On the nonreflecting boundary operators for the general two dimensional Schr\"odinger equation
Vishal Vaibhav
% insert abstract here Of the two main objectives we follow in this paper, the first one consists in the studying operators of the form $(\partial_t-i\triangle_{\Gamma)^{\alpha,\,\,\alpha=1/2,-1/2,-1,\ldots,$ where $\triangle_{\Gamma$ is the Laplace-Beltrami operator. These operators arise in the context of nonreflecting boundary conditions in the pseudo-differential approach for the general Schr\"odinger equation. The definition of such operators is discussed in various settings and a formulation in terms of fractional operators is provided. The second objective consists in deriving corner conditions for rectangular domains which are fundamentally inadmissible in the pseudo-differential approach. Stability and uniqueness of the solution is investigated for each of these novel boundary conditions.
lse_arXiv.tex
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%Title of paper
\title{On the nonreflecting boundary operators for the general two dimensional Schr\"odinger equation}

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\author{Vishal Vaibhav}
\email[]{vishal.vaibhav@gmail.com}
%\homepage[]{Your web page}
%\thanks{}
\noaffiliation{}
%\affiliation{Institute of Optics, Information and Photonics, University of Erlangen-Nuremberg, Staudtstr. 7/B2, 91058 Erlangen, Germany 
%and\\ Max Planck Institute for the Science of Light, G\"unther-Scharowsky-Strasse 1/Bau 24, 91058 Erlangen, Germany}
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\date{\today}

\begin{abstract}
% insert abstract here
Of the two main objectives we follow in this paper, the first one consists in the studying
operators of the form
$(\partial_t-i\triangle_{\Gamma})^{\alpha},\,\,\alpha=1/2,-1/2,-1,\ldots,$
where $\triangle_{\Gamma}$ is the Laplace-Beltrami operator. These operators arise in the context
of nonreflecting boundary conditions in the pseudo-differential approach for the
general Schr\"odinger equation. The definition of such operators is discussed in 
various settings and a formulation in terms of fractional operators is provided. The 
second objective consists in deriving corner conditions for rectangular domains which 
are fundamentally inadmissible in the pseudo-differential approach. Stability
and uniqueness of the solution is investigated for each of these novel boundary
conditions.
\end{abstract}

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% References should be done using the \cite, \ref, and \label commands
\section{Introduction}
In this article we consider the problem of construction of nonreflecting boundary
condition for the general two dimensional Schr\"odinger equation. In particular,
we consider the following initial-value problem (IVP):
\begin{equation}
\begin{split}\label{eq:2D-NLS}
&i\partial_tu+\triangle u+\phi(\vv{x}, t, |u|^2)u=0,\,\,(\vv{x},t)\in\field{R}^2\times\field{R}_+,\\
&u(\vv{x},0)=u_0(\vv{x}),\,\,\vv{x}\in\field{R}^2.
\end{split}
\end{equation}
The initial data is assumed to be supported within the computational domain,
$\Omega_i$, i.e., $\supp_{\vv{x}}u_0(\vv{x})\subset\Omega_i$. This problem has
been treated by several 
authors~\cite{Menza1996,Menza1997,AB2001,S2002,HH2004,ABM2004,Zheng2007,FP2011,ABK2012,ABK2013}. 
Exact formulations of the transparent boundary condition (TBC) for the free Schr\"odinger equation 
on convex domains, $\Omega_i$, with smooth boundary was provided by
Sch\"adle~\cite{S2002} in terms of a single and a double layer potential. The
special case of a circular domain was treated by Han and Huang~\cite{HH2004}.
Earlier attempts to derive an exact TBC for a rectangular domain by Menza proved to 
be problematic on account of the presence of corners~\cite{Menza1996,Menza1997}. This 
problem is resolved in a recent work by Feshchenko and Popov \cite{FP2011}. On
account of the lack of integrability, these techniques cannot be applied to 
the general Schr\"odinger equation and one has to turn to approximate methods
(refer to the review article~\cite{CiCP2008} for a comprehensive literature survey). 

For the approximate methods, we restrict ourselves to the pseudo-differential approach 
(in particular, the gauge transformation strategy) for constructing approximate 
nonreflecting boundary conditions referred to as 
the \emph{absorbing boundary conditions} or \emph{artificial boundary conditions} 
for various types of computational domains. Our goal is to
understand operators of the form
$(\partial_t-i\triangle_{\Gamma})^{\alpha},\,\,\alpha=1/2,-1/2,-1,\ldots,$ 
which appeared in the works of Menza~\cite{Menza1996,Menza1997} and Antoine et
al.~\cite{ABM2004,ABK2012}. Several aspects of such artificial boundary
conditions (ABCs) which comprises these operators are not quite well 
understood; we discuss these issues which motivate
the present work in the subsequent paragraphs. 

Contrary to the existing belief that the aforementioned operators can only be implemented via a Pad\'e
approximation of a monomial of fractional degree $\alpha$, i.e., $z^{\alpha}$,
it is shown in various settings that this operator can be expressed in terms of fractional
operators. For arbitrary functions $f(x,t),\,\,(x,t)\in\field{R}\times\field{R}_+$, the operation 
$(\partial_t-i\triangle_{x})^{\alpha}f(x,t),\,\,\alpha=1/2,-1/2,-1,\ldots,$ requires the knowledge of the
function over its entire support. If $\supp_xf(x,t)\subset\Gamma_x,\,\,\forall t\geq0$ or $f(x,t)$ is
periodic with respect to $x$ and it can be uniquely defined by its values at
$x\in\Gamma_x$ for all $t\geq0$, then it is shown that a formulation particularly convenient for
expressing the TBCs/ABCs for the IVP in \eqref{eq:2D-NLS} can be 
developed~\footnote{The TBCs derived by Feshchenko and Popov~\cite{FP2011}
happen to be a special case where this operation can be carried without
having to consider the function $f(x,t)$ over its entire support. The function 
$f(x,t)$ here refers to the restriction of the field
$u(\vv{x},t),\,\vv{x}\in\field{R}^2,$ to the segments of the boundary of the 
rectangular domain.}. Let us remark that the numerical implementation of 
such operators is not being considered in this paper; however, it can be easily seen that
the formulation developed in this paper makes these operators amenable to
convolution quadrature. It might be expected that the new scheme affords improvement in 
accuracy over the existing Pad\'e approximation based method~\footnote{A
distinction must be made between the Pad\'e approximation based methods which
are applied directly to the operator $(\partial_t-i\triangle_{x})^{\alpha}$ and
those applied to the fractional operator formulation of this operator discussed
in this paper.} reported in~\cite{ABK2013}. 

Further, it is well known that the pseudo-differential approach cannot be applied to
computational domains with corners. This precludes the rectangular domain which happens to be
a very convenient choice of the computational domain. The ABCs involving the
operators of the form $(\partial_t-i\triangle_{\Gamma})^{\alpha}$ cannot be adapted
to the rectangular domain in a straightforward manner; however, the ABCs obtained as local approximations
(with respect to $x$) or equivalently the high-frequency approximations 
of this operator admit of the possibility of constructing
corner conditions. We demonstrate this possibility for the free as well as the
general Schr\"odinger equation given by \eqref{eq:2D-NLS}. Our approach is closely related 
to the ideas presented in~\cite{BJR1990,Vacus2005}. 

Another program that we have followed in this paper is of obtaining the well-known energy 
estimate
\begin{equation}
    ||u(\vv{x},t)||_{L^2(\Omega_i)}\leq||u_0(\vv{x})||_{L^2(\Omega_i)},
\end{equation}
for the IVP in \eqref{eq:2D-NLS} with TBCs/ABCs as boundary conditions involving
operators of the form $(\partial_t-i\triangle_{\Gamma})^{\alpha}$ or high-frequency
approximations of it~\footnote{The corner conditions are also incorporated in
this equivalent formulation.}. Under the assumption that the solution exists, this 
inequality guarantees the stability as well as the uniqueness of the solution of the
equivalent initial boundary-value problem (IBVP). In 
certain cases, this result can be obtained by resorting to a general 
construct such that for any pseudo-differential operator, $P$, and a function
$u\in\fs{C}_0^{\infty}(\Omega)$ we have
\begin{equation}
2\Re\langle u|P u\rangle=\langle u|P u\rangle+\langle u|P^{\dagger} u\rangle,
\end{equation}
where $P^{\dagger}$ is adjoint of the operator $P$, $\Re$ stands for real part 
and $\langle u|v\rangle=\int_{\Omega}u^*vd\Omega$. Given the symbol
$\sigma_P$ of $P$, the symbol of the adjoint, $\sigma_{P^{\dagger}}$, can be computed 
using the following general formula: Assuming $y\in\field{R}^n$ and $\zeta$ the
covariable of $y$, then
\begin{equation}
    \sigma_{P^{\dagger}}(y,\zeta)\sim\sum_{\alpha\in\field{N}^n}
    \frac{1}{\alpha!i^{|\alpha|}}
    \partial^{\alpha}_{\zeta}[\partial^{\alpha}_{y}\sigma^*_P(y,\zeta)],
\end{equation}
where $\sigma_P^*$ stands for complex conjugate of $\sigma_P$ 
(complex conjugate of $z\in\field{C}$ is also denoted by $\overline{z}$). A more detailed 
discussion of this approach is provided in \ref{app:psido}.  

