# Searchlight Asymptotics for High-Frequency Scattering by Boundary Inflection

V.P. Smyshlyaev  Department of Mathematics
University College London
Gower Street
London
WC1E 6BT
UK<EMAIL_ADDRESS>and  I.V. Kamotski  Department of Mathematics
University College London
Gower Street
London
SW1E 6BT
UK<EMAIL_ADDRESS>
###### Abstract.

We consider an inner problem for whispering gallery high-frequency asymptotic
mode’s scattering by a boundary inflection. The related boundary-value problem
for a Schrödinger equation on a half-line with a potential linear in both
space and time appears fundamental for describing transitions from modal to
scattered asymptotic patterns, and despite having been intensively studied
over several decades remains largely unsolved. We prove that the solution past
the inflection point has a “searchlight” asymptotics corresponding to a beam
concentrated near the limit ray, and establish certain decay and smoothness
properties of the related searchlight amplitude. We also discuss further
interpretations of the above result: the existence of associated generalised
wave operator, and of a version of a unitary scattering operator connecting
the modal and scattered asymptotic regimes.

###### Key words and phrases:

diffraction, whispering gallery, boundary inflection, wave operator

##  1\. Introduction

This work is dedicated to Prof V.M. Babich on occasion of his 90-th birthday,
and is devoted to a long-standing canonical problem of high frequency
diffraction, to which area Prof Babich has made many groundbreaking
contributions. The particular problem is an inner problem for a whispering
gallery high-frequency asymptotic mode propagating along a concave part of a
boundary and then scattering at a boundary inflection point. Like Airy
equation and associated Airy function are fundamental for describing
transition from oscillatory to exponentially decaying asymptotic behaviors,
the boundary inflection problem leads to an arguably equally fundamental
canonical boundary-value problem for a special partial differential equation
(PDE) describing transition from a “modal” to a “scattered” high-frequency
asymptotic behavior.

Mathematically, the problem is for a Schrödinger equation on a half-line with
a potential linear in both space and time, which was first formulated and
analysed by M.M. Popov starting from [15] in the 1970-s. The associated
solutions have asymptotic behaviors with a discrete spectrum at one end and
with a continuous spectrum at the other end, and of central interest is to
find the map connecting the above two asymptotic regimes. The problem however
lacks separation of variables, except in the asymptotical sense at both of the
above ends. Nevertheless, as we essentially argue in the present work, a non-
standard perturbation analysis at the continuous spectrum end can be
performed, ultimately describing the desired map connecting the two asymptotic
representations.

Specifically, the problem is for a high-frequency wave process near a boundary
containing a simple inflection point, as displayed on Fig. 1.

Figure 1. Whispering gallery wave near a boundary inflection

Let the wave process be described by Helmholtz equation $\Delta U+k^{2}U=0$
over the boundary $S$ and let $(s,n)$ be a local system of coordinates
associated with the boundary, so that $s$ is the arclength along $S$ and $n$
is the normal distance to $S$. So the boundary corresponds to $n=0$, with
(say) Dirichlet boundary conditions $U(s,0)=0$. The inflection point
corresponds to $s=0$ and so the curvature $\kappa(s)$ is positive for $s<0$
where the boundary is (locally) concave and negative for $s>0$ where the
boundary becomes convex.

For the concave part of the boundary, i.e. for $s<0$ away from the inflection,
whispering gallery asymptotic solutions are known to exist for large $k$ (i.e
for high frequencies) propagating along $S$ in a thin “boundary-layer”
$\,n=O\left(k^{-2/3}\right)$. To the main order, these are of the form, see
e.g. [1] and [2],

(1) $U\sim
C\kappa^{1/6}(s)e^{iks}\exp\left\\{-i2^{-1/3}k^{1/3}\nu_{j}\int_{s_{0}}^{s}\kappa^{2/3}(s^{\prime})ds^{\prime}\right\\}\mbox{Ai}\left(\left(2\kappa(s)\right)^{1/3}k^{2/3}n-\nu_{j}\right),$

where $\mbox{Ai}(z)$ is the Airy function, solution to Airy equation
$\mbox{Ai}^{{}^{\prime\prime}}-z\mbox{Ai}=0$ exponentially decaying as
$z\to+\infty$; $-\nu_{j}$, $j=1,2,3,...$, is any of its zeros,
$0<\nu_{1}<\nu_{2}<...<\nu_{j}<\nu_{j+1}<...$, $\nu_{j}\to+\infty$ as
$j\to\infty$; $s_{0}<0$ is any reference point, and $C$ is a constant.

For the curvature $\kappa(s)=-\beta s+O(s^{2})$ near the inflection point
$s=0$, $\beta>0$, Popov has shown in [15] that the incoming whispering gallery
wave (1) has to be matched with a solution of an inner problem near $s=0$,
which in stretched coordinates $x=\beta^{1/5}k^{3/5}n$,
$t=\beta^{2/5}k^{1/5}s$, with $U(s,n)\sim e^{iks}\psi(x,t)$ is stated in
(2)–(5) below.

Since Popov’s pioneering work the problem has attracted a sustained attention
over the last several decades. Without attempting a comprehensive review, we
discuss in the next section in some detail a number of contributions
particularly relevant to our present purposes. Additionally, Kazakov has
introduced and analysed asymptotically certain integral solutions to (2) in
[9], and most recently proposed in [10] another approach for constructing a
class of particular solutions to (2) obeying boundary conditions (3).

In the present work, we aim at rigorously establishing the structure of a
“searchlight” formed past the inflection point, which corresponds to the
asymptotics of the solution to (2)–(5) as $t\to+\infty$ near the limit ray
corresponding to the straight line tangent to $S$ at $s=0$, see Fig. 1. Our
new result is stated in Theorem 1, which justifies to the main order a related
asymptotic ansatz (12) and establishes some properties of the searchlight
amplitude $G_{0}(\eta)$.

It appears that mathematically the searchlight asymptotics corresponds to a
perturbation analysis as $t\to+\infty$ of an appropriately transformed
problem. The underlying key transformation, first discovered in [3], turns out
to be closely related to a “pseudoconformal symmetry” of a free Schrödinger
equation, see e.g. [23, 22, 24]. The related unperturbed problem appears to be
one with a continuous spectrum, corresponding to a “free scattering” without a
boundary. This can be compared with Popov’s asymptotic analysis as
$t\to-\infty$ in [16], where another transformed problem is a perturbation to
an operator with a discrete spectrum corresponding to the whispering gallery
modes. Wave operators appear to exist at both “ends” $t\to\pm\infty$, and
hence so does a version of a scattering operator connecting the two asymptotic
regimes, see Theorem 2 and Corollary 1. We remark that the general idea of
transforming an operator consistently with the form of the solutions’
asymptotic expansion appears to resonate with that of generalised wave
operators, cf [6]. Notice also that other scenarios connecting asymptotic
regimes with discrete and continuous spectra are found in the literature, see
e.g. [20].

In the next section, we formulate the inner problem for $\psi(x,t)$, discuss
related background and state our main result, Theorem 1. In Section 3 we
discuss interpretations for the searchlight wave operator and for a version of
a scattering operator. Section 4 provides a detailed proof of Theorem 1.

##  2\. Formulation, background, and the main result

The inner problem for whispering gallery wave scattering near a simple
boundary inflection point was formulated by Popov [15] as follows. In the
half-plane domain $\Omega:=\\{(x,t)\,;\,x>0,\,-\infty<t<+\infty\\}$, find a
solution $\psi(x,t)$ to the Schrödinger-type equation

(2) $i\frac{\partial\psi}{\partial
t}\,=\,-\,\frac{1}{2}\,\frac{\partial^{2}\psi}{\partial x^{2}}\,-\,xt\,\psi,$

subject to Dirichlet boundary condition at $x=0$

(3) $\psi(0,t)\,=\,0,$

and with a prescribed asymptotic behaviour for large negative $t$:

(4)
$\left\|\psi(\cdot,t)\,-\,\psi^{-}_{0}(\cdot,t)\right\|_{L^{2}(0,+\infty)}\,\to\,0,\,\,{\mbox{a}s}\,t\to\,-\infty,$

where

(5)
$\psi^{-}_{0}(x,t)\,=\,D_{j}(-2t)^{1/6}\exp\left(i\nu_{j}\frac{3}{20}(-2t)^{5/3}\right)\mbox{Ai}\left(x(-2t)^{1/3}-\nu_{j}\right).$

Here $\mbox{Ai}(z)$ is the Airy function and $D_{j}$ is a constant. Formula
(5) emerges in [15] as a result of matching as $t\to-\infty$ with the
principal part of asymptotics of the incoming whispering gallery wave (1).

PDE (2) when regarded on the whole real line $\mathbb{R}$
($-\infty<x<+\infty$) is known to be reducible to a separable one, see e.g.
[5]. However the boundary condition (3) makes it non-separable, which is a
source of major challenges for its analysis.

In [16] Popov studied well-posedness of (2)–(5), whose solution
$\psi(\cdot,t)$ was a priori understood as a function from the domain of a
self-adjoint operator $A(t)$ in Hilbert space $H=L^{2}(0,+\infty)$ determined
by the right hand side of (2) with zero boundary condition at $x=0$. Function
$\psi(\cdot,t)$ is required to be strongly differentiable in $t$ in $H$ with
the left hand side of (2) understood as the strong derivative. The existence
(and uniqueness) was then established in [16] by means of wave operator
methods as $t\to-\infty$. (For an introduction to as well as for more advanced
properties of wave operators, see e.g. [8], [25].)

To that end, Popov used in [16] the following natural transformation of
(2)–(3) for identifying an “unperturbed” problem as $t\to-\infty$, for which
(5) is an exact solution. For $t<0$, introduce new variables $(\xi,\tau)$ and
function $\tilde{\psi}$ by

(6) $\xi\,=\,(-2t)^{1/3}x,\ \ \tau\,=\,-\frac{3}{20}(-2t)^{5/3},\ \ \
\psi(x,t)\,=\,(-2t)^{1/6}\tilde{\psi}(\xi,\tau).$

This transforms (2) into

(7)
$i\frac{\partial\tilde{\psi}}{\partial\tau}\,=\,-\,\,\frac{\partial^{2}\tilde{\psi}}{\partial\xi^{2}}\,+\,\xi\,\psi\,\,-\,\frac{i}{5\tau}\left(\xi\frac{\partial\tilde{\psi}}{\partial\xi}+\frac{1}{2}\tilde{\psi}\right),\
\ \xi>0,\ \tau<0,$

with similar to (3) boundary condition $\tilde{\psi}(0,\tau)=0$.

The asymptotics $\psi_{0}^{-}(x,t)$ in (5) then transforms into

(8)
$\tilde{\psi}_{0}^{-}(\xi,\tau)\,=\,D_{j}\exp\left(-i\nu_{j}\tau\right)\mbox{Ai}(\xi-\nu_{j}),$

which is an exact solution of the unperturbed version of (7) as
$\tau\to-\infty$ i.e. of the one with the last term on the right hand side of
(7) (the perturbation) omitted. Transformation (6)–(7) also helps to recover
an all-order formal asymptotic expansion to the solution of (2)–(5) as
$t\to-\infty$, initially constructed in [15]:

(9)
$\psi(x,t)\,\sim\,(-2t)^{1/6}e^{-i\nu_{j}\tau}\sum_{n=0}^{\infty}\tau^{-n}\left[P_{2n}(\xi)\mbox{Ai}\left(\xi-\nu_{j}\right)+Q_{2n-1}(\xi)\mbox{Ai}^{\prime}\left(\xi-\nu_{j}\right)\right],$

where $\xi$ and $\tau$ are given by (6); $\mbox{Ai}^{\prime}$ is the
derivative of the Airy function, and $P_{2n}$ and $Q_{2n-1}$ are polynomials
in $\xi$ of degrees $2n$ and $2n-1$ respectively, $Q_{2n-1}(0)=0$
($Q_{-1}\equiv 0$). Indeed, substitution of (9) into (7) using (6) and the
Airy equation leads to straightforward recurrence relations for $P_{2n}$ and
$Q_{2n-1}$.

