and $\xi$.

###### Proof.

We prove just the first inequality since the second one is a direct
consequence of the first one. Recall that for $\alpha\in\mathbb{R}$ the
modified Bessel function of the first kind is given by:

$J_{\alpha}(x)=\sum_{l=0}^{\infty}\frac{(-1)^{l}}{l!\,\Gamma(l+1+\alpha)}\left(\frac{x}{2}\right)^{2l+\alpha}.$
(12)

The function we encounter, and need to estimate, is given by,

$\displaystyle F_{\xi}(r,\phi,\theta,t,\rho)$
$\displaystyle=\sum_{n=-\infty}^{\infty}i^{-|n-\xi|}e^{in(\phi-\theta)}J_{|n-\xi|}\Big{(}\frac{Mr\,\rho}{2\hbar
t}\Big{)}$
$\displaystyle=\sum_{n=-\infty}^{\infty}i^{-|n-\xi|}e^{in(\phi-\theta)}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{|n-\xi|}\sum_{l=0}^{\infty}\frac{(-1)^{l}}{l!\,\Gamma(|n-\xi|+l+1)}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{2l}.$

Taking absolute values and using that
$\Gamma(\alpha+1)\Gamma(\beta+1)\leq\Gamma(\alpha+\beta+2)$ for any
$\alpha,\beta\in\mathbb{R}$ we have

$\displaystyle\left|F_{\xi}(r,\phi,\theta,t,\rho)\right|$
$\displaystyle\leq\sum_{n=-\infty}^{\infty}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{|n-\xi|}\sum_{l=0}^{\infty}\frac{1}{l!\,\Gamma(|n-\xi|+l+1)}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{2l}$
$\displaystyle\leq\sum_{n=-\infty}^{\infty}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{|n-\xi|}\sum_{l=0}^{\infty}\frac{l+1+|n-\xi|}{(l!)^{2}\,\Gamma(|n-\xi|+1)}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{2l}$ $\displaystyle\leq I_{0}\left(\frac{Mr\rho}{2\hbar
t}\right)\left\\{\sum_{n=-\infty}^{\infty}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{|n-\xi|}\frac{1+\left|n-\xi\right|}{\Gamma(\left|n-\xi\right|+1)}+\sum_{n=-\infty}^{\infty}\frac{\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{|n-\xi|}}{\Gamma(\left|n-\xi\right|+1)}\right\\},$

where we recall that $\displaystyle
I_{0}(x)=\sum_{\ell=0}^{\infty}\frac{1}{(\ell!)^{2}}\left(\frac{x}{2}\right)^{2\ell}$.
Observe that
$\displaystyle\sum_{\ell=0}^{\infty}\frac{\ell}{(\ell!)^{2}}\left(\frac{x}{2}\right)^{2\ell}\leq\sum_{\ell=0}^{\infty}\frac{1}{(\ell!)^{2}}x^{2\ell}=I_{0}(2x)$
and $I_{0}(x)\leq I_{0}(2x)$. We start estimating the first sum. Splitting the
sum in an obvious fashion, we have

$\displaystyle\sum_{n=-\infty}^{+\infty}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{|n-\xi|}\frac{1+\left|n-\xi\right|}{\Gamma(\left|n-\xi\right|+1)}$
$\displaystyle=\sum_{n>\xi}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{n-\xi}\frac{1+n-\xi}{\Gamma(n-\xi+1)}+\sum_{n\leq\xi}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{n-\xi}\frac{1-n+\xi}{\Gamma(-n+\xi+1)}.$

Let $\xi=\xi_{i}+\xi_{f}$ where $\xi_{i}\in\mathbb{Z}$ and $\xi_{f}\in[0,1]$.
It follows that

$\displaystyle\sum_{n>\xi}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{n-\xi}\frac{1+n-\xi}{\Gamma(n-\xi+1)}$
$\displaystyle=\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{-\xi_{f}}\sum_{n=\xi_{i}+1}^{\infty}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{n-\xi_{i}}\frac{1+n-\xi_{i}-\xi_{f}}{\Gamma(n-\xi_{i}+1-\xi_{f})}$
$\displaystyle=\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{-\xi_{f}}\sum_{k=1}^{\infty}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{k}\frac{1+k-\xi_{f}}{\Gamma(k+1-\xi_{f})}$
$\displaystyle\leq\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{-\xi_{f}}\sum_{k=1}^{\infty}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{k}\frac{1+k-\xi_{f}}{(k-1)!}$
$\displaystyle\leq\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{-\xi_{f}}\sum_{k=1}^{\infty}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{k}\frac{1+k}{(k-1)!}$ $\displaystyle=\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{1-\xi_{f}}\left(2+\frac{Mr\,\rho}{4\hbar
t}\right)e^{\frac{Mr\,\rho}{4\hbar t}},$

where we used the substitution $k=n-\xi_{i}$ and that
$\Gamma(k+1-\xi_{f})\geq\Gamma(k)=(k-1)!$. In a similar way, the second sum
becomes

$\displaystyle\sum_{n\leq\xi}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{\xi-n}\frac{1+\xi-n}{\Gamma(\xi-n+1)}$
$\displaystyle=\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{\xi_{f}}\sum_{n=-\infty}^{\xi_{i}}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{\xi_{i}-n}\frac{1+\xi_{f}+\xi_{i}-n}{\Gamma(\xi_{f}+\xi_{i}-n+1)}$
$\displaystyle=\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{\xi_{f}}\sum_{k=0}^{\infty}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{k}\frac{1+\xi_{f}+k}{\Gamma(\xi_{f}+k+1)}.$

As $\Gamma(k+1+\xi_{f})\geq\Gamma(k+1)=k!$ and $0\leq\xi_{f}<1$, we get that

$\displaystyle\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{\xi_{f}}\sum_{k=0}^{\infty}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{k}\frac{1+\xi_{f}+k}{\Gamma(\xi_{f}+k+1)}$
$\displaystyle\leq\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{\xi_{f}}\sum_{k=0}^{\infty}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{k}\frac{k+2}{k!}$ $\displaystyle\leq 2\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{\xi_{f}}\sum_{k=0}^{\infty}\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{k}\frac{k+1}{k!}$ $\displaystyle=2\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{\xi_{f}}\left(1+\frac{Mr\,\rho}{4\hbar
t}\right)e^{\frac{Mr\,\rho}{4\hbar t}}.$