The discussion of the primary results in this paper is broadly divided into two
sections: Section \ref{sec:fse} deals with the free Schr\"odinger
equation while Section \ref{sec:gse} deals with the general Schr\"odinger
equation. For each of these problems, we consider two types of domains, namely,
the rectangular domain (or, infinite strip with periodic boundary condition along the
unbounded direction) and convex domains with smooth boundary. The basic definition of 
the operator $(\partial_t-i\triangle_{\Gamma})^{\alpha},\,\,\alpha=1/2,-1/2,-1,\ldots,$ 
is discussed in various settings. Two families of ABCs are considered in this
paper: first one obtained via the standard pseudo-differential approach and the second one
obtained as the high-frequency approximation of the former. The derivation of
corner conditions and the study of stability and uniqueness of the solution are
carried out separately for each of these problems in the subsections.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Free Schr\"odinger equation\label{sec:fse}}

Let us start our discussion with the linear case of the equation 
\eqref{eq:2D-NLS} with null-potential, i.e.,
\begin{equation}
\begin{split}\label{eq:2D-SE}
&i\partial_tu+\triangle u=0,\,\,(\vv{x},t)\in\field{R}^2\times\field{R}_+,\\
&u(\vv{x},0)=u_0(\vv{x}),\,\,\vv{x}\in\field{R}^2.
\end{split}
\end{equation}
The problem of constructing transparent boundary condition can be exactly treated for the case of 
compactly supported initial data for a computational domain $\Omega_i$ with a general 
smooth boundary $\Gamma$ \cite{S2002}. The basic approach involves solving the 
initial boundary value problem on the exterior domain. The exterior domain is defined by 
$\Omega_{e}=\field{R}^2\setminus\overline{\Omega}_i$. It is known that a boundedness 
condition at infinity does not ensure unique solution of the above IVP; one additionally 
needs to impose a Sommerfeld-like radiation condition at infinity in order to exclude all the 
incoming waves from infinity. This condition reads as
\begin{equation}
\lim_{|\vv{x}|\rightarrow\infty}
\sqrt{|\vv{x}|}\biggl(\nabla u\cdot\frac{\vv{x}}{|\vv{x}|}
+e^{-i\frac{\pi}{4}}\partial_t^{1/2}u\biggl)=0.
\end{equation}
Using the decomposition of $u(\vv{x},t)\in L^2(\field{R}^2)=L^2(\Omega_i)\oplus L^2(\Omega_e)$ 
and introducing the fields $v(\vv{x},t)$ and $w(\vv{x},t)$ we have
\begin{equation}
\begin{split}
&\begin{cases}
\label{decomposition}
&i\partial_tv+\triangle v=0,\,\,(\vv{x},t)\in\Omega_i\times\field{R}_+,\\
&v(\vv{x},0)=u_0(\vv{x}),\,\,\vv{x}\in\Omega_i;\\
&\\
&i\partial_tw+\triangle w=0,\,\,(\vv{x},t)\in\Omega_e\times\field{R}_+,\\
&u(\vv{x},0)=0,\,\,\vv{x}\in\Omega_e,\\
&\lim_{|\vv{x}|\rightarrow\infty}\sqrt{|\vv{x}|}\biggl(\triangle w\cdot\frac{\vv{x}}{|\vv{x}|}+e^{-i\frac{\pi}{4}}\partial_t^{1/2}w\biggl)=0;
\end{cases}\\
&v(\vv{x},t)|_{\Gamma}=w(\vv{x},t)|_{\Gamma},\,
\partial_{n}v(\vv{x},t)|_{\Gamma}=\partial_{n}w(\vv{x},t)|_{\Gamma}.
\end{split}
\end{equation}
Construction of the nonreflecting boundary conditions involves solving the exterior
problem exactly. In the following sections we first consider a infinite strip 
as computational domain (or periodic boundary condition in the direction it
extends to infinity) then extend the results to a rectangular domain. 

\begin{figure}[!htbp]
\begin{center}
\def\myscale{1}
\includegraphics[scale=\myscale]{./illustration_a.eps}
\end{center}
\caption{\label{fig:rect-domain} The computational domain in the form of an
infinite strip is depicted here. This domain can be replaced by a rectangular
domain if we assume periodic boundary condition along the unbounded direction.}
\end{figure}

\subsection{Infinite strip\label{sec:DSB}}
 In this section we restrict ourself to the case of an infinite strip with the 
 boundary parallel to one of the coordinate axes, or, rectangular domain with a periodic 
 boundary condition along one of the coordinate axes. Let the coordinate axes be
 labelled as $\vv{e}_1$ and $\vv{e}_2$. For the infinite strip,
 say with boundary parallel to the axis $\vv{e}_2$, we assume that $\supp_{x_1}u_0(\vv{x})$
 is bounded. The derivation of the TBCs for the infinite straight boundary is 
 particularly simple and can be obtained using Laplace transform in time and 
 a Fourier transform in space. Let us consider the IVP in \eqref{eq:2D-SE} where 
 the domain is defined by the infinite strip between $x_1=x_l$ and $x_1=x_r$. For 
 the TBC on the right boundary, one must consider the IBVP on the right exterior domain 
 $\Omega_l=[x_r,\infty)\times\field{R}$. Let the covariable of $(x_1,x_2)$ be denoted by $(\zeta_1,\zeta_2)$,
respectively. We introduce the notation 
$\fourier_{x_1}f(x_1,x_2,t)=\fourier_{x_1}[f](\zeta_1,x_2,t)$ for one dimensional Fourier 
transform with respect to $x_1$ (similarly 
$\fourier_{x_2}f(x_1,x_2,t)=\fourier_{x_2}[{f}](x_1,\zeta_2,t)$ for Fourier transform with
respect to $x_2$). For denoting the Laplace transform of a function we use $\laplace_t
f(t)=\laplace_t[f](s)=F(s)$.