Popov’s well-posedness analysis in [16] used in fact a further transformation
of (7) via
$\tilde{\psi}(\xi,\tau)=\exp\left\\{-i\frac{\xi^{2}}{20\tau}\right\\}v(\xi,\tau)$
resulting in

(10) $i\frac{\partial
v}{\partial\tau}\,=\,-\,\frac{\partial^{2}v}{\partial\xi^{2}}\,+\,\xi\,v\,+\,\frac{1}{25}\,\frac{\xi^{2}}{\tau^{2}}v,\
\ \tau<0;\ \ \xi>0,\ \ v(0,\tau)=0,$

for which the existence of a wave operator was established. We emphasise that
the transformed problem (10) recasts the original problem as a perturbation,
when $\tau\to-\infty$, of related unperturbed problem with (self-adjoint)
“Airy” operator $A^{-}_{0}v(\xi)=-v^{{}^{\prime\prime}}+\xi v$, $\xi>0$,
$v(0)=0$, having a discrete spectrum and eigenfunctions (8) associated with
the incoming whispering gallery modes.

In [4], V.M. Babich and the first author have proved that the solution of
(2)–(5) is in fact classical, using bootstrap-type techniques based on the
asymptotic expansion (9), which has allowed to justify (9) with proved error
estimates. Moreover, [4] also proved that $\psi(x,t)$ is infinitely
differentiable in $\overline{\Omega}$, and together with all its derivatives
decays super-algebraically as $x\to+\infty$ uniformly at any bounded interval
in $t$. Namely, for all non-negative integers $\alpha$, $\beta$ and $\gamma$,
and for any real $A_{1}<A_{2}$, with $\partial$ denoting appropriate partial
derivatives,

(11) $\sup_{A_{1}\leqslant t\leqslant
A_{2}}\left|x^{\alpha}\partial_{t}^{\beta}\partial_{x}^{\gamma}\psi(x,t)\right|\,\rightarrow\,0,\
\ \ \mbox{ as }x\to+\infty.$

Similar results and some generalisations were soon thereafter obtained by
different methods in [11].

The problem of fundamental interest is to determine asymptotic behaviour of
the solution $\psi(x,t)$ of (2)–(5) as $t\to+\infty$, which would correspond
to the scattered wave field past the inflection point. In [3] a formal
asymptotic solution matching to the incoming whispering gallery wave (5) was
constructed in the shadow zone above the limit ray $x=t^{3}/6$, which curve
corresponds to the straight line tangent to the physical boundary at the
inflection point (Fig. 1). The constructed asymptotics was observed to remain
an asymptotic expansion as $t\to+\infty$ as long as $x/t^{3}-1/6=:\mu\gg
t^{-2}$, breaking down when $\mu\sim t^{-2}$. Following general boundary-layer
recipes, it was suggested in [3], see formula (4.3) of [3], to seek the
asymptotic expansion to the solution $\psi(x,t)$ as $t\to+\infty$ in the
vicinity of the limit ray, with respect to the stretched variable $\eta:=\mu
t^{2}=x/t-t^{2}/6$, in the form

(12) $\psi(x,t)\,\sim\,t^{-1/2}\exp\left\\{i\frac{7}{120}t^{5}+\frac{i}{2}\eta
t^{3}+\frac{i}{2}\eta^{2}t\right\\}\,\sum_{m=0}^{\infty}t^{-m}G_{m}(\eta),\ \
\eta:=\frac{x}{t}-\frac{1}{6}t^{2}.$

The form (12) was interpreted in [3] as a “searchlight” ansatz, for a beam
anticipated to concentrate near the limit ray. The unknown functions
$G_{m}(\eta)$ are required to have a prescribed (exponentially small)
asymptotics as $\eta\to+\infty$, to match with the shadow asymptotics over the
limit ray constructed in [3].

An accompanying curious observation made in [3] was that changing in (2) the
variables from $(x,t)$ to $(\eta,t)$ and introducing new unknown function
$G(\eta,t)$ by

(13) $\psi(x,t)\,=\,t^{-1/2}\exp\left\\{i\frac{7}{120}t^{5}+\frac{i}{2}\eta
t^{3}+\frac{i}{2}\eta^{2}t\right\\}\,G(\eta,t),\ \
\eta=\frac{x}{t}-\frac{1}{6}t^{2}$

results in the following simple PDE for $G$, see formula (4.4) of [3]:

(14) $i\,t^{2}\frac{\partial G}{\partial t}\,+\,\frac{1}{2}\,\frac{\partial
G}{\partial\eta^{2}}\,=\,0.$

Comparing (12) with (13)–(14), one concludes that all one needs is to
determine the main-order searchlight amplitudes $G_{0}(\eta)$ for

(15) $G(\eta,t)\,\sim\,\sum_{m=0}^{\infty}t^{-m}G_{m}(\eta),\ \ \mbox{ as
}t\to+\infty,$

specifying all the higher-order amplitudes via recurrence relations

(16) $G_{m+1}(\eta)\,=\,-\,\,\frac{i}{2m+2}{G_{m}^{{}^{\prime\prime}}(\eta)},\
\ \ \ j=0,1,....$

Determination of $G_{0}$ however requires additional information about the
unknown solution $\psi(x,t)$. In [17] Popov, using a Green’s function
representation for (2), has expressed searchlight amplitudes equivalent to
$G_{0}(\eta)$ up to a change of relevant variables in terms of certain
integrals involving still unknown solution’s Neumann data
$\partial_{x}\psi(0,t)=:f(t)$ at the boundary $x=0$. The formulas derived in
[17] were implemented in [18] for computing the searchlight amplitudes
numerically (where $f(t)$ was in turn computed via a finite difference method
previously developed in [19]).

The derivation in [17] was made under an assumption that the surface “flux”
$f(t)$ decays fast as $t\to+\infty$, more precisely

(17) $\int_{1}^{\infty}|f(t)|\,\,t^{p}\,dt\,<\,+\infty,\ \ \ p=1,2,3,...\,\,.$

The assumption (17), yet unproven, appeared to be partially supported by
numerical evidence, [19]. Still, it was proved thereafter by the first author
in [21] that

(18) $\int_{1}^{\infty}|f(t)|^{2}\,t^{2}\,dt\,<\,+\infty.$

Some ideas of [21], as well as those of [3] and [4], serve as important
ingredients for the present work. One of the main results of the present work
is the following theorem on the searchlight amplitude $G_{0}$, which we state
in a slightly more general form.

######  Theorem 1.

Let $\psi(x,t)\in C^{\infty}(\overline{\Omega})$ be an infinitely smooth
solution to (2) satisfying boundary condition (3) and decay condition (11); in
particular $\psi(x,t$ can be the solution to problem (2)–(5).

Then there exists (unique) $G_{0}(\eta)\in H^{1}(\mathbb{R})$ with $\eta
G_{0}(\eta)\in L^{2}(\mathbb{R})$, and with the following properties. Let, for
$t>0$,

(19)
$\psi^{+}_{0}(x,t)\,:=\,t^{-1/2}\exp\left\\{i\frac{7}{120}t^{5}+\frac{i}{2}\eta
t^{3}+\frac{i}{2}\eta^{2}t\right\\}\,G_{0}(\eta),\ \ \ \
\eta:=\,\frac{x}{t}-\frac{1}{6}t^{2}.$

Then

(20)
$\left\|\psi(\cdot,t)\,-\,\psi^{+}_{0}(\cdot,t)\right\|_{L^{2}(0,+\infty)}(t)\,\rightarrow\,0,\
\ \ \mbox{as }\,t\to+\infty.$

Moreover, for $G(\eta,t)$ introduced for $t>0$ by (13) when
$\eta\geqslant-t^{2}/6$ and extended by zero for $\eta<-t^{2}/6$, as
$t\to+\infty$,

(21) $G(\eta,t)\,\rightharpoonup\,G_{0}(\eta)\,\,\mbox{weakly in
}\,H^{1}(\mathbb{R}),\ \ \eta G(\eta,t)\,\rightharpoonup\,\eta
G_{0}(\eta)\,\,\mbox{weakly in }\,L^{2}(\mathbb{R}).$

Theorem 1 provides a rigorous justification for the main-order term in the
searchlight ansatz (12) formally constructed in [4], and establishes some
properties of the amplitude $G_{0}(\eta)$ as well as some convergence
properties as $t\to+\infty$.

######  Remark 1.

The transformation (13) leading to (14) and then to (15) and (16), as
discovered in [3], plays a key role for the purposes of the present work.
Notice that a simple further change of variable $\tau=1/t$ immediately
transforms (14) into a free Schrödinger equation on $\mathbb{R}$:

(22) $i\,\frac{\partial
G}{\partial\tau}\,-\,\frac{1}{2}\,\frac{\partial^{2}G}{\partial\eta^{2}}\,=\,0,\
\ \ \tau=t^{-1}>0.$

This appears to be closely related to a “pseudoconformal symmetry” of a free
Schrödinger equation, see e.g. [23, 22, 24]. Indeed, as observed in [12], see
also [7] p.92, an inner problem for the boundary inflection could be derived
directly in (stretched) Cartesian rather than curvilinear coordinates. The
former yields a potential-free Schrödinger equation (i.e. as (2) but with no
$xt\psi$ term) but in a domain bounded by a cubic parabola. That can then be
transformed to (2) by “straightening” the boundary. Using that transformation
“in reverse” in (2), i.e. introducing new variable $\zeta=x-\frac{1}{6}t^{3}$
and new function $\varphi$ by

(23) $\psi(x,t)\,=\,\exp\left\\{\frac{i}{2}\zeta
t^{2}\,+\,i\frac{7}{120}t^{5}\right\\}\varphi(\zeta,t)$

results in $\varphi$ solving

(24) $i\frac{\partial\varphi}{\partial
t}+\frac{1}{2}\frac{\partial^{2}\varphi}{\partial\zeta^{2}}\,=\,0,$

which is a “parabolic wave equation” as in [12]. Then the transformation of
(24) leading to (22) is:

(25) $\eta\,=\,\frac{\zeta}{t},\ \ \tau\,=\,\frac{1}{t},\ \ \
\varphi(\zeta,t)\,=\,\tau^{1/2}\exp\left(i\frac{\eta^{2}}{2\tau}\right)G(\eta,\tau).$

Therefore the free Schrödinger equation is invariant under transformation (25)
(with $\tau$ replaced by $-\tau$ or $G$ by its complex conjugate), cf. eg.
[24]. See also [13] with further reference to [14] for some further invariance
properties of (24).

Combining the transformation (23) with (25) is easily seen to yield (13) with
(22). These transformations imply in particular that (19)–(20) is held for the
analogous problem on the whole line, $x\in\mathbb{R}$. A major technical
challenge, resolved in the present work, is to prove that (19)–(20) is still
held in the presence of the boundary condition (3). $\Box$

######  Remark 2.

Similarly to (10), the transformed problem (24) in domain
$\\{(\zeta,t):\zeta>-t^{3}/6\\}$ can be viewed as a perturbation, when
$t\to+\infty$, of the unperturbed problem on the whole of the
$(\zeta,t)$-plane with related (self-adjoint) operator
$A^{+}_{0}\varphi(\xi)=-\,\frac{1}{2}\varphi^{{}^{\prime\prime}}$, having a
continuous spectrum associated with a “free scattering” i.e. with no boundary.

A detailed proof of Theorem 1 will be given in Section 4. In the next section
we give some further interpretations of Theorem 1, in the light of Remark 2.

##  3\. Further interpretations of Theorem 1: searchlight wave operator and
inflection scattering operator

In this section we give a less formal sketch, avoiding fine technical details,
of how Theorem 1 leads to further implications and interpretations in terms of
a “searchlight” wave operator as $t\to+\infty$ and of an inflection scattering
operator.

Theorem 1 applies for every incoming whispering gallery wave (5), i.e. for
$j=1,2,3,...$. Notice that it also holds for any classical solution of (2)–(3)
which is infinitely smooth, $\psi(x,t)\in C^{\infty}(\overline{\Omega})$, and
decays as $x\to+\infty$ as in (11). It immediately follows from [4], cf.
Theorem 2.1 of [4], that the latter properties are satisfied by solutions to a
Cauchy problem to (2)–(3) with initial data from for example
$C_{0}^{\infty}(0,+\infty)$ at $t_{1}=+1$:

(26) $\psi(x,t_{1})\,=\,\Psi_{1}(x),\ \ \ \ \Psi_{1}\in
C_{0}^{\infty}(0,+\infty).$

This determines a linear mapping
$U:C_{0}^{\infty}(0,+\infty)\,\to\,L^{2}(\mathbb{R})$, by
$U\Psi_{1}:=G_{0}(\eta)$, where function $G_{0}$ is associated with the
solution of (2), (3), (26) according to Theorem 1.