We take the same approach to deal with the second sum, giving

$\begin{split}\sum_{n=-\infty}^{\infty}\frac{\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{|n-\xi|}}{\Gamma(\left|n-\xi\right|+1)}&=\sum_{n\leq\xi}\frac{\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{\xi-n}}{\Gamma(\xi-n+1)}+\sum_{n>\xi}\frac{\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{n-\xi}}{\Gamma(n-\xi+1)}\\\ &=\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{\xi_{f}}\sum_{k=0}^{\infty}\frac{\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{k}}{\Gamma(\xi_{f}+k+1)}+\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{-\xi_{f}}\sum_{k=1}^{\infty}\frac{\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{k}}{\Gamma(k+1-\xi_{f})}\end{split}$

so that

$\begin{split}\sum_{n=-\infty}^{\infty}\frac{\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{|n-\xi|}}{\Gamma(\left|n-\xi\right|+1)}&\leq\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{\xi_{f}}\sum_{k=0}^{\infty}\frac{\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{k}}{k!}+\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{-\xi_{f}}\sum_{k=1}^{\infty}\frac{\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{k}}{(k-1)!}\\\ &=e^{\frac{Mr\,\rho}{4\hbar
t}}\left[\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{\xi_{f}}+\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{1-\xi_{f}}\right].\end{split}$

Collecting these estimates we finally obtain

$\displaystyle|F_{\xi}(r,\phi,\theta,t,\rho)|$ $\displaystyle\leq
e^{\frac{Mr\,\rho}{4\hbar t}}I_{0}\left(\frac{Mr\rho}{2\hbar
t}\right)\left\\{\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{1-\xi_{f}}\right.\left.\left(2+\frac{Mr\,\rho}{4\hbar
t}\right)+2\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{\xi_{f}}\left(1+\frac{Mr\,\rho}{4\hbar t}\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.+\left[\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{\xi_{f}}+\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{1-\xi_{f}}\right]\right\\}$ $\displaystyle\leq
e^{\frac{Mr\,\rho}{4\hbar t}}I_{0}\left(\frac{Mr\rho}{2\hbar
t}\right)\left\\{\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{1-\xi_{f}}\left(3+\frac{Mr\rho}{4\hbar
t}\right)+\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{\xi_{f}}\left(3+\frac{Mr\rho}{2\hbar t}\right)\right\\}.$

This gives the statement by recalling the known, alternate expression, of the
Bessel function, [49, p. 181], we have

$I_{\alpha}(x)=\frac{1}{\pi}\int_{0}^{\pi}e^{x\cos\theta}\cos\theta
d\theta-\frac{\sin(\alpha\pi)}{\pi}\int_{0}^{\infty}e^{-x\cosh(t)-\alpha
t}dt,\quad\operatorname{Re}(x)>0$

from which we deduce that there exists a constant $K>0$ such that
$|I_{0}(x)|\leq Ke^{x}$ for $x>0$. ∎

We want to use the Green’s function $K(x^{\prime\prime},x^{\prime},t)$ to
evolve the data $\psi_{0}(r,\phi)$. This involves an integral that is
difficult to compute and whose convergence is, a priori, in question. However,
using Fresnel integrals (changing the integral to one that is equivalent via
an argument in complex analysis), we arrive at an expression for the evolution
of the initial data that is easier to work with. This is the content of the
next proposition.

######  Proposition 3.2 (Fresnel integral representation of the propagator).

Let $a,b\in\mathbb{R}$ and set

$\psi_{0}(r,\phi):=e^{iax+iby}=e^{iar\cos\phi+ibr\sin\phi}.$

Then the solution

$\psi_{a,b}(r,\phi,t)=\frac{M}{2\pi\hbar
t}\int_{0}^{2\pi}\int_{0}^{\infty}e^{\frac{iM}{2\hbar
t}(\rho^{2}+r^{2})}F_{\xi}(r,\phi,\theta,t,\rho)e^{ia\rho\cos\theta+ib\rho\sin\theta}\
\rho d\rho d\theta$

can be written as a Fresnel integral as

$\psi_{a,b}(r,\phi,t)=i\frac{M}{2\pi\hbar t}e^{\frac{iM}{2\hbar
t}r^{2}}\int_{0}^{2\pi}\int_{0}^{\infty}F_{\xi}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}})e^{\frac{-M}{2\hbar
t}u^{2}}e^{iue^{i\pi/4}(a\cos\theta+b\sin\theta)}udud\theta.$ (13)

###### Proof.

We consider

$\displaystyle\psi_{a,b}(r,\phi,t)$ $\displaystyle=\frac{M}{2\pi\hbar
t}\int_{0}^{2\pi}\int_{0}^{\infty}e^{\frac{iM}{2\hbar
t}(\rho^{2}+r^{2})}F_{\xi}(r,\phi,\theta,t,\rho)e^{ia\rho\cos\theta+ib\rho\sin\theta}\
\rho d\rho d\theta$ $\displaystyle=\frac{M}{2\pi\hbar t}e^{\frac{iM}{2\hbar
t}r^{2}}\int_{0}^{2\pi}\int_{0}^{\infty}e^{\frac{iM}{2\hbar
t}\rho^{2}}F_{\xi}(r,\phi,\theta,t,\rho)e^{ia\rho\cos\theta+ib\rho\sin\theta}\
\rho d\rho d\theta.$

Fix $\theta\in[0,2\pi]$. We now interpret the integral in $\rho$ as a Fresnel
integral with the substitution $\rho\to z\in\mathbb{C}$. Define

$g(z):=e^{\frac{iM}{2\hbar
t}z^{2}}F_{\xi}(r,\phi,\theta,t,z)e^{iaz\cos\theta+ibz\sin\theta}\ z.$

The function $g(z)$ is holomorphic (and entire) and so for any closed curve
$\gamma$ we have:

$\int_{\gamma}g(z)dz=\int_{\gamma}e^{\frac{iM}{2\hbar
t}z^{2}}F_{\xi}(r,\phi,\theta,t,z)e^{iz(a\cos\theta+b\sin\theta)}\ zdz=0.$

We proceed by considering the path of integration

$\gamma=\gamma_{1}\cup\gamma_{2}\cup\gamma_{3}$

where, for $z=ue^{i\alpha}$, and $R>0$ we set

$\gamma_{1}=\\{0\leq u\leq R,\ \ \alpha=0\\},\ \ \ \ \gamma_{2}=\\{u=R,\ \
0\leq\alpha\leq\pi/4\\},\ \ \ \ -\gamma_{3}=\\{0\leq u\leq R,\ \
\alpha=\pi/4\\},$

where the sign in front of $\gamma_{3}$ takes into account the orientation. We
will exploit the fact that

$\int_{\gamma}g(z)dz=0$

to change the integral we are after into one that has Gaussian decay and is in
the conclusion of the Proposition.