Let us denote the Fourier transform with respect $x_2$ of $w(x_1,x_2,t)$ by
$\widetilde{w}(x_1,\zeta_2,t)$ and Laplace transform
with respect to $t$ of $\widetilde{w}(x_1,x_2,t)$ by $\widetilde{W}(x_1,\zeta_2, s)$, i.e.,
\begin{equation}
\begin{split}
&\widetilde{w}(x_1,\zeta_2, s) = \fourier_{x_2}w(x_1,x_2,t),\\
&\widetilde{W}(x_1,\zeta_2, s) = \laplace_t\widetilde{w}(x_1,\zeta_2,t).
\end{split}
\end{equation}
For the case of compactly supported initial data, we have
\begin{equation}
(\partial_{x_1}^2+\alpha^2)\widetilde{W}(x_1,\zeta_2,s)=0,\,\,x_1\in[x_r,\infty),
\end{equation}
where $\alpha=\sqrt{is-\zeta^2_2}$ such that $\Im(\alpha)>0$. The solution
can be worked out as follows: observing
\begin{equation*}
\begin{split}
&\partial_{x_1}\widetilde{W}(x_1,\zeta_2,s)
=i\alpha\widetilde{W}(x_1,\zeta_2,s),\\
&\laplace^{-1}[\partial_{x_1}\widetilde{W}(x_1,\zeta_2,s)]
=\laplace^{-1}[i\alpha^{-1}]\ast\laplace^{-1}[\alpha^2\widetilde{W}(x_1,\zeta_2,s)],
\end{split}
\end{equation*}
where `$\ast$' represents the convolution operation, we have
\begin{multline}
\partial_{x_1}\tilde{w}(x_1,\zeta_2,t)=e^{i\pi/4}e^{-i\zeta_2^2t}\partial_t^{-1/2}e^{i\zeta_2^2t}\\
\times\biggl[i\partial_t\tilde{w}(x_1,\zeta_2,t)+\widetilde{(\partial^2_{x_2}w)}(x_1,\zeta_2,t)\biggl].
\end{multline}
It is also easy to verify
\begin{equation}
\partial_te^{-i\zeta_2^2t}\partial_t^{-1/2}e^{i\zeta_2^2t}\widetilde{w}(x_1,\zeta_2,t) =
e^{-i\zeta_2^2t}\partial_t^{-1/2}e^{i\zeta_2^2t}\partial_t\widetilde{w}(x_1,\zeta_2,t).
\end{equation}
Now, taking the inverse Fourier transform, we obtain the Dirichlet to Neumann
map as~\cite{Menza1997}
\begin{widetext}
\begin{equation}\label{eq:IS-TBC}
\begin{split}
\partial_{x_1}{w}(\vv{x},t)
&=\frac{e^{i\pi/4}}{\sqrt{\pi}}\int_0^t\int_{\field{R}}\biggl[i\partial_{\tau}{w}(x_1,x'_2,\tau)+\partial^2_{x'_2}w(x_1,x'_2,\tau)\biggl]\frac{\mathcal{G}(x_2-x'_2,t-\tau)}{\sqrt{t-\tau}}dx_2'd\tau\\
&=-(\partial_{t}-i\partial^2_{x_2})\frac{e^{-i\pi/4}}{\sqrt{\pi}}\int_0^t\int_{\field{R}}w(x_1,x'_2,\tau)\frac{\mathcal{G}(x_2-x'_2,t-\tau)}{\sqrt{t-\tau}}dx_2'd\tau,
\end{split}
\end{equation}
\end{widetext}
where the convolution kernel is given by
\begin{equation}
\mathcal{G}(x_2,t)=\frac{e^{-i\pi/4}}{\sqrt{4\pi t}}e^{i\frac{x_2^2}{4t}},\,\,\textrm{with}\,\,\tilde{\mathcal{G}}(\zeta_2,t)=e^{-i\zeta_2^2t}.
\end{equation}
This map can be expressed compactly if we employ the notation
\begin{multline}\label{eq:Op-2D-TBC}
(\partial_t-i\partial_{x_2}^2)^{-1/2}f({x}_2,t)\\
=\frac{1}{\sqrt{\pi}}\int_0^t\int_{\field{R}}f(x_1,x'_2,\tau)\frac{\mathcal{G}(x_2-x'_2,t-\tau)}{\sqrt{t-\tau}}dx_2'd\tau,
\end{multline}
where $f$ is assumed to be an element of $\fs{C}^{\infty}(\field{R}\times\field{R}_+)$ such that it is of the 
Schwartz class with respect to $x_2$. On account of the singular nature of the
symbol, it is not a pseudo-differential operator. However, away from the points satisfying 
$\xi+\zeta_2^2=0$, it can be 
microlocally regarded as a pseudo-differential operator with a symbol $(i\xi+i\zeta_2^2)^{-1/2}$, with the 
branch cut defined by $-\pi\leq\arg{(i\xi+i\zeta_2^2)}<\pi$ where $(\zeta_2,\xi)$ are the covariables 
of $(x_2,t)$. Further, if we take the operator
$(\partial_t-i\partial_{x_2}^2)^{1/2}$ to be defined by
\begin{equation}
(\partial_t-i\partial_{x_2}^2)^{1/2}f=(\partial_t-i\partial_{x_2}^2)[(\partial_t-i\partial_{x_2}^2)^{-1/2}f],
\end{equation}
then
\begin{equation}
\partial_{x_1}{w}(\vv{x},t)+e^{-i\pi/4}(\partial_{t}-i\partial^2_{x_2})^{1/2}w(\vv{x},t)=0,\,\,(\vv{x},t)\in\Omega_l\times\field{R}_+.
\end{equation}



Next, we would like to obtain a more local approximation of this boundary condition valid for small times. 
To this end, let us consider
\begin{equation}
\laplace^{-1}[i\alpha^{-1}]=\frac{1}{2\pi}\int\frac{e^{st}}{\sqrt{is-\zeta^2_2}}ds.
\end{equation}
Setting $\xi=st$, we have
\begin{equation}\begin{split}
&\laplace^{-1}[i\alpha^{-1}]\\
&=\frac{1}{2\pi\sqrt{t}}\int_{a-i\infty}^{a+i\infty}\frac{e^{\xi}}{\sqrt{i\xi-\zeta^2_2t}}d\xi\\
&=\frac{1}{2\pi\sqrt{t}}\int_{a-i\infty}^{a+i\infty}\frac{1}{\sqrt{i\xi}}\biggl(1-\frac{\zeta^2_2t}{i\xi}\biggl)^{-1/2}e^{\xi}d\xi\\
&\sim\frac{1}{2\pi\sqrt{t}}\int_{a-i\infty}^{a+i\infty}\biggl(\frac{1}{\sqrt{i\xi}}+\frac{\zeta^2_2t}{2(i\xi)^{3/2}}+\frac{3\zeta^4_2t^2}{8(i\xi)^{5/2}}+\ldots\biggl)e^{\xi}d\xi\\
&\sim \frac{e^{i\pi/4}}{\Gamma(\tfrac{1}{2})}t^{-1/2}+\frac{e^{-i\pi/4}}{2\Gamma(\tfrac{3}{2})}t^{1/2}\zeta_2^2-\frac{3e^{i\pi/4}}{8\Gamma(\tfrac{5}{2})}t^{3/2}\zeta_2^4+\ldots
\end{split}\end{equation}
Taking the inverse Fourier transform, we obtain the asymptotic expansion as
\begin{equation}\label{eq:ABC-2D-HF}
\partial_{x_1}w+e^{-i\pi/4}\partial_t^{1/2}w-e^{i\pi/4}\frac{1}{2}\partial_{x_2}^2\partial_t^{-1/2}w=0\mod{(\partial_t^{-3/2})}.
\end{equation}