It is easy to see that $U$ is an $L^{2}$-isometry, namely

(27)
$\left\|G_{0}\right\|_{L^{2}(\mathbb{R})}\,=\,\left\|\Psi_{1}\right\|_{L^{2}(0,+\infty)}.$

Indeed, denoting henceforth $\mathbb{R}^{+}:=(0,+\infty)$, and using the fact
that a solution to (2)–(3) has a $t$-independent $L^{2}(\mathbb{R}^{+})$-norm,
given $\Psi_{1}\in C^{\infty}_{0}(\mathbb{R}^{+})$,

(28)
$\|\psi(\cdot,t)\|_{L^{2}(\mathbb{R}^{+})}\,=\,\|\Psi_{1}\|_{L^{2}(\mathbb{R}^{+})},\
\ \ \forall t\in\mathbb{R}.$

On the other hand, from (19),

$\displaystyle\|\psi^{+}_{0}(\cdot,t)\|^{2}_{L^{2}(\mathbb{R}^{+})}\,=\,\int_{0}^{+\infty}t^{-1}\left|G_{0}\left(\frac{x}{t}-\frac{1}{6}t^{2}\right)\right|^{2}dx\,\,=\,\,\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (29) $\displaystyle\,\ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \
\int_{-\frac{1}{6}t^{2}}^{+\infty}|G_{0}(\eta)|^{2}d\eta\,\,\longrightarrow\,\,\|G_{0}\|^{2}_{L^{2}(\mathbb{R})},\
\ \ \mbox{as }\,t\to+\infty.$

Hence, via passing to the limit as $t\to+\infty$ and using (20) and (28),

$\|G_{0}\|_{L^{2}(\mathbb{R})}=\lim_{t\to+\infty}\|\psi^{+}_{0}(\cdot,t)\|_{L^{2}(\mathbb{R}^{+})}=\lim_{t\to+\infty}\|\psi-(\psi-\psi^{+}_{0})(\cdot,t)\|_{L^{2}(\mathbb{R}^{+})}=\|\Psi_{1}\|_{L^{2}(\mathbb{R}^{+})},$

yielding (27).

In a standard way, as $C_{0}^{\infty}(\mathbb{R}^{+})$ is dense in
$L^{2}(\mathbb{R}^{+})$, the above isometry is uniquely extended by continuity
from $C_{0}^{\infty}(\mathbb{R}^{+})$ to $L^{2}(\mathbb{R}^{+})$. Moreover, as
immediately follows from the proof of Theorem 1, see Lemma 3 below, the range
of this extension is actually the whole of $L^{2}(\mathbb{R})$. Hence the so
constructed $U$ is in fact a unitary operator, $U:L^{2}(\mathbb{R}^{+})\to
L^{2}(\mathbb{R})$.

This all can be interpreted in terms of existence of a wave operator as
$t\to+\infty$ as follows. Fixing $t_{1}>0$, e.g. $t_{1}=+1$, consider the
Cauchy problem for (2)–(3) with initial condition (26). Its solution
$\psi(x,t)$ determines for any $t\in\mathbb{R}$ an
$L^{2}(\mathbb{R}^{+})$-isometric “propagator” operator $U(t,t_{1})$, by
$\psi(\cdot,t)=U(t,t_{1})\Psi_{1}$. Hence $U(t,t_{1})$ is uniquely extended by
density to a unitary operator in $L^{2}(\mathbb{R}^{+})$, whose inverse
$U^{-1}(t,t_{1})=U(t_{1},t)$ is similarly related to solution to the Cauchy
problem with initial data at $t$.

A physical interpretation of the searchlight ansatz (12)–(14) is that, for
large positive $t$, the effect of the boundary $x=0$ becomes increasingly
insignificant: the wave energy tends to concentrate in a neighbourhood of the
limit ray $x=\frac{1}{6}t^{3}$ away from the boundary, so further propagation
of the wave beyond the inflection point i.e. for $t\to+\infty$, to a good
approximation, is such as if there were no boundary. This intuition suggests
that for $t>0$ one can take as an unperturbed problem simply the problem for
PDE (2) on the whole line $x\in\mathbb{R}$ with a similar initial condition at
$t=t_{1}$ as (26) but for $x\in\mathbb{R}$. The latter problem is equivalently
reformulated via transformation (13) as a Cauchy problem for (14) for $t>0$.

Notice that transformation (13) leading further to (22) similarly determines,
for any $\tau,\tau_{1}>0$, a unitary propagator $\tilde{U}_{0}(\tau,\tau_{1})$
in $L^{2}(\mathbb{R})$ for (22). Importantly the point $\tau=0$, corresponding
to $t\to+\infty$, remains a regular point for (22) hence with a well-defined
(strong) limit of $\tilde{U}_{0}(\tau,\tau_{1})$ as $\tau\to 0+$. Using now
the transformation (13) in reverse, determines the unitary propagator
$U_{0}(t,t_{1})$ for the unperturbed (i.e. with no boundary) problem for (2)
with the above described asymptotic behaviour as $t\to+\infty$.

All this can be interpreted in terms of existence of a unitary analogue of
wave operator $W_{+}:L^{2}(\mathbb{R})\to L^{2}(\mathbb{R}^{+})$ as
$t\to+\infty$, which can be defined as

(30)
$W_{+}\,:=\,\mbox{s-}\lim_{t\to+\infty}U^{-1}(t,t_{1})\chi_{+}U_{0}(t,t_{1}).$

Here $\chi_{+}$ denotes restriction of a function to positive semi-axis
$\mathbb{R}^{+}$, and $\mbox{s-}\lim$ stands for a strong limit in
$L^{2}(\mathbb{R})$. Notice the role of $\chi_{+}$ in (30) for approximating
for large positive $t$ the solutions to the problem with boundary ($x=0$) in
terms of solutions of the unperturbed problem without boundary, cf e.g. (48)
in the proof of Lemma 3 below.

We state the above as the following theorem.

######  Theorem 2.

There exists wave operator $W_{+}$ defined by (30), which is a unitary
operator from $L^{2}(\mathbb{R})$ into $L^{2}(\mathbb{R}^{+})$.

As was shown in [16], and in fact also follows from [4], there exists also a
“whispering gallery” wave operator $W_{-}$ when $t\to-\infty$ for a
reformulated Cauchy problem (10) for $t<0$ in $L^{2}(\mathbb{R}^{+})$. Taken
together, this determines an analogue of a scattering operator for the
inflection problem, via determining the searchlight amplitudes corresponding
to each incoming whispering gallery wave (5), $j=1,2,...\,$. One way for such
a construction is to notice that for any fixed $j$ and constant $D_{j}$ the
$L^{2}(\mathbb{R}^{+})$-norm of $\psi^{-}_{0}(x,t)$ given by (5) is
$t$-independent for $t<0$. Normalize $D_{j}$ in (5) so that
$\|\psi^{-}_{0}(\cdot,t)\|_{L^{2}(\mathbb{R}^{+})}\equiv 1$ for all $t<0$, and
let $G_{0}^{(j)}(\eta)$ be related searchlight amplitudes for the solution
$\psi^{(j)}(x,t)$ as $t\to+\infty$ according to Theorem 1. Then, because of
(4), the $L^{2}(\mathbb{R}^{+})$-norm $t$-independence of $\psi^{(j)}(x,t)$,
and (27), $\|G_{0}^{(j)}\|_{L^{2}(\mathbb{R}^{+})}=1$.

Let $H_{-}=l^{2}$ be the standard Hilbert space of complex-valued square-
summable sequences ${\bf a}=\left(a_{j}\right)_{j=1}^{\infty}$,

$\left\|{\bf
a}\right\|_{H_{-}}\,:=\,\left(\sum_{j=1}^{+\infty}|a_{j}|^{2}\right)^{1/2}.$

Then it is natural to define a scattering operator as the map
$S:l^{2}\rightarrow L^{2}(\mathbb{R})$ by

$S{\bf a}\,:=\,\sum_{j=1}^{\infty}a_{j}G_{0}^{(j)}(\eta).$

All the above implies the following

######  Corollary 1.

The above analogue of a scattering operator $S$ is a well-defined unitary
operator from $l^{2}$ to $L^{2}(\mathbb{R})$.

The physical meaning of the scattering operator $S$ is in specifying how the
intensities of the incoming whispering gallery modes at $t\to-\infty$
transform into the searchlight amplitudes as $t\to+\infty$.

##  4\. Proof of Theorem 1

It is more convenient to work with an equivalently transformed via (13)
problem for $G(\eta,\tau)$ solving (22), which we re-write as

(31) $i\,G_{\tau}\,-\,\frac{1}{2}\,G_{\eta\eta}\,=\,0,\ \ \ \tau=t^{-1}>0,$

using henceforth subscripts as shorthands for relevant partial derivatives.
Equation (31) holds in domain
$\tilde{\Omega}=\left\\{(\eta,\tau):\,\tau>0,\,\,\eta>-\,\frac{1}{6}\tau^{-2}\right\\}$
with boundary condition on a curve $l$ as follows

(32) $G(\eta,\tau)\,=\,0,\ \ \ (\eta,\tau)\in
l:=\left\\{(\eta,\tau):\,\eta=-\,\frac{1}{6}\tau^{-2},\,\tau>0\right\\}.$

By [4] $G\in C^{\infty}(\tilde{\Omega}\cup l)$, and the decay property (11)
immediately translates into: for all non-negative integers $\alpha$, $\beta$
and $\gamma$, and for any $0<\delta<\tau_{1}<+\infty$,

(33)
$\sup_{\delta\leqslant\tau\leqslant\tau_{1}}\left|\eta^{\alpha}\partial_{\tau}^{\beta}\partial_{\eta}^{\gamma}G(\eta,\tau)\right|\,\rightarrow\,0,\mbox{as
}\eta\to+\infty.$

For proving (21), and then (20), we aim at establishing the limit behaviour of
the above solution $G(\eta,\tau)$ as $\tau\to+0$. The following lemma
establishes necessary a priori estimates for $G$ for $0<\tau<\tau_{1}$. Below,
all the norms $\|\cdot\|$ are the $L^{2}$-norms for $\eta$ above the boundary
curve $l$, i.e. in $L^{2}\left(-\frac{1}{6}\tau^{-2},+\infty\right)$.

######  Lemma 1.

Let a smooth function $G(\eta,\tau)$ solve (31) in $\tilde{\Omega}$, so that
$G\in C^{\infty}(\tilde{\Omega}\cup l)$ and (32) and (33) hold, and let
$\tau_{1}>0$. Then there exist positive constants $C_{0}$, $C_{1}$ and
$C_{2}$, such that for all $0<\tau<\tau_{1}$,

  1. (i)

(34) $\|G\|(\tau)\,\equiv\,C_{0}\,>\,0;$

  2. (ii)

(35)
$\int_{0}^{\tau_{1}}\tau^{-3}\left|G_{\eta}\left(-\,\frac{1}{6}\tau^{-2},\tau\right)\right|^{2}d\tau\,<\,+\infty;$

  3. (iii)

(36) $\left\|G_{\eta}(\eta,\tau)\right\|(\tau)\,\leqslant\,C_{1};$

  4. (iv)

(37) $\left\|\eta G(\eta,\tau)\right\|(\tau)\,\leqslant\,C_{2};$

###### Proof.

Fix $\tau_{1}>0$, e.g. $\tau_{1}=1$.

(i): (34) is standard. Multiplying (31) by $\overline{G}$ (with the overbar
denoting the complex conjugate) and integrating in $\eta$ from $-\tau^{-2}/6$
to $+\infty$, integration by parts using (32) and (33) and then taking the
imaginary part results in $\frac{d}{d\tau}\left(\|G\|^{2}\right)=0$, which
yields (34).