Firstly, let us consider the integral on $\gamma_{1}$. Here the
parameterization of $\gamma_{1}$ is given by $z=u$ and so $dz=du$ and then
using this we see that $I_{1}(R)$ becomes

$\displaystyle I_{1}(R)$ $\displaystyle:=\int_{\gamma_{1}}g(z)dz$
$\displaystyle=\int_{0}^{R}e^{\frac{iM}{2\hbar
t}u^{2}}F_{\xi}(r,\phi,\theta,t,ue^{i\alpha})e^{iu(a\cos\theta+b\sin\theta)}\
udu.$

We next consider the integral over the curve $\gamma_{2}$. Here we have the
parameterization $z=Re^{i\alpha}$, which gives $dz=Rie^{i\alpha}d\alpha$ and
gives the following for $I_{2}(R)$,

$\displaystyle I_{2}(R)$ $\displaystyle:=\int_{\gamma_{2}}g(z)dz$
$\displaystyle=iR^{2}\int_{0}^{\pi/4}e^{\frac{iM}{2\hbar
t}R^{2}e^{2i\alpha}}F_{\xi}(r,\phi,\theta,t,Re^{i\alpha})e^{iRe^{i\alpha}(a\cos\theta+b\sin\theta)}e^{2i\alpha}d\alpha$
$\displaystyle=iR^{2}\int_{0}^{\pi/4}e^{\frac{M}{2\hbar
t}R^{2}(i\cos(2\alpha)-\sin(2\alpha))}F_{\xi}(r,\phi,\theta,t,Re^{i\alpha})e^{iRe^{i\alpha}(a\cos\theta+b\sin\theta)}e^{2i\alpha}d\alpha.$

Hence,

$\displaystyle|I_{2}(R)|$
$\displaystyle:=R^{2}\int_{0}^{\pi/4}e^{-\frac{M}{2\hbar
t}R^{2}\sin(2\alpha)}\left|F_{\xi}(r,\phi,\theta,t,Re^{i\alpha})\right|e^{-R\sin(\alpha)(a\cos\theta+b\sin\theta)}d\alpha$
$\displaystyle\leq R^{2}e^{-\frac{M}{2\hbar
t}R^{2}}\int_{0}^{\pi/4}\left|F_{\xi}(r,\phi,\theta,t,Re^{i\alpha})\right|e^{-R\sin(\alpha)(a\cos\theta+b\sin\theta)}d\alpha.$

The second integral goes to zero because of the estimates on
$\left|F_{\xi}(r,\phi,\theta,t,Re^{i\alpha})\right|$ from (11) which shows
that this integral can be controlled by $q(R)e^{-cR^{2}}e^{\widetilde{c}R}$
for some function $q$ with at most polynomial growth and constants $c$ and
$\widetilde{c}$, but this clearly goes to $0$ as $R\to\infty$.

Finally, we turn to the integral over the path $\gamma_{3}$. Here we have that
$z=ue^{i\frac{\pi}{4}}$ and so $dz=e^{i\frac{\pi}{4}}du$ and then using this
parameterization we see that

$\displaystyle I_{3}(R)$
$\displaystyle:=\int_{\gamma_{3}}g(z)dz=-\int_{0}^{R}g(ue^{i\frac{\pi}{4}})e^{i\frac{\pi}{4}}du$
$\displaystyle=-e^{i\frac{\pi}{2}}\int_{0}^{R}e^{\frac{iM}{2\hbar
t}u^{2}e^{i\frac{\pi}{2}}}F_{\xi}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}})e^{iue^{i\frac{\pi}{4}}(a\cos\theta+b\sin\theta)}\
udu$ $\displaystyle=-i\int_{0}^{R}e^{\frac{iM}{2\hbar
t}u^{2}i}F_{\xi}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}})e^{iue^{i\frac{\pi}{4}}(a\cos\theta+b\sin\theta)}\
udu$ $\displaystyle=-i\int_{0}^{R}e^{\frac{-M}{2\hbar
t}u^{2}}F_{\xi}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}})e^{iue^{i\frac{\pi}{4}}(a\cos\theta+b\sin\theta)}udu.$

Using these integrals, we have that

$\int_{\gamma}g(z)dz=\sum_{\ell=1}^{3}I_{\ell}(R)=0$

and upon taking the limit as $R\to+\infty$ we have

$\lim_{R\to+\infty}I_{1}(R)=-\lim_{R\to+\infty}I_{3}(R)$

since $\displaystyle\lim_{R\to+\infty}I_{2}(R)=0$. This then gives that

$\displaystyle\lim_{R\to+\infty}I_{1}(R)$
$\displaystyle=\int_{0}^{\infty}e^{\frac{iM}{2\hbar
t}u^{2}}F_{\xi}(r,\phi,\theta,t,ue^{i\alpha})e^{iu(a\cos\theta+b\sin\theta)}\
udu$
$\displaystyle=-\lim_{R\to+\infty}I_{3}(R)=i\int_{0}^{\infty}e^{\frac{-M}{2\hbar
t}u^{2}}F_{\xi}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}})e^{iue^{i\frac{\pi}{4}}(a\cos\theta+b\sin\theta)}udu.$
(14)

We can now use (14) and integrate the resulting identity in $\theta$ to
replace the integral:

$\psi_{a,b}(r,\phi,t):=\frac{M}{2\pi\hbar t}e^{\frac{iM}{2\hbar
t}r^{2}}\int_{0}^{2\pi}\int_{0}^{\infty}e^{\frac{iM}{2\hbar
t}\rho^{2}}F_{\xi}(r,\phi,\theta,t,\rho)e^{ia\rho\cos\theta+ib\rho\sin\theta}\
\rho d\rho d\theta$

with the Fresnel integral, given by

$\psi_{a,b}(r,\phi,t)=i\frac{M}{2\pi\hbar t}e^{\frac{iM}{2\hbar
t}r^{2}}\int_{0}^{2\pi}\int_{0}^{\infty}F_{\xi}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}})e^{\frac{-M}{2\hbar
t}u^{2}}e^{iue^{i\frac{\pi}{4}}(a\cos\theta+b\sin\theta)}udud\theta$

giving the statement of the Proposition. ∎

##  4\. Aharonov-Bohm Infinite Order Differential Operators and
the Supershift Property

Based on the Fresnel integral representation of the solution of the
Schrödinger equation, given in Proposition 3.2, see (13), we further have to
manipulate this solution to represent it as a power series of the coefficients
$a$ and $b$ appearing in the superoscillatory function associated with the
initial datum.