For the periodic case, we may take the computational domain to be 
$\Omega_i =[x_l,x_r]\times[0,2\pi]$ so that
\begin{align}
&w(x_1,x_2,t)=\sum_{m\in\field{Z}}\widetilde{w}_m(x_1,t)e^{im x_2},\\
&W(x_1,x_2,s)=\sum_{m\in\field{Z}}\widetilde{W}_m(x_1,s)e^{im x_2}.
\end{align}
with $\widetilde{W}_m(x_1,s)=\laplace[\widetilde{w}_m(x_1,t)]$. For simplicity we demonstrate 
the procedure for the $m$-th Fourier component. The
complete result is obtained by superposing all the components. Putting
$\alpha_m=\sqrt{is-m^2}$, the following results can be obtained in the same
manner as before to obtain
\begin{multline}
%&\partial_{x_1}\widetilde{W}_m(x_1,s)
%=i\alpha_m\widetilde{W}(x_1,s),\\
%&\laplace^{-1}[\partial_{x_1}\widetilde{W}_m(x_1,s)]
%=\laplace^{-1}[i\alpha_m^{-1}]\ast\laplace^{-1}[\alpha_m^2\widetilde{W}_m(x_1,s)],\\
\partial_{x_1}\widetilde{w}_m(x_1,t)=e^{i\pi/4}
e^{-i m^2t}\partial_t^{-1/2}e^{i m^2t}\\
\times\biggl[i\partial_t\widetilde{w}_m(x_1,t)-m^2\widetilde{w}_m(x_1,t)\biggl].
\end{multline}
It is also easy to verify the following relations:
\begin{align}
&\partial_te^{-im^2t}\partial_t^{-1/2}e^{im^2t}\widetilde{w}_m(x_1,t)\nonumber\\
&=e^{-im^2t}\partial_t^{-1/2}e^{im^2t}\partial_t\widetilde{w}_m(x_1,t),\\
&e^{-i m^2t}\partial_t^{-1/2}e^{i m^2t}
\biggl[i\partial_t\widetilde{w}_m(x_1,t)-m^2\widetilde{w}_m(x_1,t)\biggl]\nonumber\\
&=ie^{-im^2t}\partial_t^{1/2}e^{im^2t}\widetilde{w}_m(x_1,t)\label{eq:ISP-Op2}.
\end{align}
Using these relations, one can derive an equation analogous to \eqref{eq:IS-TBC} 
\begin{widetext}
\begin{equation}\label{eq:IPS-TBC}% Infinite periodic strip
\begin{split}
\partial_{x_1}{w}(\vv{x},t)
&=\frac{e^{i\pi/4}}{\sqrt{\pi}}\int_0^t\int_{0}^{2\pi}\biggl[i\partial_{\tau}{w}(x_1,x'_2,\tau)+\partial^2_{x'_2}w(x_1,x'_2,\tau)\biggl]\frac{\mathcal{G}(x_2-x'_2,t-\tau)}{\sqrt{t-\tau}}dx_2'd\tau\\
&=-(\partial_{t}-i\partial^2_{x_2})\frac{e^{-i\pi/4}}{\sqrt{\pi}}\int_0^t\int_{0}^{2\pi}w(x_1,x'_2,\tau)\frac{\mathcal{G}(x_2-x'_2,t-\tau)}{\sqrt{t-\tau}}dx_2'd\tau,
\end{split}
\end{equation}
\end{widetext}
with the convolution kernel given by
\begin{equation}\label{eq:kernel-periodic}
\mathcal{G}(x_2,t)=\sum_{m\in\field{Z}}e^{im x_2-im^2t},
\end{equation}
which is defined only in a distributional sense. For periodic functions
the operator defined in \eqref{eq:Op-2D-TBC} becomes 
\begin{multline}\label{eq:Op-ISP-TBC}
(\partial_t-i\partial_{x_2}^2)^{-1/2}f({x}_2,t)\\
=\frac{1}{\sqrt{\pi}}\int_0^t\int_{0}^{2\pi}f(x_1,x'_2,\tau)
\frac{\mathcal{G}(x_2-x'_2,t-\tau)}{\sqrt{t-\tau}}dx_2'd\tau,
\end{multline}
where the kernel is defined by \eqref{eq:kernel-periodic}. We now turn our
attention to obtaining a form of the operator
$(\partial_t-i\partial_{x_2}^2)^{1/2}$ which can be numerically implemented.
Using the relation \eqref{eq:ISP-Op2}, we may write
\begin{equation}
\begin{split}
\partial_{x_1}{w}(\vv{x},t)
&=-{e^{-i\pi/4}}\sum_{m\in\field{Z}}e^{-im^2t}\partial^{1/2}_t
\biggl[e^{im^2t}\widetilde{w}_m(x_1,t)e^{imx_2}\biggl],\\
&=-{e^{-i\pi/4}}\partial^{1/2}_{t'}\sum_{m\in\field{Z}}
\biggl[e^{-im^2(t-t')}\widetilde{w}_m(x_1,t')e^{imx_2}\biggl]_{t'=t}.
\end{split}
\end{equation}
Introducing the auxiliary function $\varphi(x_1,x_2,t,t')$ such that
\begin{equation}
\begin{split}
&\varphi(x_1,x_2,t,t')
=\sum_{m\in\field{Z}}\biggl[e^{-im^2(t-t')}\widetilde{w}_m(x_1,t')e^{imx_2}\biggl],\\
&\partial_{x_1}{w}(\vv{x},t)=-{e^{-i\pi/4}}\partial^{1/2}_{t'}\varphi(x_1,x_2,t,t')|_{t'=t}.
\end{split}
\end{equation}
In order to determine all the values of the function needed to compute the non-local
fractional derivative, the following relations can be used:
\begin{equation}\label{eq:FSE-auxi-ivp}
\begin{split}
&[i\partial_t+\partial^2_{x_2}]\varphi(x_1,x_2,t,t')=0,\,\,x_1\in[x_r,\infty),\\
&\varphi(x_1,x_2,t',t')=w(x_1,x_2,t').
\end{split}
\end{equation}
For all possible values of $t'$ satisfying $0\leq t'\leq t$ and $x_2\in[0,2\pi]$, one must 
solve the above initial value problem starting from the initial condition at
$t=t'$ under the periodic boundary condition in $x_2$. This process is
schematically shown in Fig. \ref{fig:IVP-auxi-periodic}. The history of the
field is needed along the vertical line up to the diagonal in the $(t,t')$-plane.
\begin{figure}[!htbp]
\begin{center}
\def\myscale{1}
\includegraphics[scale=\myscale]{./illustration_d1.eps}
%\subfloat[]{\includegraphics[scale=\myscale]{./illustration_e1.eps}}
\end{center}
\caption{\label{fig:IVP-auxi-periodic} Schematic illustrating how the auxiliary
equation \eqref{eq:FSE-auxi-ivp} will be solved in order to provide the
history of the field needed in the TBCs. The field is known along the diagonal
which serves as the initial conditions to obtain the values of the field needed
in the TBCs (arrow in the line depicts the evolution direction in time).}
\end{figure}
\begin{rmk}\label{rmk:periodic}
Such a procedure can be used to compute the action of any operator of the form
$(\partial_t-i\partial^2_{x})^{-n/2},\,\,n=1,2,\ldots,$ on any function periodic in $x\in\field{R}$, say,
$f(x,t)$. Introducing an auxiliary function $\varphi(x,t,t')$ such that
\begin{equation}
\begin{split}
&\varphi(x,t,t')
=\sum_{m\in\field{Z}}\biggl[e^{-im^2(t-t')}\widetilde{f}_m(t')e^{imx}\biggl]_{t'=t},\\
&(\partial_t-i\partial^2_{x})^{-n/2}f(x,t)=\partial^{-n/2}_{t'}\varphi(x,t,t')|_{t'=t}.
\end{split}
\end{equation}
The associated IVP is given by
\begin{equation}
\begin{split}
&[i\partial_t+\partial^2_{x}]\varphi(x,t,t')=0,\\
&\varphi(x,t',t')=f(x,t').
\end{split}
\end{equation}
\end{rmk}

\subsubsection{Stability and uniqueness}
An equivalent formulation of the IVP \eqref{eq:2D-SE} on the computational
domain $\Omega_i = [x_l,x_r]\times[0, 2\pi]$ with periodic boundary condition
along the axis $\vv{e}_2$ is given by
\begin{equation}
\begin{split}\label{eq:2D-SE-Omegai}
&i\partial_tu+\triangle u=0,\,\,(\vv{x},t)\in\Omega_i\times\field{R}_+,\\
&u(\vv{x},0)=u_0(\vv{x})\in L^2(\Omega_i),\,\,\supp\, u_0(\vv{x})\subset\Omega_i,\\
&u(x_1, x_2+2\pi,t)=u(x_1, x_2,t),\,\,t>0,\\
&\partial_{n}{u}(\vv{x},t)+
e^{-i\pi/4}(\partial_{t}-i\partial^2_{x_2})^{1/2}u(\vv{x},t)=0,\,\,\vv{x}\in\Gamma_l\cup\Gamma_r.
\end{split}
\end{equation}
Assuming that the solution $u(\vv{x},t)$ exists, we have
\begin{equation}
\begin{split}
&\int_{\Omega_i}(\partial_t|u|^2)d^2\vv{x}
=2\Re\int_{\Gamma_l\cup\Gamma_r}(u^*i\nabla u)\cdot d\vbs{\varsigma},\\
&||u(\vv{x},T)||^2_{L^2(\Omega_i)}-||u_0(\vv{x})||^2_{L^2(\Omega_i)}
=\Re\int_0^T dt\int_{\Gamma_l\cup\Gamma_r}(u^*i\nabla u)\cdot
d\vbs{\varsigma},
\end{split}
\end{equation}
where $d\vbs{\varsigma}=\vv{e}_n|d\vv{x}|$ for $\vv{x}\in\Gamma$.
Further noting $\tilde{u}_m(x_1,t)\in H^{1/4}([0, T])$, we have
\begin{equation}
\begin{split}
&\mathcal{I}_R\\
&=-2\Re\int_{\field{R}_+}
dt\int_{\Gamma_r}\biggl(u^*e^{i\pi/4}\sqrt{\partial_t-i\partial^2_{x_2}}u\biggl)dx_2,\\
&=-4\pi\Re\sum_{m\in\field{Z}}\int_{\field{R}_+}dt
\overline{\biggl(\tilde{u}_m(x_r,t)e^{-im^2t}\biggl)}
\biggl(\partial_t^{1/2}\tilde{u}_m(x_r,t)e^{-im^2t}\biggl),\\
&=-2\sum_{m\in\field{Z}}\int_{\{\xi<0\}}d\xi|\xi|^{1/2}
\biggl|\fourier_t\biggl[\tilde{u}_m(x_r,t)e^{-im^2t}\biggl](x_r,\xi)\biggl|^2\leq0.\\
\end{split}
\end{equation}
Similarly, it can be shown that
\begin{equation}
\mathcal{I}_L
=-2\Re\int_{\field{R}_+}
dt\int_{\Gamma_l}\biggl(u^*e^{i\pi/4}\sqrt{\partial_t-i\partial^2_{x_2}}u\biggl)dx_2<0.\\
\end{equation}
Therefore, we have 
\begin{equation}
||u(\vv{x},T)||_{L^2(\Omega_i)}\leq||u_0(\vv{x})||_{L^2(\Omega_i)}.
\end{equation}
This result also guarantees the uniqueness of the solution.  