(ii) & (iii): This follows a similar proof of (18) in [21]. Multiplying (31)
by $\overline{G_{\eta\eta}}$ and taking the imaginary part results in
$\overline{G_{\eta\eta}}G_{\tau}+{G_{\eta\eta}}\overline{G_{\tau}}=0$, or
equivalently

$\left(\overline{G_{\eta}}G_{\tau}+{G_{\eta}}\overline{G_{\tau}}\right)_{\eta}\,-\,\left(\left|G_{\eta}\right|^{2}\right)_{\tau}\,=\,0.$

Integrate the latter in $\eta$ from $-\tau^{-2}/6$ to $+\infty$ using (33),
and notice that

$\left(\left\|G_{\eta}\right\|^{2}\right)_{\tau}\,=\,\frac{d}{d\tau}\left(\int_{-\tau^{-2}/6}^{+\infty}\left|G_{\eta}(\eta,\tau)\right|^{2}d\eta\right)=\int_{-\tau^{-2}/6}^{+\infty}\left(\left|G_{\eta}\right|^{2}\right)_{\tau}d\eta-\frac{1}{3}\tau^{-3}\left(\left|G_{\eta}\right|^{2}\right)_{l},$

where $(\cdot)_{l}$ denotes the value of the expression in the brackets on the
curve $l$ i.e. for $\eta=-\tau^{-2}/6$. As a result,

(38)
$-\,\left(\overline{G_{\eta}}G_{\tau}+{G_{\eta}}\overline{G_{\tau}}\right)_{l}\,-\,\left(\left\|G_{\eta}\right\|^{2}\right)_{\tau}-\frac{1}{3}\tau^{-3}\left(\left|G_{\eta}\right|^{2}\right)_{l}\,=\,0.$

Now as $(G)_{l}=0$ for all $\tau$, the tangential derivative of $G$ along $l$
is zero, implying $G_{\tau}+\frac{1}{3}\tau^{-3}G_{\eta}=0$ on $l$ or
$G_{\tau}=-\frac{1}{3}\tau^{-3}G_{\eta}$. Using the latter in the first term
of (38) results in
$\frac{1}{3}\tau^{-3}\left(\overline{G}_{\eta}G_{\eta}+G_{\eta}\overline{G}_{\eta}\right)_{l}-\left(\left\|G_{\eta}\right\|^{2}\right)_{\tau}-\frac{1}{3}\tau^{-3}\left(\left|G_{\eta}\right|^{2}\right)_{l}\,=\,0$
or

(39)
$\tau^{-3}\left(\left|G_{\eta}\right|^{2}\right)_{l}\,=\,{3}\left(\left\|G_{\eta}\right\|^{2}\right)_{\tau}.$

Denoting $g(\tau):=\left(G_{\eta}\right)_{l}=G_{\eta}(-\tau^{-2}/6,\tau)$, and
integrating (39) from some $\tau>0$ to $\tau_{1}$, $\tau<\tau_{1}$, results in

(40)
$\int_{\tau}^{\tau_{1}}{\tau^{\prime}}^{-3}|g(\tau^{\prime})|^{2}d\tau^{\prime}\,+\,{3}\left\|G_{\eta}\right\|^{2}(\tau)\,=\,{3}\left\|G_{\eta}\right\|^{2}(\tau_{1}).$

For fixed $\tau_{1}>0$, the right hand side of (40) is constant, which
immediately yields (36) for $0<\tau<\tau_{1}$. For (35), it only remains to
pass to the limit in (40) as $\tau\to+0$.

(iv): Multiply now (31) by $\eta^{2}\overline{G}$ and take the imaginary part,
giving

$\eta^{2}\left(\overline{G}G_{\tau}+{G}\overline{G_{\tau}}\right)-\frac{1}{2i}\eta^{2}\left(\overline{G}G_{\eta\eta}-{G}\overline{G_{\eta\eta}}\right)\,=\,0,$

which can be equivalently rewritten as

(41)
$\left(\eta^{2}|G|^{2}\right)_{\tau}\,+\,\frac{i}{2}\left[\eta^{2}\left(\overline{G}G_{\eta}-{G}\overline{G_{\eta}}\right)\right]_{\eta}\,-\,i\eta\left(\overline{G}G_{\eta}-{G}\overline{G_{\eta}}\right)\,=\,0.$

Integrating (41) over $\eta$, the middle term vanished due to (32) and (33),
and as a result

(42) $\left(\left\|\eta
G\right\|^{2}\right)_{\tau}\,=\,i\int_{-\frac{1}{6}\tau^{-2}}^{+\infty}\left(\eta\overline{G}G_{\eta}\,-\,\eta{G}\overline{G_{\eta}}\right)d\eta.$

Denoting, for $0<\tau\leqslant\tau_{1}$, $I(\tau):=\left\|\eta
G\right\|(\tau)=\left(\int_{-\frac{1}{6}\tau^{-2}}^{+\infty}\eta^{2}|G(\eta,\tau)|^{2}d\eta\right)^{1/2}$,
and applying Cauchy-Schwartz inequality to the right hand side of (42) results
in $|2II_{\tau}|\leqslant\,2I\|G_{\eta}\|$. As a result, via (36),
$|I_{\tau}(\tau)|\leqslant\|G_{\eta}\|(\tau)\leqslant C_{1}$, and hence

$\|\eta G\|(\tau)=I(\tau)\,\leqslant\,I(\tau_{1})+C_{1}\tau_{1}\,=:\,C_{2},$

yielding (37). ∎

We regard, for $0<\tau\leqslant\tau_{1}$, $G(\cdot,\tau)$ as defined on the
whole of the real line $\mathbb{R}$ by extending it by zero for
$\eta<-\tau^{-2}/6$. Lemma 1 provides boundedness of $G(\eta,\tau)$ in the
$H^{1}(\mathbb{R})$ norm (via (34) and (36)), and of $\eta G(\eta,\tau)$ in
the $L^{2}(\mathbb{R})$ norm, via (37). The following well-known lemma
establishes analogues of compactness results under these conditions. As it
plays an important role for our proof of Theorem 1 we give its short proof for
the readers’ convenience.

######  Lemma 2.

Let, for all $0<\tau<\tau_{1}$, $G(\eta,\tau)\in H^{1}(\mathbb{R})$ with $\eta
G(\eta,\tau)\in L^{2}(\mathbb{R})$ be such that that the respective $H^{1}$
and $L^{2}$ norms are bounded, i.e. (34), (36) and (37) hold. Then there
exists a subsequence $\tau_{n}\to+0$ and $G_{0}\in H^{1}(\mathbb{R})$ with
$\eta G_{0}(\eta)\in L^{2}(\mathbb{R})$, such that

(43) $G(\eta,\tau_{n})\,\rightharpoonup\,G_{0}(\eta)\,\mbox{weakly in
}\,H^{1}(\mathbb{R}),\ \eta G(\eta,\tau_{n})\,\rightharpoonup\,\eta
G_{0}(\eta)\,\mbox{weakly in }L^{2}(\mathbb{R}).$

(44) $G(\eta,\tau_{n})\,\rightarrow\,G_{0}(\eta)\,\,\mbox{strongly in
}\,L^{2}(\mathbb{R}).$

###### Proof.

(i) By the lemma’s assumptions, for all $0<\tau<\tau_{1}$, $G(\cdot,\tau)$
belongs to Hilbert space $H$ of functions from $H^{1}(\mathbb{R})$ such that
$\|u\|^{2}_{H}:=\|u\|^{2}_{H^{1}(\mathbb{R})}+\|\eta
u\|^{2}_{L^{2}(\mathbb{R})}<\infty$, which specifies the norm in $H$. Then it
is easy to see that the theorem on weak compactness of a unit ball implies the
existence of $G_{0}\in H$ and of a subsequence $\tau_{n}\to 0+$, such that
(43) holds.

(ii) For a subsequence chosen in (i), for any $N=1,2,...$, (43) and the
Rellich compactness theorem ensure that

(45) $G(\eta,\tau_{n})\,\rightarrow\,G_{0}(\eta)\,\,\mbox{ strongly in
}\,L^{2}(-N,N).$

Now, denoting here for brevity $G_{n}(\eta):=G(\eta,\tau_{n})$,

$\|G_{n}-G_{0}\|_{L^{2}(\mathbb{R})}^{2}\,=\,\int_{-N}^{N}|G_{n}-G_{0}|^{2}d\eta+\int_{\eta>N}|G_{n}-G_{0}|^{2}d\eta\,\,\leqslant$
(46)
$\int_{-N}^{N}|G_{n}-G_{0}|^{2}d\eta+N^{-2}\int_{|\eta|>N}\eta^{2}|G_{n}-G_{0}|^{2}d\eta\leqslant\int_{-N}^{N}|G_{n}-G_{0}|^{2}d\eta\,+\,CN^{-2},\,\forall
n,N,$

with a constant $C$ independent of $n$ and $N$. (With latter inequality
following from (37) and (i) .) Now taking limit-supremum as $n\to\infty$ of
inequality (46) for every $N$ and using (45) results in
$\limsup_{n\to\infty}\|G_{n}-G_{0}\|^{2}\leqslant CN^{-2}$ for all
$N\in\mathbb{N}$. Finally, (44) follows from the latter as $N$ is arbitrary. ∎

To complete the proof of Theorem 1 we essentially need to show that for the
actual solution satisfying (31)–(33) the choice of $G_{0}$ according to Lemma
2 is actually independent of the subsequence and therefore the convergences in
(43) and (44) actually hold for the whole of $t\to+0$. The latter is being
achieved below by adapting and refining certain technical arguments from [4],
as follows.

To this end, we first aim at showing that, given a well-behaved function
$G_{0}(\eta)$, there exists a well-behaved solution to (31)–(32) for $\tau>0$,
such that $G_{0}$ is its strong $L^{2}$-limit as $\tau\to+0$. To begin with,
we can take as such $G_{0}\in C_{0}^{\infty}(\mathbb{R})$. The following key
lemma holds:

######  Lemma 3.

Let $G_{0}\in C_{0}^{\infty}(\mathbb{R})$. Then there exists unique
$G(\eta,\tau)\in C^{\infty}(\tilde{\Omega}\cup l)$ a solution to (31) in
$\tilde{\Omega}$, satisfying (32) and (33), such that

(47) $\|G(\cdot,\tau)-G_{0}\|\,\to\,0\ \ \ \mbox{ as }\tau\to+0.$

Equivalently, there exists a unique solution $\psi(x,t)\in
C^{\infty}(\overline{\Omega})$ to (2), satisfying (3), (11), such that (20)
holds for $\psi^{+}_{0}(x,t)$ defined by (19).

###### Proof.

The proof follows the ideas from Lemmas 3.7–3.9 and Theorem 3.1 of [4],
readjusting the argument as necessary from the asymptotic condition (4) for
$t\to-\infty$ to (20) as $t\to+\infty$. In this respect, it is more convenient
to deal with the latter statement of the lemma, with the stated equivalence
following via the transformation (13), (22). Indeed, let $G_{0}(\eta)\in
C_{0}^{\infty}(\mathbb{R})$ with associated $G(\eta,\tau)$, and related
$\psi_{0}^{+}(x,t)$ and $\psi(x,t)$ determined by respectively (19) and (13)
(with $\tau=t^{-1}$ in the latter). Then, for the left hand side of (20), for
large enough $t$,

$\displaystyle\|\psi(x,t)-\psi^{+}_{0}(x,t)\|^{2}_{L^{2}(\mathbb{R}^{+})}\,\,=\,\,\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\displaystyle\int_{0}^{+\infty}t^{-1}\left|G\left(\frac{x}{t}-\frac{1}{6}t^{2},t\right)-G_{0}\left(\frac{x}{t}-\frac{1}{6}t^{2}\right)\right|^{2}dx\,=$
$\displaystyle\int_{-\frac{1}{6}\tau^{-2}}^{+\infty}\left|G(\eta,\tau)-G_{0}(\eta)\right|^{2}d\eta\,=\,\int_{-\infty}^{+\infty}\left|G(\eta,\tau)-G_{0}(\eta)\right|^{2}d\eta\,=$
(48) $\displaystyle\,\,\,\,\ \ \ \ \ \ \ \ \ \
\left\|G(\cdot,\tau)\,-\,G_{0}\right\|^{2}_{L^{2}(\mathbb{R})}.$

(Since the support of $G_{0}(\eta)$ is finite and $G(\eta,\tau)$ is defined to
vanish for $\eta<-\frac{1}{6}\tau^{-2}$, for small enough $\tau$ the second
integral in (48) could be replaced by that over the whole of the real line.)
Thus (48) establishes the stated equivalence.