######  Lemma 4.1 (Series expansion of the solution in the parameters $a$ and
$b$).

Let $\psi_{a,b}(r,\phi,t)$ be the solution of the Schrödinger equation given
by (13). Then we have

$\displaystyle\psi_{a,b}(r,\phi,t)=i\frac{M}{2\pi\hbar t}e^{\frac{iM}{2\hbar
t}r^{2}}\sum_{m=0}^{\infty}\frac{\left(ie^{i\pi/4}\right)^{m}}{m!}\sum_{\ell=0}^{m}\binom{m}{\ell}a^{m-\ell}b^{\ell}c_{m,\ell}(r,\phi,t),$

where

$c_{m,\ell}(r,\phi,t):=\int_{0}^{2\pi}\int_{0}^{\infty}e^{\frac{-M}{2\hbar
t}u^{2}}F_{\xi}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}})\cos^{m-\ell}\theta\sin^{\ell}\theta
u^{m+1}dud\theta.$ (15)

###### Proof.

We consider

$\psi_{a,b}(r,\phi,t)=i\frac{M}{2\pi\hbar t}e^{\frac{iM}{2\hbar
t}r^{2}}\int_{0}^{2\pi}\int_{0}^{\infty}F_{\xi}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}})e^{\frac{-M}{2\hbar
t}u^{2}}e^{iue^{i\pi/4}(a\cos\theta+b\sin\theta)}udud\theta$

and we expand the exponential $e^{iue^{i\pi/4}(a\cos\theta+b\sin\theta)}$ to
get

$\psi_{a,b}(r,\phi,t)=i\frac{M}{2\pi\hbar t}e^{\frac{iM}{2\hbar
t}r^{2}}\int_{0}^{2\pi}\int_{0}^{\infty}e^{\frac{-M}{2\hbar
t}u^{2}}F_{\xi}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}})\sum_{m=0}^{\infty}\frac{\left(iue^{i\pi/4}(a\cos\theta+b\sin\theta)\right)^{m}}{m!}udud\theta.$

We want to exchange the infinite sum and the integrals involved. This will be
accomplished with the Lebesgue Dominated Convergence Theorem and the estimates
on the function $F_{\xi}(r,\phi,\theta,t,\rho)$ from (11). To do this, for
$L\in\mathbb{N}$, define

$\displaystyle
G_{\xi,L}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}}):=e^{\frac{-M}{2\hbar
t}u^{2}}F_{\xi}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}})\sum_{m=0}^{L}\frac{\left(iue^{i\pi/4}(a\cos\theta+b\sin\theta)\right)^{m}}{m!}u.$

By the computations above, in particular those connected to proving (11), one
can show that for $u$ large and for all $\theta\in[0,2\pi)$:

$\displaystyle|G_{\xi,L}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}})|$
$\displaystyle\lesssim e^{\frac{-M}{2\hbar
t}u^{2}}e^{\tilde{c}u}q(u)\sum_{m=0}^{L}\frac{u^{m}(|a|+|b|)^{m}}{m!}$
$\displaystyle\lesssim e^{\frac{-M}{2\hbar
t}u^{2}}e^{\tilde{c}u}q(u)e^{u(|a|+|b|)}=e^{\frac{-M}{2\hbar
t}u^{2}}e^{\tilde{c}u}q(u),$

for some constant $\tilde{c}$ and for some function $q$ with at most
polynomial growth in $u$. This function is then clearly integrable on
$[0,2\pi]\times[0,\infty)$. The Lebesgue Dominated Convergence Theorem then
allows for the interchange of the sum and the double integral, so the function
$\psi_{a,b}=\psi_{a,b}(r,\phi,t)$, turns out to be

$\displaystyle\psi_{a,b}$ $\displaystyle=i\frac{M}{2\pi\hbar
t}e^{\frac{iM}{2\hbar
t}r^{2}}\sum_{m=0}^{\infty}\frac{\left(ie^{i\pi/4}\right)^{m}}{m!}\int_{0}^{2\pi}\int_{0}^{\infty}e^{\frac{-M}{2\hbar
t}u^{2}}F_{\xi}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}})\left(a\cos\theta+b\sin\theta\right)^{m}u^{m+1}dud\theta.$

Next, expand the binomial
$\displaystyle\left(a\cos\theta+b\sin\theta\right)^{m}=\sum_{\ell=0}^{m}\binom{m}{\ell}a^{m-\ell}b^{\ell}\cos^{m-\ell}\theta\sin^{\ell}\theta$
and obtain

$\displaystyle\psi_{a,b}(r,\phi,t)$ $\displaystyle=i\frac{M}{2\pi\hbar
t}e^{\frac{iM}{2\hbar
t}r^{2}}\sum_{m=0}^{\infty}\frac{\left(ie^{i\pi/4}\right)^{m}}{m!}\sum_{\ell=0}^{m}\binom{m}{\ell}a^{m-\ell}b^{\ell}c_{m,\ell}(r,\phi,t),$

where

$c_{m,\ell}(r,\phi,t):=\int_{0}^{2\pi}\int_{0}^{\infty}e^{\frac{-M}{2\hbar
t}u^{2}}F_{\xi}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}})\cos^{m-\ell}\theta\sin^{\ell}\theta
u^{m+1}dud\theta$

as claimed in (15) and completing the proof of the Lemma. ∎

######  Remark 4.2.

The superoscillatory function $Y_{n}(x,y,a)$ defined in (4), with coefficients
$C_{j}(n,a)$ in (2), contains the entire functions $g$ and $h$, monotone
increasing in $a$, with

$g(\lambda)=\sum_{u=0}^{\infty}g_{u}\lambda^{u},\ \ \
h(\lambda)=\sum_{v=0}^{\infty}h_{v}\lambda^{v}.$ (16)

This function satisfies

$\lim_{n\to\infty}Y_{n}(x,y,a)=e^{ig(a)x}e^{ih(a)y}$ (17)

pointwise in $x$ and $y$ on compact sets in $\mathbb{R}^{2}$. The important
fact in our considerations is that the function $F_{n}(x,a)$ defined in (1),
extended to an entire complex function, converges in the space $A_{1}$.

The crucial problem is now to represent the solution of the Schrödinger
equation with the superoscillatory initial datum given in Remark 4.2 via an
infinite order differential operators. The series expansions of the entire
functions $g$ and $h$ given in (16) will be now used to define the infinite
order differential operator associated with the Aharonov-Bohm field.