\begin{figure}[!htbp]
\begin{center}
\def\myscale{1}
\includegraphics[scale=\myscale]{./illustration_b.eps}
\end{center}
\caption{\label{fig:rect-domain-periodic} Rectangular domain}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Rectangular domains\label{sec:rect-domain}}
Let us consider a rectangular domain given by 
\begin{equation}
\Omega_i=\{(x_1,x_2):x_b<x_2<x_t, x_l<x_1<x_r\},
\end{equation}
and denote the boundaries as $\Gamma_{l,r}=\{(x_1,x_2)\in\partial\Omega_i: x_1=x_{l,r}\}$, 
respectively and $\Gamma_{b,t}=\{(x_1,x_2)\in\partial\Omega_i: x_1=x_{b,t}\}$, respectively.
Assuming $\supp_{\vv{x}} u_0(\vv{x})$ bounded in $\Omega_i$, the TBCs for the
infinite strip cannot be taken to be the transparent boundary operators at the
straight edges because the corresponding operator requires knowledge of the entire 
support of the field along the tangential direction at the boundary. This clearly cannot 
be achieved because after a certain time the field would have spread outside the
domain defined by any segment of the rectangular domain, say, $\Gamma_r$. This issue can be resolved in the
following way: Using the same notation as in the last section and observing that
\begin{equation}
\begin{split}
&\partial_{x_1}\widetilde{w}(x_1,\zeta_2,t)\\
&=e^{i\pi/4}e^{-i\zeta_2^2t}\partial_t^{-1/2}e^{i\zeta_2^2t}
\biggl[i\partial_t\widetilde{w}(x_1,\zeta_2,t)
+\widetilde{(\partial^2_{x_2}w)}(x_1,\zeta_2,t)\biggl],\\
&\partial_{x_1}\widetilde{w}(x_1,\zeta_2,t)
=-e^{-i\pi/4}\partial_{t\rq{}}^{1/2}e^{-i\zeta_2^2(t-t\rq{})}
\widetilde{w}(x_1,\zeta_2,t\rq{})\biggl|_{t\rq{}=t},\\
&\partial_{x_1}{w}(x_1,x_2,t)
=-e^{-i\pi/4}\partial_{t\rq{}}^{1/2}{\varphi}(x_1,x_2,t,t\rq{})
\biggl|_{t\rq{}=t}.
\end{split}
\end{equation}
where the auxiliary function is defined by 
$\fourier_{x_2}{\varphi}(x_1,x_2,t,t\rq{})=e^{-i\zeta_2^2(t-t\rq{})}
\widetilde{w}(x_1,\zeta_2,t\rq{})$. It is easy to verify that the function $\varphi$ satisfies 
the following IVP
\begin{equation}
\begin{split}
&[i\partial_{t}+\partial^2_{x_2}]\varphi(x_1,x_2,t,t\rq{})=0,\\
&\varphi(x_1,x_2,t',t')=w(x_1,x_2,t').
\end{split}
\end{equation}
The boundary conditions at the endpoints of the segment $\Gamma_r$ are not
known because the original problem is defined on the infinite domain $x_2\in \field{R}$.
However it can be shown that the value of the auxiliary function $\varphi(x_1,x_2,t,t')$ 
at $t=0$ is compactly supported in $\Gamma_r$ for $x_1\in[x_r,\infty)$. This
would allow one to impose the transparent boundary conditions at the endpoints of
$\Gamma_r$. To this end, let us consider the IVP defined by \eqref{eq:2D-SE}, it can 
be solved using Fourier transform in $(x_1,x_2)$. Putting 
$\tilde{u}_0(\vbs{\zeta})=\fourier_{x_1}\fourier_{x_2}u_0(\vv{x})$, we have
\begin{equation}
\begin{split}
&\fourier_{x_1}\fourier_{x_2}u(\vv{x},t')=e^{-i(\zeta_1^2+\zeta_2^2)t\rq{}}
\tilde{u}_0(\vbs{\zeta}),\\
&\varphi(x_1,x_2,t,t')=\frac{1}{(2\pi)^2}\int_{\field{R}^2}e^{i\vbs{\zeta}\cdot\vv{x}-i\zeta_1^2t'-i\zeta_2^2t}
\tilde{u}_0(\vbs{\zeta})d^2\vbs{\zeta}.
\end{split}
\end{equation}
\begin{figure*}[!htbp]
\begin{center}
\def\myscale{1}
\subfloat[Evolution of $\varphi$ for $\vv{x}\in\Gamma_r\cup\Gamma_l$.]
{\includegraphics[scale=\myscale]{./illustration_d.eps}}
\subfloat[Evolution of $\varphi$ for $\vv{x}\in\Gamma_t\cup\Gamma_b$.]
{\includegraphics[scale=\myscale]{./illustration_e.eps}}
\end{center}
\caption{\label{fig:IVP-auxi}Schematic depiction of the evolution problem for
the auxiliary field $\varphi(x_1,x_2,t,t')$ in the $(t,t')$-plane. The filled
circle corresponds to the entire domain while the empty circle denotes the end points.
The evolution over the interior of the domain is carried either above  or below the
diagonal starting from the diagonal values (direction of evolution is
depicted by the arrow head). The end points are evolved
on either side of the diagonal. Note that the vertical/horizontal lines where
the arrows end corresponds to the history of the auxiliary field needed for
TBCs in the current time-step. The TBCs for auxiliary field require the history 
of the auxiliary field at the endpoints, depicted by broken lines; these values
are taken from the adjacent segment of the boundary.}
\end{figure*}

Therefore, $\supp_{x_2}\varphi(x_1,x_2,0,t')\subset\Gamma_r$ for
$x_1\in\field{R}$ and 
\begin{equation}
[i\partial_{t'}+\partial^2_{x_1}]\varphi(x_1,x_2,t,t')=0.
\end{equation}
Hence, the boundary condition at the endpoints of
$\Gamma_r$ is given by
\begin{equation}
\partial_{x_2}\varphi(x_1,x_2,t,t')\pm e^{-i\pi/4}\partial^{1/2}_t\varphi(x_1,x_2,t,t')=0,
\end{equation}
$x_1\in\Gamma_r, x_2\in\{x_t, x_b\}$, respectively. This requires the knowledge of 
$\varphi(x_1,x_2,t,t')$ for $x_2=\partial\Gamma_r$ and $0\leq t\leq t'$. The algorithm 
can be explained by means of the Fig. \ref{fig:IVP-auxi}. There two IVPs for the auxiliary field,
$\varphi(x_1,x_2,t,t')$ each of which evolve the field either above or below the
diagonal in the $(t,t')$-plane starting from the values at the diagonal in their
respective domains. These points are represented by filled circles and the
arrows denote the direction of evolution. All diagonal points are evolved along
$t$ or $t'$-axis in order to provide the history needed for TBCs in the current time step (represented by
horizontal or vertical line in $(t,t')$-plane). The empty circles correspond 
to the corner points. The values of the auxiliary field at the corners are needed 
for the TBCs satisfied by the auxiliary fields; this relationship is depicted by the broken
lines. Note that these values at the corners can be taken from the adjacent segment of 
the boundary where it is already being computed. 