1\. Let now $G_{0}\in C_{0}^{\infty}(\mathbb{R})$ have a support in
$[-\Lambda,\Lambda]$ for some $\Lambda>0$. Choose some large enough
$N\in\mathbb{N}$ and define an asymptotic solution to (2) as $t\to+\infty$, cf
(12),

(49)
$\psi^{(N)}(x,t)\,:=\,t^{-1/2}\exp\left\\{i\frac{7}{120}t^{5}+\frac{i}{2}\eta
t^{3}+\frac{i}{2}\eta^{2}t\right\\}\,\sum_{m=0}^{N}t^{-m}G_{m}(\eta),\ \
\eta=\frac{x}{t}-\frac{1}{6}t^{2},$

where $G_{m}$, $m=1,2,...,N$, are determined from $G_{0}$ by (16). It then
follows that $\psi^{(N)}(x,t)\in C^{\infty}(\overline{\Omega}_{*})$ and the
support of $\psi^{(N)}$ in $\Omega_{*}$ is separated from the boundary $x=0$
of $\Omega$, where $\Omega_{*}:=\\{(x,t):\,x>0,\
t>(6\Lambda)^{1/2}+1=:t_{*}\\}$. In particular, as $\psi^{(N)}(x,t)$ vanishes
for $\frac{x}{t}-\frac{1}{6}t^{2}<-\Lambda$, i.e. for
$x<\frac{1}{6}t(t^{2}-6\Lambda)$, $\psi^{(N)}$ satisfies the boundary
condition (3) in $\Omega_{*}$. Moreover, as $\psi^{(N)}$ also vanishes for
$\frac{x}{t}-\frac{1}{6}t^{2}>\Lambda$, the decay condition (11) is trivially
satisfied by $\psi^{(N)}$ for any $t_{*}<A_{1}<A_{2}<+\infty$.

We aim at constructing solutions to (2)–(3) with Cauchy data given by (49) at
large enough $t=T$ and ultimately showing that these solutions converge to the
sought solution $\psi(x,t)$ as $T\to+\infty$. So, following p.1570 of [4], for
all $\xi>t_{*}$ we define $\psi^{[N]}(x,t;\xi)$ as solution to the Cauchy
problem for (2)–(3) in $\Omega_{*}$ with initial data at $t=\xi$:

(50)
$L\psi^{[N]}:=\,i\psi^{[N]}_{t}\,+\,\frac{1}{2}\psi^{[N]}_{xx}\,+\,xt\,\psi^{[N]}\,=\,0\
\ \ \mbox{in }\,\Omega_{*},$ (51) $\psi^{[N]}(0,t;\xi)\,=\,0,\ \ \
\psi^{[N]}(x,\xi;\xi)\,=\,\psi^{(N)}(x,\xi).$

By Theorem 2.1 of [4] and its proof, for any $\xi>t_{*}$, there exists such a
unique function $\psi^{[N]}(x,t;\xi)\in C^{\infty}(\overline{\Omega_{*}})$
satisfying (11) for any $t_{*}<A_{1}<A_{2}<+\infty$.

2\. Following [4] further, we next argue that for any $(x,t)\in\Omega_{*}$ and
$\xi>t_{*}$, $\psi^{[N]}(x,t;\xi)$ is continuously differentiable in $\xi$,

(52) $\frac{\partial\psi^{[N]}(x,t;\xi)}{\partial\xi}\,=:\,V_{N}(x,t;\xi),$

and the derivative $V_{N}(x,t;\xi)$ is a unique solution in $\Omega_{*}$ to
another Cauchy problem

(53) $LV_{N}\,=\,0,\ \ \ V_{N}(0,t;\xi)\,=\,0,\ \ \
V_{N}(x,\xi;\xi)\,=\,-iL\psi^{(N)}.$

Again by Theorem 2.1 of [4], as defined by (53), $V_{N}\in
C^{\infty}(\overline{\Omega_{*}})$ and satisfies (11) for any
$t_{*}<A_{1}<A_{2}<+\infty$.

The relation (52) between the solutions of (50)–(51) and (53) is proved in
Lemma 3.7 of [4].111Formally, (52)–(53) can be shown by differentiating the
Cauchy data in (51) with respect to $\xi$:
$\psi^{[N]}_{t}(x,t;\xi)+\psi^{[N]}_{\xi}(x,t;\xi)=\psi^{(N)}_{\xi}(x,\xi)$
when $t=\xi$, and then as $\psi^{[N]}$ solves (50),
$\psi^{[N]}_{t}(x,t;\xi)=\frac{i}{2}\psi^{[N]}_{xx}(x,t;\xi)+ixt\psi^{[N]}(x,t;\xi)=\frac{i}{2}\psi^{(N)}_{xx}(x,\xi)+ix\xi\psi^{(N)}(x,\xi),\
\ t=\xi$ (having in the last equality used (51) again), which leads to (53).
This derivation has however to be adjusted as the differentiability of
$\psi^{[N]}$ with respect to the parameter $\xi$ is not known in advance, see
proof of Lemma 3.7 of [4] for the details.  Therefore, for any
$t_{*}<T_{1}<T_{2}<+\infty$ and for any $(x,t)\in\Omega_{*}$,

(54)
$\psi^{[N]}(x,t;T_{2})\,-\,\psi^{[N]}(x,t;T_{1})\,=\,\int_{T_{1}}^{T_{2}}V_{N}(x,t;\xi)\,d\xi.$

Further following [4], given a function $\psi(x,t)\in
C^{\infty}(\overline{\Omega})$ obeying (11), for any non-negative integers
$\alpha$, $\beta$ and $\gamma$, we introduce semi-norms

$\|\psi\|_{\alpha\beta\gamma}(t)\,:=\,\left(\int_{0}^{+\infty}x^{2\alpha}\left|\partial_{t}^{\beta}\partial_{x}^{\gamma}\psi(x,t)\right|^{2}dx\right)^{1/2}.$

The technical argument in the proof of Lemma 3.7 of [4] ensures that (52) is
infinitely differentiable in $x$ and $t$ with
$\partial^{\beta}_{t}\partial^{\gamma}_{x}V_{N}(x,t;\xi)$ continuous in $\xi$
for any $\beta$ and $\gamma$. As a result, the identity (54) is also
infinitely differentiable in $x$ and $t$, and yields Cook’s type estimates,
cf. e.g. [8]

(55)
$\left\|\psi^{[N]}(x,t;T_{2})\,-\,\psi^{[N]}(x,t;T_{1})\right\|_{\alpha\beta\gamma}(t)\,\leqslant\,\int_{T_{1}}^{T_{2}}\left\|V_{N}(x,t;\xi)\right\|_{\alpha\beta\gamma}(t)\,d\xi.$

3\. As $\psi^{(N)}$ is an asymptotic solution to (2) as $t\to+\infty$, it
directly follows from its construction, see (12), (14)–(16), that for
$t>t_{*}$

(56)
$\left\|L\psi^{(N)}\right\|_{\alpha\beta\gamma}(t)\,\leqslant\,C(\alpha,\beta,\gamma,N)\,t^{-2+3\alpha+4\beta+2\gamma-N}$

with $t$-independent constants $C(\alpha,\beta,\gamma,N)$.

Next, as problem (53) has initial data satisfying estimates (56) at $t=\xi$,
[4] implies similar estimate for $V_{N}$, cf Lemma 3.8 of [4]. Namely,
generally for a function $\psi(x,t)$ solving (2)–(3) and satisfying (11),
according to inequality (3.1) of [4], for any $t_{1},t_{2}\in\mathbb{R}$,

(57)
$\|\psi\|_{\alpha\beta\gamma}(t_{1})\,\leqslant\,\sum_{|J^{\prime}|\leqslant|J|,\beta^{\prime}=0}C_{JJ^{\prime}}\left(|t_{1}|^{|J|-|J^{\prime}|}+|t_{2}|^{|J|-|J^{\prime}|}+1\right)\|\psi\|_{\alpha^{\prime}0\gamma^{\prime}}(t_{2}).$

Here for multiindices $J=(\alpha,\beta,\gamma)$ and
$J^{\prime}=(\alpha^{\prime},\beta^{\prime},\gamma)$, we denote
$|J|:=3\alpha+4\beta+2\gamma$ and
$|J^{\prime}|:=3\alpha^{\prime}+4\beta^{\prime}+2\gamma^{\prime}$. The
constants $C_{JJ^{\prime}}$ are independent of $t_{1}$ and $t_{2}$.

Applying (57) to $\psi(x,t)=V_{N}(x,t;\xi)$ with $t_{1}=t$ and $t_{2}=\xi$,
for $t_{*}<A\leqslant t\leqslant B<+\infty$ and $\xi>t_{*}$, we obtain via
(53) and (56)

(58)
$\left\|V_{N}(x,t;\xi)\right\|_{\alpha\beta\gamma}(t)\,\leqslant\,C(\alpha,\beta,\gamma,A,B,N)\,\xi^{-2+3\alpha+4\beta+2\gamma-N}$

with constant $C(\alpha,\beta,\gamma,A,B,N)$ independent from $\xi$ and $t$.

Given $N$ and $1\leqslant t_{*}<A<B<+\infty$, (55) and (58) imply that, for
all $\alpha$, $\beta$ and $\gamma$ with $|J|\leqslant N$, $A\leqslant
t\leqslant B$, and $t_{*}<T_{1}<T_{2}$,

$\left\|\psi^{[N]}(x,t;T_{2})-\psi^{[N]}(x,t;T_{1})\right\|_{\alpha\beta\gamma}(t)\,\leqslant\,C(\alpha,\beta,\gamma,A,B,N)\int_{T_{1}}^{T_{2}}\xi^{-2}d\xi\,\longrightarrow\,0,$

as $T_{1},T_{2}\to+\infty$. Therefore, as $A$ and $B$ such as
$t_{*}<A<B<+\infty$ are arbitrary, there exists a function $\psi^{[N]}(x,t)$
such that $x^{\alpha}\partial^{\beta}_{t}\partial^{\gamma}_{x}\psi^{[N]}\in
L^{2}(\mathbb{R}^{+})$ for all $\alpha$, $\beta$ and $\gamma$ with
$|J|\leqslant N$ and for all $t>t_{*}$, and

(59)
$\left\|\psi^{[N]}(x,t;T)\,-\,\psi^{[N]}(x,t)\right\|_{\alpha\beta\gamma}(t)\,\,\rightarrow\,0\
\ \mbox{as }\,T\,\to+\infty.$

4\. The rest of the proof follows proof of Lemma 3.9 of [4] and shows that for
large enough $N$ the above constructed $\psi^{[N]}(x,t)$ does not actually
depend on $N$ and is the sought unique smooth solution $\psi(x,t)$.

First, (59) implies via the embedding theorems that, for large enough $N$,
$\psi^{[N]}(x,t)$ is a classical solution of (2) satisfying boundary condition
(3). Further, since a solution to (2)–(3) has a $t$-independent
$L^{2}(\mathbb{R}^{+})$ norm, the uniqueness would immediately follow from
(20). So all what remains to show is that, for large enough $N$,
$\psi^{[N]}(x,t)$ obeys (20), i.e. (with $\|\cdot\|$ denoting the
$L^{2}(\mathbb{R}^{+})$ norm)

(60)
$\left\|\psi^{[N]}(x,t)\,-\,\psi^{+}_{0}(x,t)\right\|(t)\,\rightarrow\,0,\ \ \
\mbox{as }\,t\to+\infty.$

We have, for all $t,T>t_{*}$,

$\displaystyle\left\|\psi^{[N]}(x,t)\,-\,\psi^{+}_{0}(x,t)\right\|\,\,\leqslant\,\,\left\|\psi^{[N]}(x,t)-\psi^{[N]}(x,t;T)\right\|\,\,+\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (61) $\displaystyle\,\ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \
\left\|\psi^{[N]}(x,t;T)-\psi^{(N)}(x,t)\right\|\,+\,\left\|\psi^{(N)}(x,t)-\psi^{+}_{0}(x,t)\right\|.$

Keeping first $t>t_{*}$ fixed and passing to the limit as $T\to+\infty$, the
first term on the right hand side vanishes due to (59). Next, as the second
term vanishes for $t=T$ by (51) and $L\psi^{[N]}=0$, due to a general estimate
$\left|\frac{d}{dt}\|\varphi\|\,\right|\leqslant\|L\varphi\|$, for $T>t$,

$\left\|\psi^{[N]}(x,t;T)-\psi^{(N)}(x,t)\right\|\,\leqslant\,\left|\int_{t}^{T}\left\|L\psi^{(N)}(x,t^{\prime})\right\|dt^{\prime}\right|\,\,\,\,\leqslant\,\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (62) $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \
\int_{t}^{+\infty}\left\|L\psi^{(N)}(x,t^{\prime})\right\|dt^{\prime}\,\,\leqslant\,\,C_{N}t^{-1-N}$

with some $t$-independent constant $C_{N}$. (The last inequality has used (56)
with $\alpha=\beta=\gamma=0$.) Passing finally to the limit as $t\to+\infty$,
the second term vanishes by (62). Finally, the third term on the right hand
side of (61) vanishes as $t\to+\infty$ as the $L^{2}$-norm of every “higher-
order” term in (49), i.e. for $m\geqslant 1$, vanishes in the limit.