######  Definition 4.3 (Infinite order differential operator for the Aharonov-
Bohm field).

Let $w\in\mathbb{C}$ be an auxiliary variable. Denote by $\mathcal{D}_{w}$ the
complex derivative with respect to the variable $w$ and define the formal
infinite order differential operators, for $m\geq\ell$, where
$m,\ell\in\mathbb{N}_{0}$ as

$\mathcal{G}_{m,\ell}(\mathcal{D}_{w}):=\left(\sum_{u=0}^{\infty}\frac{g_{u}}{i^{u}}\mathcal{D}_{w}^{u}\right)^{m-\ell},$
(18)

and

$\mathcal{H}_{\ell}(\mathcal{D}_{w}):=\left(\sum_{v=0}^{\infty}\frac{h_{v}}{i^{v}}\mathcal{D}_{w}^{v}\right)^{\ell},$
(19)

where the coefficients $g_{u}$ and $h_{v}$ are given by (16). We define the
infinite order differential operator associated with the Aharonov-Bohm field
as

$\mathcal{U}(r,\phi,t;\mathcal{D}_{w}):=i\frac{M}{2\pi\hbar
t}e^{\frac{iM}{2\hbar
t}r^{2}}\sum_{m=0}^{\infty}\frac{\left(ie^{i\pi/4}\right)^{m}}{m!}\sum_{\ell=0}^{m}\binom{m}{\ell}c_{m,\ell}(r,\phi,t)\mathcal{G}_{m,\ell}(\mathcal{D}_{w})\mathcal{H}_{\ell}(\mathcal{D}_{w}),$
(20)

where $\mathcal{H}_{\ell}(\mathcal{D}_{w})$ and
$\mathcal{H}_{\ell}(\mathcal{D}_{w})$ are defined in (18) and (19)
respectively.

######  Theorem 4.4 (Representation of the solution via the Aharonov-Bohm
infinite order differential operator).

The solution of the Schrödinger equation in the Aharonov-Bohm field with
superoscillatory initial datum as in Remark 4.2 can be written in terms of the
infinite order differential operators $\mathcal{G}_{m,\ell}(\mathcal{D}_{w})$
and $\mathcal{H}_{\ell}(\mathcal{D}_{w})$ as

$\begin{split}\psi_{g(a),h(a)}(r,\phi,t)&=\mathcal{U}(r,\phi,t;\mathcal{D}_{\xi})e^{iaw}\Big{|}_{w=0}\\\
&=i\frac{M}{2\pi\hbar t}e^{\frac{iM}{2\hbar
t}r^{2}}\sum_{m=0}^{\infty}\frac{\left(ie^{i\pi/4}\right)^{m}}{m!}\sum_{\ell=0}^{m}\binom{m}{\ell}c_{m,\ell}(r,\phi,t)\mathcal{G}_{m,\ell}(\mathcal{D}_{w})\mathcal{H}_{\ell}(\mathcal{D}_{w})e^{iaw}\Big{|}_{w=0}.\end{split}$
(21)

###### Proof.

In the function $\psi_{a,b}(r,\phi,t)$, given in Lemma 4.1, we replace the
parameter $a$ by the entire function $g(a)$ and $b$ by the entire function
$h(a)$ so that we have

$\displaystyle\psi_{g(a),h(a)}(r,\phi,t)$ $\displaystyle=i\frac{M}{2\pi\hbar
t}e^{\frac{iM}{2\hbar
t}r^{2}}\sum_{m=0}^{\infty}\frac{\left(ie^{i\pi/4}\right)^{m}}{m!}\sum_{\ell=0}^{m}\binom{m}{\ell}c_{m,\ell}(r,\phi,t)(g(a))^{m-\ell}(h(a))^{\ell}.$
(22)

Recalling the series expansions in (16) for the functions $g(a)$ and $h(a)$
and substituting this into (22) gives

$\displaystyle\psi_{g(a),h(a)}(r,\phi,t)$ $\displaystyle=i\frac{M}{2\pi\hbar
t}e^{\frac{iM}{2\hbar
t}r^{2}}\sum_{m=0}^{\infty}\frac{\left(ie^{i\pi/4}\right)^{m}}{m!}\sum_{\ell=0}^{m}\binom{m}{\ell}c_{m,\ell}(r,\phi,t)\left(\sum_{u=0}^{\infty}g_{u}a^{u}\right)^{m-\ell}\left(\sum_{v=0}^{\infty}h_{v}a^{v}\right)^{\ell}.$

Observe that using the auxiliary complex variable $w$ we have

$\lambda^{\ell}=\frac{1}{i^{\ell}}\mathcal{D}_{w}^{\ell}e^{iw\lambda}\Big{|}_{w=0}\
\ \ {\rm for}\ \ \ \lambda\in\mathbb{C},\ \ \ \ell\in\mathbb{N},$

where $\mathcal{D}_{w}$ is the derivative with respect to $w$ and $|_{w=0}$
denotes evaluation at $w=0$. This coupled with the computations above gives
(21), and proves the statement of the theorem. ∎

######  Theorem 4.5.

Let $A_{1}$ be the space of entire functions as in Definition 2.6. Then the
infinite order differential operator for the Aharonov-Bohm field
$\mathcal{U}(r,\phi,t;\mathcal{D}_{w})$, defined in (19), acts continuously
from $A_{1}$ to $A_{1}$.

###### Proof.

We split the proof in steps.

Step 1: Estimates for the operators $\mathcal{G}_{m,\ell}(\mathcal{D}_{w})$
and $\mathcal{H}_{\ell}(\mathcal{D}_{w})$. For $m\geq\ell$ for
$m,\ell\in\mathbb{N}$, using the property