\begin{rmk}
From the discussion above we obtain the following useful definition of the operator
$(\partial_t-i\partial^2_{x})^{1/2},
(x,t)\in\field{R}\times \field{R}_+$:
\begin{multline}
(\partial_t-i\partial_{x}^2)^{1/2}f(x,t)\\=\frac{1}{2\pi}\iint d\zeta dx'
e^{-i\zeta(x-x')}[\partial_{t'}^{1/2}e^{-i\zeta^2(t-t')}f(x',t')]_{t'=t}.
\end{multline}
This definition makes it explicit that one has to consider the function
$f(x,t)$ over its entire support with respect to $x\in\field{R}$ in order to
compute the expression; however, in special cases this can be avoided.
Similarly, a formal definition of the operator $(\partial_t-i\partial^2_{x})^{-m/2},
(x,t)\in\field{R}\times \field{R}_+,\,\,m>0,$ for any arbitrary function $f(x,t)$ with sufficient
smoothness property can be given. This operator can be defined as
\begin{multline}
(\partial_t-i\partial_{x}^2)^{-m/2}f(x,t)\\=
\frac{1}{2\pi}\iint d\zeta dx'e^{-i\zeta(x-x')}[\partial_{t'}^{-m/2}e^{-i\zeta^2(t-t')}f(x',t')]_{t'=t}.
\end{multline}
The integration with respect to $\zeta$ can be easily performed. Defining the convolution kernel $\mathcal{K}(x,t)$ by
\begin{equation}
\mathcal{K}(x,t)=\begin{cases}
\frac{e^{-i\pi/4}}{2\sqrt{\pi}\Gamma(m/2)}t^{-\frac{m+3}{2}}\exp\biggl[i\frac{x^2}{4t}\biggl],&t>0,\\
0,&t<0,
\end{cases}
\end{equation}
we obtain 
\begin{equation}
    (\partial_t-i\partial_{x}^2)^{-m/2}f(x,t)=\iint dx'dt'\mathcal{K}(x-x',t-t')f(x',t').
\end{equation}
From this definition it is clear that the operator cannot be defined on a
compact domain with respect to $x$ for arbitrary functions. Let us introduce
$\varphi(x,t,t')$ defined by
\begin{equation}
\varphi(x,t,t')=\frac{1}{2\pi}\iint d\zeta dx'
e^{-i\zeta(x-x')}e^{-i\zeta^2(t-t')}f(x',t'),
\end{equation}
so that 
\begin{equation}\label{eq:associated-IVP}
\begin{split}
&(\partial_t-i\partial_{x}^2)^{-m/2}f(x,t)=\partial^{-m/2}_{t'}\varphi(x,t,t')|_{t'=t},\\
&i\partial_t\varphi(x,t,t')+\partial_x^2\varphi(x,t,t')=0,\\
&\varphi(x,t',t') = f(x,t').
\end{split}
\end{equation}
The possibility of defining this operator on a compact domain with respect to $x$
depends on the initial condition $\varphi(x,0,t')$ given by
\begin{equation}
\varphi(x,0,t')=\frac{1}{2\pi}\iint d\zeta dx'
e^{-i\zeta(x-x')}e^{i\zeta^2t'}f(x',t').
\end{equation}
If $\supp_x\varphi(x,0,t')$ is compact then the PDE in \eqref{eq:associated-IVP}
can be solved on any domain $\Omega$ such that
$\supp_x\varphi(x,0,t')\subset\Omega,\,\,\forall t'\geq0$. This allows us to
impose TBCs for \eqref{eq:associated-IVP} at the boundaries; however, the value of the function 
$\varphi(x,t,t')$ at the boundaries for $t\in[0,t']$ needs to be computed.
\end{rmk}


\subsubsection{High-frequency approximation\label{sec:HF-rect}}
The high-frequency approximation affords the possibility of simplifying the TBC
by making them ``local'' in terms of the spatial variable. The asymptotic
expansion worked out in \eqref{eq:ABC-2D-HF} can be carried out for each of the edges 
of the rectangular domain to obtain the following ABCs:
\begin{equation}\label{eq:ABC-2D-HF-rect}
\begin{split}
&\partial_{n}u+e^{-i\pi/4}\partial_t^{1/2}u
-e^{i\pi/4}\frac{1}{2}\partial_{x_2}^2\partial_t^{-1/2}u=0,\,\,\vv{x}\in\Gamma_r\cup\Gamma_l,\\
&\partial_{n}u+e^{-i\pi/4}\partial_t^{1/2}u
-e^{i\pi/4}\frac{1}{2}\partial_{x_1}^2\partial_t^{-1/2}u=0,\,\,\vv{x}\in\Gamma_b\cup\Gamma_t.
\end{split}
\end{equation}
These boundary conditions become problematic at the corners of the rectangular
domain. This aspect can be illustrated by the considering the weak formulation
of the original IVP as follows: Consider a test function $\psi(\vv{x})\in
H^{1}(\Omega_i)$; taking the inner product with the equation \eqref{eq:2D-SE} we
have
\begin{multline}
\int_{\Omega_i}(i\partial_tu+\nabla^2u)\psi d^2\vv{x}\\
=i\partial_t\int_{\Omega_i}u\psi d^2\vv{x}
-\int_{\Omega_i}(\nabla u)\cdot(\nabla\psi)d^2\vv{x}
+\int_{\partial\Omega_i}\psi(\nabla u)\cdot d\vbs{\varsigma}.
\end{multline}
Let us consider the top and right boundaries. The boundary integrals are given by
\begin{widetext}
\begin{equation}
\begin{split}
&\mathcal{I}_R+\mathcal{I}_T\\
&= \int_{\Gamma_r}\psi\partial_{{x_1}}u dx_2
   +\int_{\Gamma_t}\psi\partial_{{x_2}}u dx_1\\
&=-e^{-i\pi/4}\int_{\Gamma_r\cup\Gamma_l}\psi\partial^{1/2}_tu
  +\frac{1}{2}e^{i\pi/4}\int_{\Gamma_r}\psi\partial^2_{{x_2}}\partial^{-1/2}_tu dx_2
  +\frac{1}{2}e^{i\pi/4}\int_{\Gamma_t}\psi\partial^2_{{x_1}}\partial^{-1/2}_tu dx_1\\
&=-e^{-i\pi/4}\int_{\Gamma_r\cup\Gamma_l}\psi\partial^{1/2}_tu
+\frac{1}{2}e^{i\pi/4}\biggl[\psi\partial_{x_2}\partial^{-1/2}_tu\biggl|_{x_2=x_b}^{x_t}
-\int_{\Gamma_r}(\partial_{x_2}\psi)(\partial_{{x_2}}\partial^{-1/2}_tu) dx_2+\psi\partial_{x_1}\partial^{-1/2}_tu\biggl|_{x_1=x_l}^{x_r}
-\int_{\Gamma_t}(\partial_{x_1}\psi)(\partial_{{x_1}}\partial^{-1/2}_tu)dx_1\biggl].
\end{split}
\end{equation}
\end{widetext}
The following terms corresponding to the top-right corner in the above equation 
is problematic as the boundary conditions in the current form cannot be used to
evaluate them:
\begin{equation}
\begin{split}
\biggl(\partial_{x_2}\partial^{-1/2}_tu
+\partial_{x_1}\partial^{-1/2}_tu\biggl)_{\Gamma_r\cap\Gamma_t}
=\partial_t^{-1/2}\biggl(\partial_{x_2}u
+\partial_{x_1}u\biggl)_{\Gamma_r\cap\Gamma_t}.
\end{split}
\end{equation}
In order to evaluate this term, we proceed as follows. Carrying out the
fractional integration, $\partial^{-1/2}_t$, of the evolution
equation \eqref{eq:2D-SE}, we have
\begin{equation}
i\partial_t^{1/2}u+(\partial^2_{x_1}+\partial^2_{x_2})\partial_t^{-1/2}u=0,
\,\,(x_1,x_2)\in\Gamma_r\cap\Gamma_t.
\end{equation}
Here, the fact that the field is zero at the corner at $t=0$ is explicitly used
to arrive at the fractional derivative. Using BCs \eqref{eq:ABC-2D-HF-rect} and
the last equation, we obtained the following corner condition:
\begin{equation}
\partial_{x_1}u+\partial_{x_2}u+\frac{3}{2}e^{-i\pi/4}\partial_t^{1/2}u=0,
\,\,(x_1,x_2)\in\Gamma_r\cap\Gamma_t.
\end{equation}
A similar procedure can be used to the construct corner conditions for the other
corners of the rectangular domain:
\begin{equation}
\partial_{n}u|_{\Gamma_i}+\partial_{n}u|_{\Gamma_j}+\frac{3}{2}e^{-i\pi/4}\partial_t^{1/2}u=0,
\,\,(x_1,x_2)\in \Gamma_i\cap\Gamma_j,
\end{equation}
where $i\neq j$ and $i,j\in\{r,t,l,b\}.$