This proves (60), and for completing the proof of the lemma it remains to
observe that by the uniqueness $\psi(x,t):=\psi^{[N]}(x,t)$ is, for large
enough $N$, independent of $N$. Therefore, taking $N$ large enough,
$x^{\alpha}\partial^{\beta}_{t}\partial^{\gamma}_{x}\psi=x^{\alpha}\partial^{\beta}_{t}\partial^{\gamma}_{x}\psi^{[N]}\in
L^{2}(\mathbb{R}^{+})$ for all non-negative $\alpha$, $\beta$ and $\gamma$,
and so $\psi(x,t)\in C^{\infty}(\overline{\Omega_{*}})$ with (11) held. As a
solution to (2), (3), by e.g. Theorem 2.1 of [4], it can be uniquely continued
for all $t\in\mathbb{R}$ with all the required properties, so the lemma is
proved. ∎

Lemma 3 allows to complete the proof of Theorem 1 by implying that the $G_{0}$
constructed in Lemma 2 does not actually depend on the choice of the
subsequence $\tau_{n}$ and the stated convergences therefore hold for the
whole of $\tau\to+0$, as follows.

######  Lemma 4.

Let $\psi(x,t)\in C^{\infty}(\overline{\Omega})$ be a solution to (2),
satisfying the boundary condition (3) and the decay condition (11), and let
function $G(\eta,\tau)$ be associated with it via the transformation (13),
(22).

Then the convergences in (43) and (44) are held for the whole of $\tau\to+0$,
and such $G_{0}$ is therefore unique.

For this $G_{0}$ assertions (20) and (21) are valid, and therefore Theorem 1
holds.

###### Proof.

Let $\psi(x,t)$ be a solution to (2) satisfying (3) and (11), and let
$G(\eta,\tau)$ be associated with it via (13), (22). Then $G(\eta,\tau)$
satisfies assumptions of Lemma 1 and therefore of Lemma 2. Applying to
$G(\eta,\tau)$ Lemma 2, let $\tau_{n}\to+0$ be a related subsequence, and let
$G_{0}\in H^{1}(\mathbb{R})$ with $\eta G_{0}(\eta)\in L^{2}(\mathbb{R})$ be
the associated limit, see (43) and (44). Using the transformation (13), (22)
in reverse, we conclude that for $t_{n}=\tau_{n}^{-1}\to+\infty$ as
$n\to\infty$,

(63)
$\left\|\psi(x,t_{n})\,-\,\psi^{+}_{0}(x,t_{n})\right\|_{L^{2}(\mathbb{R}^{+})}(t_{n})\,\rightarrow\,0,\
\ \ \mbox{as }\,n\to+\infty,$

where $\psi_{0}^{+}$ is given by (19). This means that (20) is held for the
subsequence $t_{n}\to+\infty$, and we need to show that it actually holds for
the whole of $t\to+\infty$.

Since $C_{0}^{\infty}(\mathbb{R})$ is dense in $L^{2}(\mathbb{R})$, for any
$\varepsilon>0$ choose $G_{0}^{\varepsilon}\in C_{0}^{\infty}(\mathbb{R})$
such that $\|G_{0}^{\varepsilon}-G_{0}\|_{L^{2}(\mathbb{R})}<\varepsilon$.
Applying for every such $G_{0}^{\varepsilon}$ Lemma 3, let
$\psi^{\varepsilon}(x,t)\in C^{\infty}(\overline{\Omega})$ be associated
solution to (2), satisfying (3) and (11), so that (20) and (19) read

(64)
$\left\|\psi^{\varepsilon}(x,t)\,-\,\psi^{\varepsilon}_{0}(x,t)\right\|_{L^{2}(\mathbb{R}^{+})}(t)\,\rightarrow\,0,\
\ \ \mbox{as }\,t\to+\infty,$

where

(65)
$\psi^{\varepsilon}_{0}(x,t)\,=\,t^{-1/2}\exp\left\\{i\frac{7}{120}t^{5}+\frac{i}{2}\eta
t^{3}+\frac{i}{2}\eta^{2}t\right\\}\,G^{\varepsilon}_{0}(\eta),\ \ \ \
\eta:=\,\frac{x}{t}-\frac{1}{6}t^{2}.$

Now (with $\|\cdot\|$ denoting the $L^{2}(\mathbb{R}^{+})$ norm)

$\displaystyle\left\|\psi(x,t)\,-\,\psi^{+}_{0}(x,t)\right\|\,\,\leqslant\,\,\left\|\psi(x,t)-\psi^{\varepsilon}(x,t)\right\|\,\,\,\,+\
\ \ \ \ \ \ \ \ \ $ (66) $\displaystyle\,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \
\left\|\psi^{\varepsilon}(x,t)-\psi^{\varepsilon}_{0}(x,t)\right\|\,\,+\,\,\left\|\psi^{\varepsilon}_{0}(x,t)-\psi^{+}_{0}(x,t)\right\|.$

For the first term on the right hand side of (66), notice that
$\tilde{\psi}^{\varepsilon}(x,t):=\psi(x,t)-\psi^{\varepsilon}(x,t)$ is a
$C^{\infty}(\overline{\Omega})$-solution of (2) satisfying (3) and (11), as so
are both $\psi(x,t)$ and $\psi^{\varepsilon}(x,t)$. Hence
$\tilde{\psi}^{\varepsilon}(x,t)$ has a $t$-independent
$L^{2}(\mathbb{R}^{+})$-norm, $\|\tilde{\psi}^{\varepsilon}(x,t)\|\equiv
c_{\varepsilon}$, $\forall t$. On the other hand, taking $t=t_{n}$ and passing
to the limit as $n\to+\infty$, via (63) and (64),

$\displaystyle\left\|\psi(x,t)-\psi^{\varepsilon}(x,t)\right\|\,\equiv\,c_{\varepsilon}\,=\,\lim_{n\to+\infty}\left\|\psi(x,t_{n})-\psi^{\varepsilon}(x,t_{n})\right\|\,\,=$
(67) $\displaystyle\,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\lim_{n\to+\infty}\left\|\psi_{0}^{+}(x,t_{n})-\psi_{0}^{\varepsilon}(x,t_{n})\right\|\,=\,\left\|G_{0}-G_{0}^{\varepsilon}\right\|_{L^{2}(\mathbb{R})}\,\,<\,\varepsilon.$

The last limit in (67) is evaluated similarly to that in (29), with
$\psi_{0}^{+}(x,t)$ replaced by
$\psi_{0}^{+}(x,t)-\psi_{0}^{\varepsilon}(x,t)$ and $G_{0}$ by
$G_{0}-G_{0}^{\varepsilon}$. This straightforward modification of (29) also
immediately implies that for the third term on the right hand side of (66),

(68)
$\left\|\psi^{\varepsilon}_{0}(x,t)-\psi^{+}_{0}(x,t)\right\|\,\leqslant\,\left\|G_{0}-G_{0}^{\varepsilon}\right\|_{L^{2}(\mathbb{R})}\,=\,c_{\varepsilon}\,<\,\varepsilon.$

Hence, employing in (66) estimates (67) and (68),

(69)
$\left\|\psi(x,t)\,-\,\psi^{+}_{0}(x,t)\right\|\,\,\leqslant\,\,2\varepsilon\,+\,\left\|\psi^{\varepsilon}(x,t)-\psi^{\varepsilon}_{0}(x,t)\right\|,\
\ \ \forall t>0,\,\forall\varepsilon>0.$

Fixing first in (69) $\varepsilon>0$, and taking the limit-supremum as
$t\to+\infty$ using (64) yields

$\limsup_{t\to+\infty}\left\|\psi(x,t)\,-\,\psi^{+}_{0}(x,t)\right\|\,\leqslant\,\,2\varepsilon,\
\ \forall\,\varepsilon>0.$

Finally, (20) follows as $\varepsilon>0$ is arbitrary.

To complete the proof of the lemma (and with it of Theorem 1) it remains to
notice that via the transformation (13), (22), (20) immediately implies that
(44) does hold for the whole of $\tau\to+0$. Hence, by the uniqueness of the
limits, for any other subsequence $\tilde{\tau}_{n}\to+0$ the application of
Lemma 2 must produce the same limit $G_{0}$ and hence (43) is also held for
the whole of $\tau\to+0$, resulting in (21). ∎

##  5\. Concluding remarks

We have proved that the solution $\psi(x,t)$ to the inner scattering problem
(2)–(5) has a searchlight asymptotics as $t\to+\infty$ in the form (19) with
convergences (20) and (21) held. The searchlight amplitude $G_{0}(\eta)$ has
certain smoothness and decay properties, as also proved in Theorem 1. The
resulting analogue of a scattering operator is a well-defined unitary
operator, Corollary 1, and provides a full information of how the intensities
of the incoming whispering gallery modes transform into the outgoing
searchlight amplitudes. Still, the searchlight asymptotics as $t\to+\infty$ is
non-uniform in the sense that an additional effort would be needed for
understanding how exactly the solution behaves for large positive $t$ near the
boundary $x=0$. This would require a separate investigation.

## References

  * [1] Babič V. M., Buldyrev V. S., _Short-Wavelength Diffraction Theory_ , Springer, Berlin, 1991.
  * [2] Babič V. M., Kirpičnikova N. Y., _The Boundary-Layer Method in Diffraction Problems, Springer_ , Springer, Berlin, 1979.
  * [3] Babich V. M., Smyshlyaev V. P., _The scattering problem for the Schrödinger equation with a potential linear in time and in space. I: Asymptotics in the shadow zone_ , J. Sov. Math. 32 (1986), 103–111.
  * [4] Babich V. M., Smyshlyaev V. P., _The scattering problem for the Schrödinger equation with a potential linear in time and in space. II: Correctness, smoothness, behavior of the solution at infinity_ , J. Sov. Math. 38 (1987), 1562–1576.
  * [5] Baz A. I, Perelomov A. M. Zel’dovich Y. B., _Scattering, Reactions and Decay in Nonrelativistic Quantum Mechanics_ , Israel Program for Scientific Translations, Jerusalem, 1969.
  * [6] Buslaev V. S., Matveev V. B., _Wave operators for the Schrödinger equation with a slowly decreasing potential_ , Theor. Math. Phys. 2 (1970), 266–274.
  * [7] Hewett D. P., Ockendon J. R., Smyshlyaev V. P., _Contour integral solutions of the parabolic wave equation_ , Wave Motion 84 (2019), 90–109.
  * [8] Kato T., _Perturbation Theory of Linear Operators_ , Springer, 1966.
  * [9] Kazakov A. Y., _Special Function related to the concave-convex boundary problems of the diffraction theory_ , J. Phys. A: Math. Gen. 36 (2003), 4127–4141.
  * [10] Kazakov A. Ya., _“Separation of variables” in the Model Problems of the Diffraction Theory. A Formal Scheme_ , J. Math. Sci. (NY) 243 (2019), 715–725.
  * [11] Nakamura G., Shibata Y., Tanuma K., _Whispering gallery waves in a neighborhood of a higher order zero of the curvature of the boundary_ , Publ. Res. Inst. Math. Sci. 25 (1989), 605–629.
  * [12] Ockendon J. R., Tew R. H, _Thin-layer solutions of the Helmholtz and related equations_ , SIAM Rev. 54 (2012), 3–51.
  * [13] Ockendon J. R., Tew R. H, _Thin-layer solutions of the Helmholtz equation_ , Eur. J. Appl. Math. (2020), published online 6 October 2020.
  * [14] Olver P. J., _Applications of Lie Groups to Partial Differential Equations_ , Springer, 1991.
  * [15] Popov M. M., _The problem of whispering gallery waves in a neighbourhood of a simple zero of the effective curvature of the boundary_ , J. Sov. Math. 11 (1979), 791–797.
  * [16] Popov M. M., _Correctness of the problem of whispering gallery waves in a neighborhood of points of zero curvature of the boundary_ , J. Sov. Math. 19 (1982), 1487–1493.
  * [17] Popov M. M., _Whispering-gallery waves in a neighborhood of an inflection point of the boundary. Asymptotics of the wave field as $t\to\infty$_, J. Sov. Math. 32 (1986), 205–214.
  * [18] Popov M. M., Krasavin V. G. _Direction diagram of radiation in the problem of an inflection point of the boundary_ , J. Sov. Math. 32 (1986), 215–219.
  * [19] Popov M. M., Pshenchik I., _Numerical solution of the problem on whispering gallery waves in a neighborhood of a simple zero of the effective curvature of the boundary_ , J. Sov. Math. 11 (1979), 797–804.
  * [20] Smirnov A. B., Fedotov A. A., _Adiabatic evolution generated by a Schrödinger operator with discrete and continuous spectra_ , Functional Analysis and Its Applications 50 (2016), 76–79.
  * [21] Smyshlyaev V. P., _Concentration of the solution near a limit ray in the neighbourhood of an Inflection point of the boundary_ , J. Sov. Math. 55 (1991), 1757–1760.
  * [22] Sulem C., Sulem P. L., _The nonlinear Schrödinger equation. Self-focusing and wave collapse_ , Springer, New York, 1999.
  * [23] Talanov V. I., _Focusing of light in cubic media_ , J. Experimental Theor. Phys. Letters 11 (1970), 199–201.
  * [24] Tao T., _The pseudoconformal and conformal transformations_ , Online note (2013), https://terrytao.wordpress.com/2013/02/14/the-pseudoconformal-and-conformal-transformations/
  * [25] Yafaev D. R., _Mathematical scattering theory. General theory_ , American Mathematical Society, 1992.