$\mathcal{D}_{w}^{v}\sum_{j=0}^{\infty}f_{j}w^{j}=\sum_{j=v}^{\infty}f_{j}\frac{j!}{(j-v)!}w^{j-v}=\sum_{k=0}^{\infty}f_{v+k}\frac{(v+k)!}{k!}w^{k}$

and iterating, $\ell$ times, using the relation

$\left(\sum_{v_{1}=0}^{\infty}\frac{h_{v_{1}}}{i^{v_{1}}}\mathcal{D}_{w}^{v_{1}}\right)\sum_{j=0}^{\infty}f_{j}w^{j}=\sum_{v_{1}=0}^{\infty}\frac{h_{v_{1}}}{i^{v_{1}}}\sum_{k=0}^{\infty}f_{v_{1}+k}\frac{(v_{1}+k)!}{k!}w^{k}$

we have

$\displaystyle\mathcal{H}_{\ell}(\mathcal{D}_{w})f(w)$
$\displaystyle=\left(\sum_{v=0}^{\infty}\frac{h_{v}}{i^{v}}\mathcal{D}_{w}^{v}\right)^{\ell}f(w)$
$\displaystyle=\sum_{v_{1}=0}^{\infty}\frac{h_{v_{1}}}{i^{v_{1}}}\ldots\sum_{v_{\ell}=0}^{\infty}\frac{h_{v_{\ell}}}{i^{v_{\ell}}}\sum_{k=0}^{\infty}f_{\sum_{j=1}^{\ell}v_{j}+k}\frac{(\sum_{j=1}^{\ell}v_{j}+k)!}{k!}w^{k}$
$\displaystyle=\prod_{j=1}^{\ell}\sum_{v_{j}=0}^{\infty}\frac{h_{v_{j}}}{i^{v_{j}}}\sum_{k=0}^{\infty}f_{\sum_{j=1}^{\ell}v_{j}+k}\frac{(\sum_{j=1}^{\ell}v_{j}+k)!}{k!}w^{k}.$

Similarly considering the operators $\mathcal{G}_{m,\ell}(\mathcal{D}_{w})$,
and performing analogous computations we have

$\begin{split}&\mathcal{G}_{m,\ell}(\mathcal{D}_{w})\mathcal{H}_{\ell}(\mathcal{D}_{w})f(w)=\left(\sum_{u=0}^{\infty}\frac{g_{u}}{i^{u}}\mathcal{D}_{w}^{u}\right)^{m-\ell}\left(\sum_{v=0}^{\infty}\frac{h_{v}}{i^{v}}\mathcal{D}_{w}^{v}\right)^{\ell}f(w)\\\
&=\prod_{p=1}^{m-\ell}\sum_{u_{p}=0}^{\infty}\frac{g_{u_{p}}}{i^{u_{p}}}\prod_{j=1}^{\ell}\sum_{v_{j}=0}^{\infty}\frac{h_{v_{j}}}{i^{v_{j}}}\sum_{k=0}^{\infty}f_{\sum_{p=1}^{m-\ell}u_{p}+\sum_{j=1}^{\ell}v_{j}+k}\frac{\left(\sum_{p=1}^{m-\ell}u_{p}+\sum_{j=1}^{\ell}v_{j}+k\right)!}{k!}w^{k}.\end{split}$

We now estimate
$\mathcal{G}_{m,\ell}(\mathcal{D}_{w})\mathcal{H}_{\ell}(\mathcal{D}_{w})f(w)$
taking the modulus

$\begin{split}&\Big{|}\mathcal{G}_{m,\ell}(\mathcal{D}_{w})\mathcal{H}_{\ell}(\mathcal{D}_{w})f(w)\Big{|}\\\
&\leq\prod_{p=1}^{m-\ell}\sum_{u_{p}=0}^{\infty}|g_{u_{p}}|\prod_{j=1}^{\ell}\sum_{v_{j}=0}^{\infty}|h_{v_{j}}|\sum_{k=0}^{\infty}\big{|}f_{\sum_{p=1}^{m-\ell}u_{p}+\sum_{j=1}^{\ell}v_{j}+k}\big{|}\frac{\left(\sum_{p=1}^{m-\ell}u_{p}+\sum_{j=1}^{\ell}v_{j}+k\right)!}{k!}\
|w|^{k}\end{split}$

and using the estimate in Lemma 2.7 for the coefficients
$f_{\sum_{p=1}^{m-\ell}u_{p}+\sum_{j=1}^{\ell}v_{j}+k}$ we have for some
$C_{f}>0$ and $b>0$ that

$\left|f_{\sum_{p=1}^{m-\ell}u_{p}+\sum_{j=1}^{\ell}v_{j}+k}\right|\leq
C_{f}\frac{b^{\sum_{p=1}^{m-\ell}u_{p}+\sum_{j=1}^{\ell}v_{j}+k}}{\Gamma\left(\sum_{p=1}^{m-\ell}u_{p}+\sum_{j=1}^{\ell}v_{j}+k+1\right)}.$

By elementary properties of the Gamma function we have

$\begin{split}&\Big{|}\mathcal{G}_{m,\ell}(\mathcal{D}_{w})\mathcal{H}_{\ell}(\mathcal{D}_{w})f(w)\Big{|}\\\
&\leq
C_{f}\prod_{p=1}^{m-\ell}\sum_{u_{p}=0}^{\infty}|g_{u_{p}}|\prod_{j=1}^{\ell}\sum_{v_{j}=0}^{\infty}|h_{v_{j}}|\sum_{k=0}^{\infty}\frac{\,b^{\sum_{p=1}^{m-\ell}u_{p}+\sum_{j=1}^{\ell}v_{j}+k}\,\left(\sum_{p=1}^{m-\ell}u_{p}+\sum_{j=1}^{\ell}v_{j}+k\right)!}{\Gamma\left(\sum_{p=1}^{m-\ell}u_{p}+\sum_{j=1}^{\ell}v_{j}+k+1\right)\,k!}|w|^{k}.\end{split}$

Simplifying we have

$\displaystyle\Big{|}\mathcal{G}_{m,\ell}(\mathcal{D}_{w})\mathcal{H}_{\ell}(\mathcal{D}_{w})f(w)\Big{|}$
$\displaystyle\leq
C_{f}\prod_{p=1}^{m-\ell}\sum_{u_{p}=0}^{\infty}|g_{u_{p}}|b^{\sum_{p=1}^{m-\ell}u_{p}}\prod_{j=1}^{\ell}\sum_{v_{j}=0}^{\infty}|h_{v_{j}}|b^{\sum_{j=1}^{\ell}v_{j}}\
\sum_{k=0}^{\infty}\frac{b^{k}}{k!}\ |w|^{k}$
$\displaystyle=C_{f}\left(\sum_{u=0}^{\infty}|g_{u}|b^{u}\right)^{m-\ell}\left(\sum_{v=0}^{\infty}|h_{v}|b^{v}\right)^{\ell}\
e^{b|w|}$ (23)

and we observe that since the function $g$ and $h$ are entire

$\sum_{u=0}^{\infty}|g_{u}|b^{u}<\infty\ \ {\rm and}\ \
\sum_{v=0}^{\infty}|h_{v}|b^{v}<\infty$

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

Step 2: Estimates for the coefficients $c_{m,\ell}(r,\phi,t)$. We consider the
modulus of the coefficients in (15)