\subsubsection{Stability and uniqueness}
Let us write the equivalent formulation of the IVP \eqref{eq:2D-SE} for a
rectangular domain $\Omega_i$ using the TBCs derived in the last section:
\begin{equation}
\begin{split}\label{eq:2D-SE-Omegai-rect}
&i\partial_tu+\triangle u=0,\,\,(\vv{x},t)\in\Omega_i\times\field{R}_+,\\
&u(\vv{x},0)=u_0(\vv{x})\in L^2(\Omega_i),\,\,\supp\, u_0(\vv{x})\subset\Omega_i,\\
&\partial_{n}{u}(\vv{x},t)+
e^{-i\pi/4}(\partial_{t}-i\partial^2_{x_2})^{1/2}u(\vv{x},t)=0,\,\,\vv{x}\in\Gamma_l\cup\Gamma_r,\\
&\partial_{n}{u}(\vv{x},t)+
e^{-i\pi/4}(\partial_{t}-i\partial^2_{x_1})^{1/2}u(\vv{x},t)=0,\,\,\vv{x}\in\Gamma_b\cup\Gamma_t.
\end{split}
\end{equation}
Assuming that the solution $u(\vv{x},t)$ exits for $t\in[0,T]$, we have
\begin{multline}\label{eq:Energy-estimate}
||u(\vv{x},T)||^2_{L^2(\Omega_i)}-||u_0(\vv{x})||^2_{L^2(\Omega_i)}\\
=2\Re\int_0^T dt\biggl[\int_{\Gamma_l\cup\Gamma_r}(u^*i\nabla u)\cdot
d\vbs{\varsigma}+\int_{\Gamma_b\cup\Gamma_t}(u^*i\nabla u)\cdot
d\vbs{\varsigma}\biggl].
\end{multline}
In the above equation, the fields in the boundary integral can be extended to
$\field{R}_+$ without changing the value of the integral by setting them
identically to zero outside the interval $[0,T]$. For the right boundary, we have
\begin{equation}
\mathcal{I}_R
=-2\Re\int_0^Tdt
\int_{\Gamma_r}\biggl(u^*e^{i\pi/4}\sqrt{\partial_t-i\partial^2_{x_2}}u\biggl)dx_2.
\end{equation}
Again, without changing the value of the integral with respect to $x_2$ the fields can be extended to
whole line by setting them identically to zero outside $\Gamma_r$. We these
considerations, one can write
\begin{equation}
\begin{split}
&\mathcal{I}_R\\
&=-2\Re\int_{\field{R}_+}dt
\int_{\field{R}}\biggl(u^*e^{i\pi/4}\sqrt{\partial_t-i\partial^2_{x_2}}u\biggl)dx_2,\\
&=-\frac{1}{\pi}\Re\int_{\field{R}_+}dt\int_{\field{R}}d\zeta_2
\overline{\biggl(\tilde{u}(x_r,\zeta_2,t)e^{-i\zeta_2^2t}\biggl)}
\biggl(\partial_t^{1/2}\tilde{u}(x_r,\zeta_2,t)e^{-i\zeta_2^2t}\biggl),\\
&=-\frac{1}{2\pi^2}\int_{\{\xi<0\}}d\xi\int_{\field{R}}d\zeta_2
|\xi|^{1/2}
\biggl|\fourier_t\biggl[\tilde{u}(x_r,\zeta_2,t)e^{-i\zeta_2^2t}\biggl](x_r,\zeta_2,\xi)\biggl|^2\\
&\leq0.
\end{split}
\end{equation}
Similarly, it can be shown that the other boundary integrals namely
$\mathcal{I}_{L}$, $\mathcal{I}_{B}$ and $\mathcal{I}_{T}$ corresponding to the left,
bottom and top boundary, respectively, also satisfy the same inequality as
that of $\mathcal{I}_{R}$. Therefore, we have 
\begin{equation}
||u(\vv{x},T)||_{L^2(\Omega_i)}\leq||u_0(\vv{x})||_{L^2(\Omega_i)}.
\end{equation}
Since $T\in\field{R}_+$ is arbitrary, one replace $T$ with $t$ in the above
inequality. This result solution also guarantees the uniqueness of the solution.  

For the high-frequency approximation, we have the following equivalent
formulation on $\Omega_i$: Putting $\Gamma=\partial\Omega_i$ and $\Gamma_C$, 
the set of four corner points, define the Laplace-Beltrami operator as
$\triangle_{\Gamma}$ so that
\begin{equation}
\begin{split}\label{eq:2D-SE-Omegai-rect-HF}
&i\partial_tu+\triangle u=0,\,\,(\vv{x},t)\in\Omega_i\times\field{R}_+,\\
&u(\vv{x},0)=u_0(\vv{x})\in L^2(\Omega_i),\,\,\supp\, u_0(\vv{x})\subset\Omega_i,\\
&\partial_{n}u+e^{-i\pi/4}\partial_t^{1/2}u
    -e^{i\pi/4}\frac{1}{2}\triangle_{\Gamma}\partial_t^{-1/2}u=0,\,\,\vv{x}\in\Gamma\setminus\Gamma_C,\\
    &\partial_{n}u|_{\Gamma_i}+\partial_{n}u|_{\Gamma_j}+\frac{3}{2}e^{-i\pi/4}\partial_t^{1/2}u=0,
    \,\,(x_1,x_2)\in \Gamma_i\cap\Gamma_j,
\end{split}
\end{equation}
where $i\neq j$, and $i,j\in\{r,t,l,b\}$. Taking equation \eqref{eq:Energy-estimate} 
and following the standard approach, we have
\begin{widetext}
\begin{equation}
\begin{split}
&\int_{\Gamma}(u^*i\nabla u)\cdot d\vbs{\varsigma}\\
&=-e^{i\pi/4}\int_{\Gamma}u^*\partial^{1/2}_{t}u|d\vv{x}|
-\frac{1}{2}e^{-i\pi/4}\int_{\Gamma\setminus\Gamma_C}u^*\triangle_{\Gamma}\partial_t^{-1/2}u
|d\vv{x}|\\
&=-e^{i\pi/4}\int_{\Gamma}u^*\partial^{1/2}_{t}u|d\vv{x}|
+\frac{1}{2}e^{-i\pi/4}\int_{\Gamma\setminus\Gamma_C}(\partial_{\Gamma}u)^*(\partial_{\Gamma}\partial_t^{-1/2}u)
|d\vv{x}|
+\frac{3}{4}i|u(x_r, x_t)|^2
-\frac{3}{4}i|u(x_l, x_t)|^2
+\frac{3}{4}i|u(x_l, x_b)|^2
-\frac{3}{4}i|u(x_r, x_b)|^2.
\end{split}
\end{equation}
\end{widetext}
Taking the real part and plugging the result back into equation
\eqref{eq:Energy-estimate} yields
\begin{widetext}
\begin{equation}
\begin{split}
||u(\vv{x},T)||^2_{L^2(\Omega_i)}-||u_0(\vv{x})||^2_{L^2(\Omega_i)}&=2\Re\int_0^T dt\biggl[-e^{i\pi/4}\int_{\Gamma}u^*\partial^{1/2}_{t}u|d\vv{x}|
+\frac{1}{2}e^{-i\pi/4}\int_{\Gamma\setminus\Gamma_C}(\partial_{\Gamma}u)^*(\partial_{\Gamma}\partial_t^{-1/2}u)
|d\vv{x}| \biggl]\\
    &=2\int_{\{\xi<0\}}
    d\xi\sum_{\Gamma_i}\int_{\field{R}}d\zeta
    \biggl(-|\xi|^{1/2}+\frac{1}{2}|\xi|^{-1/2}|\zeta|^2\biggl)
    |\fourier_t\fourier_{\sigma}[u(\vv{x},t)]|^2,
\end{split}
\end{equation}
where $\varsigma$ is taken as the tangential variable to the boundary so that
$\partial_{\Gamma}=\partial_{\varsigma}$. In the region defined by
$|\xi|\geq|\zeta|^2/2$, we obtain the condition
\begin{equation}
||u(\vv{x},T)||_{L^2(\Omega_i)}\leq||u_0(\vv{x})||_{L^2(\Omega_i)},
\end{equation}
\end{widetext}
which guarantee stability and uniqueness of the solution on the bounded domain.