# Imaging through a square multimode fiber by scanning focused spots with the
memory effect

Sylvain Mezil  Univ. Grenoble Alpes, CNRS, LIPhy, F-38000 Grenoble, France
Institut Langevin, ESPCI Paris, Université PSL, CNRS, 75005 Paris, France
Irène Wang  Univ. Grenoble Alpes, CNRS, LIPhy, F-38000 Grenoble, France
Corresponding author<EMAIL_ADDRESS>Emmanuel Bossy  Univ.
Grenoble Alpes, CNRS, LIPhy, F-38000 Grenoble, France

###### Abstract

The existence of a shift-shift memory effect, whereby any translation of the
input field induces translations in the output field in four symmetrical
directions, has been observed in square waveguides by correlation
measurements. Here we demonstrate that this memory effect is also observed in
real space and can be put to use for imaging purposes. First, a focus is
created at the output of a square-core multimode fiber, by wavefront shaping
based on feedback from a guide-star. Then, thanks to the memory effect, four
symmetrical spots can be scanned at the fiber output by shifting the wavefront
at the fiber input. We demonstrate that this property can be exploited to
perform fluorescence imaging through the multimode fiber, without requiring
the measurement of a transmission matrix.

Imaging at the tip of an endoscopic probe allows to see deeply, albeit
invasively, into living tissues and organisms. In order to reduce the damage
caused by its insertion, minimizing the diameter of endoscopic probes is an
important issue. Multimode fibers (MMFs) present an advantageous ratio between
the information content they carry and their footprint. However, optical
signals transmitted by MMFs are scrambled by modal dispersion. To recover this
information, most methods are based on prior knowledge of the scrambling
process by either measuring the transmission matrix [1, 2, 3, 4, 5, 6] or
learning the output speckle patterns [7, 8, 9]. Whatever the calibration
method, access to both the proximal and the distal end of the fiber is
required. This, combined with the sensitivity of the calibration results to
small perturbations of the fiber (deformations, temperature change), limits
the potential applications of these methods.

By many aspects MMFs bear similarities with scattering layers: as light
propagates through such layers, spatial information are also distorted or
scrambled. To look inside a scattering environment, a range of calibration-
free methods have emerged, based on the optical memory effect: in spite of
multiple scattering, a transformation of the input field can be transmitted by
the medium, resulting in the same transformation of the output field [10, 11].
In this case, even if the output field is unknown, its correlations can be
exploited for imaging [12, 13, 14, 15, 16]. Although memory-effect-based
imaging has been reported in fiber bundles [17], similar methods have not been
extended to single MMFs, due to their general lack of a proper memory effect.
A rotational memory effect has been observed in circular MMFs [18], but it
alone cannot allow 2D imaging. When combining the feedback signal from a
guide-star with memory-effect considerations, it has been shown that enough
information on the fiber transmission can be gathered, so that a small region
at the fiber output can be imaged, using measurements from only one end [19].
However, due to the absence of a real radial memory effect, the size and shape
of this region varies with the location of the guide-star and reduces to a
single point at the center of the fiber.

Recently, a special type of shift-shift memory effect in square-core multimode
fibers (SqMMFs) has been observed [20]. It has the ability to produce two-
dimensional (2D) translations of the output pattern, as opposed to the sole
azimuthal rotation reported in circular MMFs. According to this effect, that
we will now systematically refer to as the memory effect, any field at the
input of a SqMMF results in four speckle patterns shifting along four
symmetrical directions at its output. These four directions are mirror
reflections of the input shift direction relative to the edges of the square
core. This memory effect was demonstrated by cross-correlating speckle
patterns in the output plane for different shifts induced in the input plane,
producing four peaks in the cross-correlation space that correspond to four
speckle patterns shifting in the output plane.

The aim of the present work is to exploit this memory effect to perform
fluorescence imaging in SqMMFs by scanning focused spots in the sample plane.
In a parallel work, we also recently demonstrated that the memory effect in
sqMMFs can be also exploited for imaging without any prior assistance, through
a correlation-imaging approach with speckle patterns in the sample plane [21].
This however requires both significant averaging over speckle realizations and
solving an ill-posed inverse problem. The method proposed here has the
advantage to provide a straightforward reconstruction from a single scan,
directly through a focus-scanning microscopy approach. Starting from a focused
spot that we first obtain by use of a guide-star, we indeed demonstrate that
the memory effect allows to simply scan four foci across the sample plane, as
opposed to the previous observation that reported shifting peaks in the
correlation domain [20, 21]. These spots can be scanned around the guide-star
generated focus, over what we will call an isoplanetic patch, defined by the
observed range of the memory effect, typically $\sim$10 µin our experiment. By
exploiting the memory effect in SqMMFs to scan the focused spots in a sample
plane, we experimentally demonstrate imaging of fluorescence bead.

Our proposed approach consists of two steps: first, the field at the proximal
fiber facet is optimized to generate a focus at the fiber output using as
feedback the intensity at a single point (’guide-star’). Then, the optimized
input field is translated in 2D in the transverse plane, which induces the
scan of four foci over the sample, placed at the distal facet, and enables the
acquisition of a fluorescence image using a bucket detector on the proximal
side.

Based on symmetry arguments in the case of an ideal square waveguide with
mirror-like walls, we can show that, translating an optimized input field by
$\Delta\mathbf{r}_{a}=(x_{a},y_{a})$, leads, at the fiber output, to splitting
of the initial focus into four spots, each of them moving in symmetrical
directions: $\Delta\mathbf{r}_{a}\\!=\\!(x_{a},y_{a})$,
$\Delta\mathbf{r}_{a}^{\pm}=(x_{a},-y_{a})$,
$\Delta\mathbf{r}_{a}^{\mp}=(-x_{a},y_{a})$ and
$\Delta\mathbf{r}_{a}^{-}=(-x_{a},-y_{a})$. We experimentally demonstrate this
property using a setup for wavefront optimization and fluorescence imaging, as
depicted in Fig. 1.

Figure 1: Experimental setup. A 532-nm laser beam is sent onto a DMD. The
pattern displayed on the DMD diffracts a phase-modulated field in its first
order, which is selected by the iris, reflected by a dichroic mirror (DM) and
imaged by a lens and an objective (O1) onto the fiber input facet. The
fluorescent sample is placed at the fiber output; its emission is collected
back through the fiber, transmitted by the dichroic mirror and detected by a
photomultiplier (PMT). A camera (CMOS) at the fiber distal end provides an
artificial guide-star (using the intensity of only one pixel) for wavefront
shaping and transmission images of the sample.

The wavefront at the input of a SqMMF (CeramOptec, 100$\times$100 µm2 section
and 30-mm long) is shaped using a DMD (Digital Micromirror Device) that
displays binary Lee holograms [22]. Input modes consist of plane waves in the
DMD plane, corresponding to focused spots at the fiber entrance. They have a
spacing of 1 µm and cover a 80$\times$80 µm2 square centered on the fiber
facet (6400 input modes), the 10 µm margin on each side allowing the input
pattern to translate while remaining inside the fiber core. To focus light at
the fiber output, the relative phases of input modes can then be optimized
using the feedback signal from a guide-star collected on any kind of bucket
detector, provided here by the intensity measured at one single pixel of the
camera plane conjugated to the fiber output facet. Using the DMD, the phase of
each input mode is varied relative to that of a reference spot (located in the
corner of the fiber facet). Five levels of phase, spanning the [0,2$\pi$]
range, are sequentially acquired, while monitoring the feedback signal; the
optimal phase for each mode is determined using a sinusoidal fit. Finally, the
optimized wavefront is obtained by summing all input modes, each with its
optimal phase. This leads to a single focus at the fiber output (Fig. 2). The
measured enhancement factor (computed as the ratio of the peak intensity of
the focus divided by the average intensity of the background speckle over the
remaining area of the fiber [5]) exceeds 2000.

After optimization, we shift the pattern in the input plane of the fiber by
adding a tilt to the optimal phase map on the DMD. The input scan range is
typically 20$\times$20 µm2 with a step of 0.5 µm to 0.25 µm in either
direction.

Figure 2: Memory-effect induced scanning through a square multimode fiber. The
left column shows camera images of the input facet of the fiber (the fiber
core is delimited by a solid white line) and the right column the
corresponding images of the fiber output. After optimisation of the input
wavefront in a 80$\times$80 $\mu$m region in the fiber core (dotted write
line), the fiber output is a focused spot (top row). By shifting the input
field by 4 $\mu$m in the $x$ direction, the output becomes two spots displaced
in opposite directions (middle). When the input field is shifted in both $x$
and $y$ directions, four spots are translated in symmetrical directions
(bottom). Red arrows show the translation vector of the input field,
$\Delta\mathbf{r}_{a}$, in both image spaces. Figure 3: Extension of the
isoplanetic patch. The maximum value of the four peaks in the four directions
$\Delta{r_{a}}^{\\{\;\;,-,\pm,\mp\\}}$ are extracted for each input shift (a),
producing a map for each peak showing its intensity as a function of the input
shift ($\Delta x_{a}$,$\Delta y_{a}$) (b). Line profiles at 0∘, (c) and at 45∘
(d) are extracted from these maps. Experimental values (symbols),
corresponding fits (colored lines) and the case of an infinite memory effect
(dashed line) are plotted. Estimated ranges for the memory effect
$\ell_{\text{ME}}$ are 8.1, 6.7, 5.6 and 6.8 µm for the spots at
$\Delta\mathbf{r}_{a}$, $\Delta\mathbf{r}_{a}^{\mp}$,
$\Delta\mathbf{r}_{a}^{\pm}$,$\Delta\mathbf{r}_{a}^{-}$, respectively. All
intensities values are normalized by the optimized spot maximum intensity.
Scale bar: 5 µm.

We first verified that foci are indeed scanned at the fiber output by
acquiring a camera image of the output intensity for each input shift. Fig. 2
shows that, when the input shift direction is along an edge of the square
core, the focus splits into two spots, while, in the general case, it splits
into four spots moving in symmetrical directions. These results confirms the
existence of the memory effect directly from the translation of a focal spot,
which was to date only demonstrated via cross-correlation of speckle patterns
[20, 21]. The intensities of these spots, normalized by the initial focus
intensity, are lower than the values predicted for the ideal case ($1/4$ for
the two-spots case and $1/16$ for the four-spots case, although the ratio
between two and four spots is preserved. Moreover, we observe approximately
equal intensities on the four spots, as expected.

The spots intensities, as a function of the input translation distance, are
presented on Fig. 3, where the maximum intensity of each peak is extracted
from the images and displayed on 2D maps as a function of input shift $(\Delta
x_{a},\Delta y_{a})$ (Fig. 3(b)). As predicted, higher intensity values are
observed for shift directions along the fiber core edge (horizontal and
vertical). Line profiles extracted from these maps show that the measured
intensities approach the ideal case at very small input shifts, but they
quickly drop from $||\Delta\mathbf{r}_{a}||\sim 3$ µm, and finally vanish
around $||\Delta\mathbf{r}_{a}||\sim 10$ µm. This maximum translation distance
is in agreement with the extent of the memory effect observed in Ref. [20]
from computed correlations.