$\begin{split}|c_{m,\ell}(r,\phi,t)|&=\left|\int_{0}^{2\pi}\int_{0}^{\infty}e^{\frac{-M}{2\hbar
t}u^{2}}F_{\xi}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}})\cos^{m-\ell}\theta\sin^{\ell}\theta
u^{m+1}du\ d\theta\right|\\\
&\leq\int_{0}^{2\pi}\int_{0}^{\infty}e^{\frac{-M}{2\hbar
t}u^{2}}\big{|}F_{\xi}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}})\cos^{m-\ell}\theta\sin^{\ell}\theta
u^{m+1}\big{|}du\\\ &\leq\int_{0}^{2\pi}\int_{0}^{\infty}e^{\frac{-M}{2\hbar
t}u^{2}}\big{|}F_{\xi}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}})\big{|}u^{m+1}dud\theta.\end{split}$

Now, thanks to Lemma 3.1 for $u$ real and non-negative there exists a positive
constant $c$ such that

$\left|F_{\xi}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}})\right|\lesssim
q(u)e^{{c}u}$ (24)

where $q(u)$ is a function that has at most polynomial growth in $u$ and the
coefficients in the function and in ${c}$ depend upon $r,M,t,\theta,\phi$ and
$\xi$. More explicitly,

$\displaystyle\left|F_{\xi}(r,\phi,\theta,t,\rho)\right|\leq
e^{\frac{Mr\,\rho}{4\hbar t}}I_{0}\left(\frac{Mr\rho}{2\hbar
t}\right)\left\\{\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{1-\xi_{f}}\left(3+\frac{Mr\rho}{4\hbar
t}\right)+\left(\frac{Mr\,\rho}{4\hbar
t}\right)^{\xi_{f}}\left(3+\frac{Mr\rho}{2\hbar t}\right)\right\\},$

where $\rho=|ue^{i\frac{\pi}{4}}|=u$. So, for $t>0$, we have

$\begin{split}|c_{m,\ell}(r,\phi,t)|&=\int_{0}^{2\pi}\int_{0}^{\infty}e^{\frac{-M}{2\hbar
t}u^{2}}\big{|}F_{\xi}(r,\phi,\theta,t,ue^{i\frac{\pi}{4}})\big{|}u^{m+1}du\
d\theta\\\ &\leq\int_{0}^{2\pi}\int_{0}^{\infty}e^{\frac{-M}{2\hbar
t}u^{2}}q(u)e^{{c}u}u^{m+1}du\ d\theta\\\ &\leq
2\pi\int_{0}^{\infty}e^{\frac{-M}{4\hbar t}u^{2}}\Big{(}e^{\frac{-M}{4\hbar
t}u^{2}}q(u)e^{{c}u}\Big{)}u^{m+1}du\\\ &\leq
K\int_{0}^{\infty}e^{\frac{-M}{4\hbar t}u^{2}}u^{m+1}du\end{split}$

where we set $\displaystyle
K:=2\pi\sup_{u\in[0,\infty]}\Big{(}e^{\frac{-M}{4\hbar
t}u^{2}}q(u)e^{{c}u}\Big{)}<\infty$. Using the following well known integral

$\int_{0}^{\infty}x^{n}e^{-\gamma
x^{2}}dx=\frac{\Gamma\Big{(}\frac{n+1}{2}\Big{)}}{2\gamma^{(n+1)/2}},\ \ \
{\rm for}\ \ \ \gamma>0.$

we conclude that

$|c_{m,\ell}(r,\phi,t)|\leq
K\frac{\Gamma\Big{(}\frac{m+2}{2}\Big{)}}{2\Big{(}\frac{M}{4\hbar
t}\Big{)}^{(m+2)/2}}.$ (25)

Observe that the estimate does not depend on $\ell$.

Step 3: The continuity estimate. Based on the previous points, for $t>0$, we
now consider

$\begin{split}&\left|\mathcal{U}(r,\phi,t;\mathcal{D}_{w})f(w)\right|=\left|i\frac{M}{2\pi\hbar
t}e^{\frac{iM}{2\hbar
t}r^{2}}\sum_{m=0}^{\infty}\frac{\left(ie^{i\pi/4}\right)^{m}}{m!}\sum_{\ell=0}^{m}\binom{m}{\ell}c_{m,\ell}(r,\phi,t)\mathcal{G}_{m,\ell}(\mathcal{D}_{w})\mathcal{H}_{\ell}(\mathcal{D}_{w})f(w)\right|\\\
&\leq\frac{M}{2\pi\hbar
t}\sum_{m=0}^{\infty}\frac{1}{m!}\sum_{\ell=0}^{m}\binom{m}{\ell}\big{|}c_{m,\ell}(r,\phi,t)\big{|}\
\big{|}\mathcal{G}_{m,\ell}(\mathcal{D}_{w})\mathcal{H}_{\ell}(\mathcal{D}_{w})f(w)\big{|}\\\
&\leq\frac{M}{2\pi\hbar
t}\sum_{m=0}^{\infty}\frac{1}{m!}\sum_{\ell=0}^{m}\binom{m}{\ell}K\frac{\Gamma\Big{(}\frac{m+2}{2}\Big{)}}{2\Big{(}\frac{M}{4\hbar
t}\Big{)}^{(m+2)/2}}C_{f}\left(\sum_{u=0}^{\infty}|g_{u}|b^{u}\right)^{m-\ell}\left(\sum_{v=0}^{\infty}|h_{v}|b^{v}\right)^{\ell}\
e^{b|w|}\\\ &=\frac{M}{2\pi\hbar
t}KC_{f}\sum_{m=0}^{\infty}\frac{1}{m!}\frac{\Gamma\Big{(}\frac{m+2}{2}\Big{)}}{2\Big{(}\frac{M}{4\hbar
t}\Big{)}^{(m+2)/2}}\sum_{\ell=0}^{m}\binom{m}{\ell}\left(\sum_{u=0}^{\infty}|g_{u}|b^{u}\right)^{m-\ell}\left(\sum_{v=0}^{\infty}|h_{v}|b^{v}\right)^{\ell}\
e^{b|w|}\\\ &=\frac{M}{2\pi\hbar
t}KC_{f}\sum_{m=0}^{\infty}\frac{1}{m!}\frac{\Gamma\left(\frac{m+2}{2}\right)}{2\left(\frac{M}{4\hbar
t}\right)^{(m+2)/2}}\left(\sum_{u=0}^{\infty}|g_{u}|b^{u}+\sum_{v=0}^{\infty}|h_{v}|b^{v}\right)^{m}\
e^{b|w|},\end{split}$

where we have utilized the estimates (25) and (23) in the course of the
estimation above. Now setting

$\Lambda:=\sum_{m=0}^{\infty}\frac{1}{m!}\frac{\Gamma\Big{(}\frac{m+2}{2}\Big{)}}{2\Big{(}\frac{M}{4\hbar
t}\Big{)}^{(m+2)/2}}\left(\sum_{u=0}^{\infty}|g_{u}|b^{u}+\sum_{v=0}^{\infty}|h_{v}|b^{v}\right)^{m}$

using the estimate in Lemma 2.8, for $q=2$:

$\Gamma\left(\frac{m}{2}+1\right)\leq(m!)^{1/2}$

we have

$\begin{split}\Lambda&\leq\sum_{m=0}^{\infty}\frac{1}{m!}\frac{(m!)^{1/2}}{2\Big{(}\frac{M}{4t}\Big{)}^{(m+2)/2}}\left(\sum_{u=0}^{\infty}|g_{u}|b^{u}+\sum_{v=0}^{\infty}|h_{v}|b^{v}\right)^{m}\\\
&=\frac{2t}{M}\sum_{m=0}^{\infty}\frac{1}{(m!)^{1/2}}\left(\left(\frac{4t}{M}\right)^{1/2}\left(\sum_{u=0}^{\infty}|g_{u}|b^{u}+\sum_{v=0}^{\infty}|h_{v}|b^{v}\right)\right)^{m}<\infty.\end{split}$

All together this gives the estimate

$\big{|}\mathcal{U}(r,\phi,t;\mathcal{D}_{w})f(w)\big{|}\leq\frac{M}{2\pi\hbar
t}KC_{f}\Lambda\ e^{b|w|}$

which gives us the statement of the theorem. Note that the constants involved
depend upon $M,t,r,\phi$ and $\xi$. ∎

Due to the continuity theorem proved above now we can show that solutions of
the Schrödinger equation with superoscillatory datum under the Aharonov-Bohm
field possess the supershift property.

######  Theorem 4.6.

The solution of the Schrödinger equation, with the propagator (3) with
superoscillatory initial datum given by (4) can be written as

$\Psi_{n}(r,\phi,t)=\sum_{j=0}^{n}C_{j}(n,a)\psi_{g\left(1-\frac{2j}{n}\right),h\left(1-\frac{2j}{n}\right)}(r,\phi,t)$

for all $r\in(0,\infty)$ $\phi\in[0,2\pi]$, $t>0$, where
$\psi_{g(a),h(a)}(r,\phi,t)$ is given by (21) and $C_{j}(n,a)$ are given by
(2). Moreover, $\psi_{a,b}(r,\phi,t)$ has the supershift property, that is

$\lim_{n\to\infty}\Psi_{n}(r,\phi,t)=\lim_{n\to\infty}\sum_{j=0}^{n}C_{j}(n,a)\psi_{g\left(1-\frac{2j}{n}\right),h\left(1-\frac{2j}{n}\right)}(r,\phi,t)=\psi_{g(a),h(a)}(r,\phi,t).$

###### Proof.

Since the solution $\psi_{g(a),h(a)}(r,\phi,t)$ can be represented via the
Aharonov-Bohm infinite order differential operator, from Theorem 4.4,
$\psi_{g\left(1-\frac{2j}{n}\right),h\left(1-\frac{2j}{n}\right)}(r,\phi,t)$
is given in terms of the infinite order differential operators
$\mathcal{G}_{m,\ell}(\mathcal{D}_{w})$ and
$\mathcal{H}_{\ell}(\mathcal{D}_{w})$ as

$\begin{split}&\psi_{g\left(1-\frac{2j}{n}\right),h\left(1-\frac{2j}{n}\right)}(r,\phi,t)=\mathcal{U}(r,\phi,t;\mathcal{D}_{\xi})e^{i\left(1-\frac{2j}{n}\right)w}\Big{|}_{w=0}\\\
&=i\frac{M}{2\pi\hbar t}e^{\frac{iM}{2\hbar
t}r^{2}}\sum_{m=0}^{\infty}\frac{\left(ie^{i\pi/4}\right)^{m}}{m!}\sum_{\ell=0}^{m}\binom{m}{\ell}c_{m,\ell}(r,\phi,t)\mathcal{G}_{m,\ell}(\mathcal{D}_{w})\mathcal{H}_{\ell}(\mathcal{D}_{w})e^{i\left(1-\frac{2j}{n}\right)w}\Big{|}_{w=0}.\end{split}$

By linearity, the evolution of the superoscillatory initial datum is given by

$\Psi_{n}(r,\phi,t)=\sum_{j=0}^{n}C_{j}(n,a)\psi_{g\left(1-\frac{2j}{n}\right),h\left(1-\frac{2j}{n}\right)}(r,\phi,t).$

By Theorem 4.5, it is immediate that

$\begin{split}&\lim_{n\to\infty}\Psi_{n}(r,\phi,t)=\lim_{n\to\infty}\sum_{j=0}^{n}C_{j}(n,a)\psi_{g\left(1-\frac{2j}{n}\right),h\left(1-\frac{2j}{n}\right)}(r,\phi,t)\\\
&=i\frac{M}{2\pi\hbar t}e^{\frac{iM}{2\hbar
t}r^{2}}\sum_{m=0}^{\infty}\frac{\left(ie^{i\pi/4}\right)^{m}}{m!}\sum_{\ell=0}^{m}\binom{m}{\ell}c_{m,\ell}(r,\phi,t)\mathcal{G}_{m,\ell}(\mathcal{D}_{w})\mathcal{H}_{\ell}(\mathcal{D}_{w})\lim_{n\to\infty}\sum_{j=0}^{n}C_{j}(n,a)e^{i\left(1-\frac{2j}{n}\right)w}\Big{|}_{w=0}\\\
&=i\frac{M}{2\pi\hbar t}e^{\frac{iM}{2\hbar
t}r^{2}}\sum_{m=0}^{\infty}\frac{\left(ie^{i\pi/4}\right)^{m}}{m!}\sum_{\ell=0}^{m}\binom{m}{\ell}c_{m,\ell}(r,\phi,t)\mathcal{G}_{m,\ell}(\mathcal{D}_{w})\mathcal{H}_{\ell}(\mathcal{D}_{w})e^{iaw}\Big{|}_{w=0}\\\
&=\psi_{g(a),h(a)}(r,\phi,t)\end{split}$

since the sequence
$\displaystyle\sum_{j=0}^{n}C_{j}(n,a)e^{i\left(1-\frac{2j}{n}\right)w}$
converges to $e^{iaw}$ in $A_{1}$, concluding the proof of the Theorem. ∎

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