To quantify more precisely the extent of the isoplanetic patch, we fitted
these 2D maps by assuming four diffraction-limited Airy spots (at their
theoretical positions), which intensity decreases exponentially as a function
of $||\Delta\mathbf{r}_{a}||$ with a characteristic length $\ell_{\text{ME}}$,
which is hereafter referred to as the range of the memory effect. This
empirical expression describes reasonably well the experimental measurements,
as shown on the line profiles of Fig. 3(c) and (d), and the fits yield
$\ell_{\text{ME}}$ values ranging between 5 and 8 µm.

By measuring $\ell_{\text{ME}}$ for different guide-star location on the fiber
output facet, we observed that $\ell_{\text{ME}}$ remains mostly unchanged
within the fiber core, and is reduced only around the corners. Interestingly,
unlike circular fibers in which case the isoplanetic patch is highly elongated
at the fiber edge and reduces to a point at the center [19], SqMMFs exhibit a
much more stable behaviour across the whole fiber facet, although a maximal
$\ell_{\text{ME}}$ is observed at the center.

Figure 4: Imaging results. (a) Acquisition of the output facet for a shift of
$\Delta_{x}$=$\Delta_{y}$=3 µm (from the case in (c,h,m)). (b) Representation
of the scan direction of each quadrant associated with
$\Delta{r_{a}}^{\\{\;\;,-,\pm,\mp\\}}$. (c-g) Bright field image of the
sample. Red square delimits the scanned area. Dashed lines separate the
quadrants of each focus point. (h-l) Reconstruction of the expected image.
(m-q) Maximum of the fluorescent signal acquired on the PMT. Cases are
associated per line (i.e., (c)-(h)-(m)). Scale bar: 2 µm.

This memory effect was then exploited for fluorescence imaging by placing a
coverslip with dispersed 1 µm-diameter fluorescent beads at the fiber output.
Results are displayed in Fig. 4. The first column of Fig. 4 (c-g) displays the
bright-field images of the sample. Each scanned area contains 2 to 3
fluorescent beads; cases (c)-(f) having two beads in only one quadrant, while
case (g) have three beads in three different ones. For this experiment, the
optimised input field is shifted from 0 to 10 µm (in $x$ and $y$ directions)
so each output spot scans one quadrant of the sample (each spot scans a 10x10
µm2 area, making the total scanned area of 20x20 µm2). The four quadrants and
the associated spots are represented in Fig. 4(a-b). We chose to represent the
fluorescence image in the input translation frame $\Delta\mathbf{r}_{a}$. The
resulting 10x10 µm2 image is hence the superposition of the four quadrants
that are simultaneously scanned. Since the scan direction of the spot differs
in each quadrant (see axes in Fig. 4(b)), each contribution provides a flipped
image of the real sample.

The second column of Fig. 4 (i.e., (h)-(l)) corresponds to the expected
results: they are obtained by folding the bright-field images and take into
account the exponential decay of the spots intensities. The center of the scan
is determined from images of the output spots acquired during the scan. For
the cases (c)-(f), the sample is moved so that the same two beads are
successively in each quadrant of the scanned area. In the case of Fig. 4(c)
where the beads are scanned by the spot associated to $\Delta\mathbf{r}_{a}$,
the expected image is in the same direction as the bright-field image. The one
from the second case (Fig. 4(d), associated with $\Delta\mathbf{r}_{a}^{\mp}$)
preserves the vertical axis but flips the horizontal one, while the reverse is
observed from the third case (Fig. 4(e), associated with
$\Delta\mathbf{r}_{a}^{\pm}$)). Finally, the one from the fourth case (Fig.
4(g), associated with $\Delta\mathbf{r}_{a}^{-}$)) flips both axis. (The fifth
case will be discussed later.) Furthermore, due to the limited memory effect
(see Fig. 3), the expected beads intensities decrease with distance from the
center.

The third column (Fig. 4(m-q)) corresponds to the images reconstructed from
the photomultiplier (PMT). One can see the very good agreement between the
expected images and the actual reconstructed ones. For the cases Fig. 4(m-p),
the two beads are retrieved in the expected position. A small shift can be
observed for the fourth case, which is attributed to a shift of the sample
during acquisition. A background signal is also observed, due to the fraction
of light that is not focused at the fiber output and forms a background
speckle (part of this speckle can be observed in Fig. 4(a)). This speckle
illuminates the sample on a 100x100 µm2 area (the fiber core) where several
other beads are present. However, while this background decreases the signal-
to-noise ratio, it clearly remains possible to detect and the localize of the
fluorescent beads.

Since the signal detected by the PMT is the fluorescence excited
simultaneously by the four foci, it is impossible to retrieve from which
quadrant the signal originates from a single image. As long as all the beads
are within one quadrant, this ambiguity affects only the direction of the
image (which can undergo mirror symmetries). Conversely, if several beads are
in different quadrants, this can lead to a wrong image reconstruction as
illustrated in Fig. 4(g,l,q). Here, there are three beads dispersed in the
upper-right, in the lower-right and, further, in the lower-left quadrants
(associated with $\Delta\mathbf{r}_{a}$, $\Delta\mathbf{r}_{a}^{\pm}$ and
$\Delta\mathbf{r}_{a}^{-}$, respectively). On the PMT image (which matches the
expected image), the first two beads appear to be distant of $\sim$3 µm while
they actually are $\sim$7 µm apart. (Note that the third bead is below the
background level.) From such image, the two beads could be either 3, 5, 7 or 8
µm apart. However, this ambiguity could be easily removed by slightly
translating the sample while observing the image, in $x$ and $y$: if a bead is
in the $\Delta\mathbf{r}_{a}$ quadrant, it shifts in the same direction as the
sample in both $x$ and $y$; in the $\Delta\mathbf{r}_{a}^{\pm}$ quadrant, it
shifts in the sample direction in $x$ and the opposite direction in $y$; etc.

In conclusion, we have demonstrated that, after optimizing the wavefront at
the entrance of a SqMMF using the feedback from a single guide-star, four foci
can be scanned at the fiber output by simply translating the input wavefront.
This property originates from the previously observed memory effect in SqMMFs.
We have shown that this effect can be exploited for fluorescence imaging in
the vicinity of the guide-star, an approach that only requires access to the
fiber proximal end. Although the persistence of the memory effect presently
limits the field-of-view to $\sim$10$\times$10 µm2, this area remains stable
across most of the fiber section, which is not the case for circular fibers.
We anticipate that by combining a priori models about the transmission of a
SqMMF with information provided by a guide-star, imaging fidelity and field-
of-view could be improved, making SqMMFs a promising alternative for micro-
endoscopy.

Funding H2020 European Research Council (COHERENCE 681514); H2020 Marie
Sklodowska-Curie Actions (DARWIN 750420).

Acknowledgments The authors thank Dorian Bouchet and Antonio Caravaca-Aguirre
for fruitful discussions and insightful advice.

## References

  * [1] Youngwoon Choi, Changhyeong Yoon, Moonseok Kim, Taeseok Daniel Yang, Christopher Fang-Yen, Ramachandra R Dasari, Kyoung Jin Lee, and Wonshik Choi.  Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber.  Physical review letters, 109(20):203901, 2012.
  * [2] Tomáš Čižmár and Kishan Dholakia.  Exploiting multimode waveguides for pure fibre-based imaging.  Nature communications, 3(1):1–9, 2012.
  * [3] Damien Loterie, Salma Farahi, Ioannis Papadopoulos, Alexandre Goy, Demetri Psaltis, and Christophe Moser.  Digital confocal microscopy through a multimode fiber.  Opt. Express, 23(18):23845–23858, 2015.
  * [4] Shay Ohayon, Antonio Miguel Caravaca-Aguirre, Rafael Piestun, and James J. Di Carlo.  Minimally invasive multimode optical fiber microendoscope for deep brain fluorescence imaging.  Biomed. Opt. Express, 9(4):1492–1509, 2018.
  * [5] Sergey Turtaev, Ivo T. Leite, Tristan Altwegg-Boussac, Janelle M. P. Pakan, Nathalie L. Rochefort, and Tomáš Čižmár.  High-fidelity multimode fibre-based endoscopy for deep brain in vivo imaging.  Light Sci. Appl., 7(92), 2018.
  * [6] Sebastian A. Vasquez-Lopez, Raphaël Turcotte, Vadim Koren, Martin Plöschner, Zahid Padamsey, Martin J. Booth, Tomáš Čižmár, and Nigel J. Emptage.  Subcellular spatial resolution achieved for deep-brain imaging in vivo using a minimally invasive multimode fiber.  Light Sci. Appl., 7(110), 2018.
  * [7] Reza Nasiri Mahalati, Ruo Yu Gu, and Joseph M. Kahn.  Resolution limits for imaging through multi-mode fiber.  Opt. Express, 54(1), 2013.
  * [8] Lyubov V. Amitonova and Johannes F. de Boer.  Compressive imaging through a multimode fiber.  Opt. Lett., 43(21):5427–5430, 2018.
  * [9] Babak Rahmani, Damien Loterie, Georgia Konstantinou, Demetri Psaltis, and Christophe Moser.  Multimode optical fiber transmission with a deep learning network.  Light: Science & Applications, 7(1):69, 2018.
  * [10] Shechao Feng, Charles Kane, Patrick A Lee, and A Douglas Stone.  Correlations and fluctuations of coherent wave transmission through disordered media.  Physical review letters, 61(7):834, 1988.
  * [11] Benjamin Judkewitz, Roarke Horstmeyer, Ivo M Vellekoop, Ioannis N Papadopoulos, and Changhuei Yang.  Translation correlations in anisotropically scattering media.  Nature physics, 11(8):684–689, 2015.
  * [12] Jacopo Bertolotti, Elbert G Van Putten, Christian Blum, Ad Lagendijk, Willem L Vos, and Allard P Mosk.  Non-invasive imaging through opaque scattering layers.  Nature, 491(7423):232–234, 2012.
  * [13] Ori Katz, Pierre Heidmann, Mathias Fink, and Sylvain Gigan.  Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations.  Nature photonics, 8(10):784–790, 2014.
  * [14] Sam Schott, Jacopo Bertolotti, Jean-Francois Léger, Laurent Bourdieu, and Sylvain Gigan.  Characterization of the angular memory effect of scattered light in biological tissues.  Opt. Express, 23(10):13505–13516, 2015.
  * [15] Dong Wang, Sujit K. Sahoo, Xiangwen Zhu, Giorgio Adamo, and Cuong Dang.  Non-invasive super-resolution imaging through dynamic scattering media.  Nature Communications, 12(1):3150, 2021.
  * [16] Lei Zhu, Fernando Soldevila, Claudio Moretti, Alexandra d’Arco, Antoine Boniface, Xiaopeng Shao, Hilton B. de Aguiar, and Sylvain Gigan.  Large field-of-view non-invasive imaging through scattering layers using fluctuating random illumination.  Nature Communications, 13(1):1447, 2022.
  * [17] Amir Porat, Esben Ravn Andresen, Hervé Rigneault, Dan Oron, Sylvain Gigan, and Ori Katz.  Widefield lensless imaging through a fiber bundle via speckle correlations.  Optics express, 24(15):16835–16855, 2016.
  * [18] Lyubov V Amitonova, Allard P Mosk, and Pepijn WH Pinkse.  Rotational memory effect of a multimode fiber.  Optics express, 23(16):20569–20575, 2015.
  * [19] Shuhui Li, Simon A. R. Horsley, Tomáš Tyc, Tomáš Čižmár, and David B. Phillips.  Memory effect assisted imaging through multimode optical fibres.  Nature Communications, 12(1):3751, 2021.
  * [20] Antonio M. Caravaca-Aguirre, Adrien Carron, Sylvain Mezil, Irène Wang, and Emmanuel Bossy.  Optical memory effect in square multimode fibers.  Opt. Lett., 46(19):4924–4927, 2021.
  * [21] Dorian Bouchet, Antonio M Caravaca-Aguirre, Philippe Moreau, Irène Wang, and Emmanuel Bossy.  Speckle-correlation imaging through a kaleidoscopic multimode fiber.  arXiv preprint arXiv:2212.14765, 2022.
  * [22] Wai-Hon Lee.  Computer-generated holograms: Techniques and applications.  Progr. Opt., 16:119–232, 1978.