%==========================================================================
\section{General Schr\"odinger equation\label{sec:gse}}
The linear Schr\"odinger equation with time-dependent potential and the
nonlinear case are treated in the same fashion. Let us consider the IVP 
corresponding to the general Schr\"{o}dinger equation given by
\begin{equation}
\begin{split}\label{eq:2D-NL}
&i\partial_tu+\triangle u+\phi(\vv{x},t,|u|^2)u=0,\,\,(\vv{x},t)\in\field{R}^2\times\field{R}_+,\\
&u(\vv{x},0)=u_0(\vv{x}),\,\,\vv{x}\in\field{R}^2.
\end{split}
\end{equation}
The potential function, $\phi$ is assumed to be real valued.
\begin{figure}[!htbp]
\begin{center}
\def\myscale{1}
\includegraphics[scale=\myscale]{./illustration_c.eps}
\end{center}
\caption{\label{fig:computational-domain} Parametrization of the boundary
$\Gamma=\partial\Omega_i$}
\end{figure}
\subsection{Convex domain with smooth boundary}
The computational domain is taken to be a convex set $\Omega_i\subset\field{R}^2$ with a 
smooth boundary $\Gamma=\partial\Omega_i$. Let $\vbs{\chi}(s)$ be the parametrization of 
the curve $\Gamma$ where $s$ is the length along the curve. Introducing the tangent vector 
$\vv{e}_{\tau}$, also a function of $s$ only, and taking into account the convexity of the domain 
we have
\begin{equation}
d\vv{x}=dr\vv{e}_n+(1+r\kappa)ds\vv{e}_{\tau},
\end{equation}
where $\kappa$ is the curvature of the boundary $\Gamma$ given by
\begin{equation}
    \kappa(s)=\biggl|{\frac{d\vv{e}_{\tau}}{ds}}\biggl|
    =\frac{|\det(\vbs{\chi}',\vbs{\chi}'')|}{|\vbs{\chi}'|^3}.
\end{equation}
The initial data is assumed to be compactly supported in the computational domain $\Omega_i$.
Carrying out the gauge transformation $u=ve^{i\Phi}$, the evolution operator 
$L\equiv i\partial_t+\triangle+\phi$ in the curvilinear system is given by
\begin{equation}\label{eq:gauge-transform}
L(r,s,t,\partial_r,\partial_s,\partial_t)
=i\partial_t+\partial_r^2+A\partial_r+h^{-2}\partial^2_s+B\partial_s+C,
\end{equation}
where $A=(2i\Phi_r+h^{-1}\kappa)$, $B=(2ih^{-2}\Phi_s+h^{-1}\partial_sh^{-1})$ and $C=i\triangle\Phi-(\nabla\Phi)^2$.
The pseudo-differential approach allows us to construct various order ABCs given
by\\
$\ABC_{1a}$:
\begin{equation}
\partial_{{n}}u+e^{-i\pi/4}e^{i\Phi}(\partial_t-i\partial_s^2)^{1/2}e^{-i\Phi}u=0;
\end{equation}
$\ABC_{2a}$:
\begin{multline}
\partial_{{n}}u+e^{-i\pi/4}e^{i\Phi}(\partial_t-i\partial_s^2)^{1/2}e^{-i\Phi}u\\
+\frac{1}{2}\kappa u+e^{-i\pi/4}\Phi_se^{i\Phi}(\partial_t-i\partial_s^2)^{-1/2}\partial_s(e^{-i\Phi}u)\\
+\frac{1}{2}i\kappa e^{i\Phi}(\partial_t-i\partial_s^2)^{-1}\partial_s^2(e^{-i\Phi}u)=0.
\end{multline}
where 
\begin{equation}
(\partial_t-i\partial^2_s)^{1/2}f(s,t)=(\partial_t-i\partial^2_s)(\partial_t-i\partial^2_s)^{-1/2}f(s,t),
\end{equation}
assuming $f(s,0)=0$.

\begin{rmk}
    For the field $u(\vv{x},t)$, the action of operators of the form
    $(\partial_t-i\partial^2_s)^{\alpha},\alpha=1/2, -1/2, -1,\ldots,$ can be easily
    computed by observing that the field can be given a periodic extension in
    terms of the parametrization variable, $s$, so that the method discussed in
    Remark \ref{rmk:periodic} can be employed. The smoothness of the boundary is
    therefore a necessary condition for this method to be applicable. From the previous sections, we 
    know that the operator represented by
$(\partial_t-i\partial^2_s)^{-m/2}$, where $m$ is a positive integer, is defined as
\begin{multline}
(\partial_t-i\partial^2_s)^{-m/2}f(s,t)\\
=\frac{1}{2\pi}\iint{d\zeta}\,ds'\,e^{i\zeta (s-s')}[\partial_{t'}^{-m/2}e^{-i\zeta^2(t-t')}{f}(s',t')]_{t'=t}.
\end{multline}
Let $\fourier_{s}f(s,t)=\tilde{f}(\zeta,t)$, then the operation $(\partial_t-i\partial^2_s)^{-m/2}$ 
involves the inverse Fourier transform of $\tilde{f}(\zeta,t')e^{-i\zeta^2(t-t')}$ 
with respect to $\zeta$. Therefore, if $f(s,t)$ is of the Schwartz class (with respect to $s\in\field{R}$) 
then so is $\tilde{f}(\zeta,t')e^{-i\zeta^2(t-t')}$, so that $(\partial_t-i\partial^2_s)^{-m/2}f(s,t)$ 
will also be of the Schwartz class (with respect to $s\in\field{R}$). Further, it is straightforward to show that 
$(\partial_t-i\partial^2_s)^{-m/2}f(s,t)$ is continuous at $t=0$. 

Next, our aim is to define the operator $(\partial_t-i\partial^2_s)^{-m/2}$ for 
tempered distributions $f(s,t)$ such that $\supp_tf\subset[0,\infty)$. To this end, 
let us observe that the transpose $\tp(\partial_t-i\partial^2_s)^{-m/2}$ is given by
\begin{multline}
\tp(\partial_t-i\partial^2_s)^{-m/2}g(s,t)\\
=\frac{1}{2\pi}\iint{d\zeta}\,ds'\,e^{i\zeta (s-s')}[\tp\partial_{t'}^{-m/2}e^{-i\zeta^2(t-t')}{g}(s',t')]_{t'=t},
\end{multline}
where $\tp\partial_{t'}^{-m/2}$ denotes the Weyl fractional integral. The domain of 
definition of this operator is evident. Now, for any $g(s,t)\in\Schwartz(\field{R}\times\field{R})$ 
with $\supp_t\,g\subset[0,\infty)$, the operation $(\partial_t-i\partial^2_s)^{-m/2}f(s,t)$ 
for distributions $f(s,t)\in\Schwartz'(\field{R}\times\field{R})$ 
with $\supp_t\,f\subset[0,\infty)$ can be defined by introducing a Schwartz class function $g_1(s,t)$ 
such that it agrees with $\tp(\partial_t-i\partial^2_s)^{-m/2}g(s,t)$ for $t\in[0,\infty)$ so that
\begin{equation}
\begin{split}
&\langle g(s,t),(\partial_t-i\partial^2_s)^{-m/2}f(s,t)\rangle\\
&=\langle \tp(\partial_t-i\partial^2_s)^{-m/2}g(s,t),f(s,t)\rangle\\
&=\langle g_1(s,t),f(s,t)\rangle.
\end{split}
\end{equation}
Therefore, $(\partial_t-i\partial^2_s)^{-m/2}f(s,t)$ is also tempered such that
$\supp_t\,(\partial_t-i\partial^2_s)^{-m/2}f(s,t)\subset[0,\infty)$. Note that
any function of the tangential variable, $s$, can be extended periodically on
the whole line. Thus periodic functions being a tempered distribution 
can be easily included in the domain of definition of
$(\partial_t-i\partial^2_s)^{-m/2}$. It is interesting to note that by applying the Leibniz formula 
for the fractional integrals one can obtain local approximation of the ABCs with
respect to $s$:
\begin{widetext}
\begin{equation}\begin{split}
(\partial_t-i\partial^2_s)^{-m/2}f(s,t)=&\frac{1}{2\pi}\iint{d\zeta}\,ds'\,e^{i\zeta (s-s')}[\partial_{t'}^{-m/2}e^{-i\zeta^2(t-t')}{f}(s',t')]_{t'=t}\\
=&\frac{1}{2\pi}\iint{d\zeta}\,ds'\,e^{i\zeta (s-s')}
    \biggl[\sum_{j\in\field{N}}\binom{-m/2}{j}(\partial^j_{t'}e^{-i\zeta^2(t-t')})\partial_{t'}^{-m/2-j}{f}(s',t')\biggl]_{t'=t}\\
=&\frac{1}{2\pi}\iint{d\zeta}\,ds'\,e^{i\zeta (s-s')}
    \biggl[\sum_{j\in\field{N}}\binom{-m/2}{j}(-i\zeta^{2})^j\partial_{t}^{-m/2-j}{f}(s',t)\biggl]\\
=&\sum_{j\in\field{N}}\binom{-m/2}{j}(i\partial_s^{2})^j\partial_{t}^{-m/2-j}{f}(s,t).
\end{split}\end{equation}
\end{widetext}
\end{rmk}

The second family of various order ABCs are obtained under high-frequency assumption with respect to the
temporal frequency. These are given by\\
$\ABC_{1b}$:
\begin{equation}
\partial_{{n}}u+e^{-i\pi/4}e^{i\Phi}\partial_t^{1/2}e^{-i\Phi}u=0.
\end{equation}
$\ABC_{2b}$:
\begin{multline}
\partial_{{n}}u+e^{-i\pi/4}e^{i\Phi}\partial_t^{1/2}e^{-i\Phi}u+\frac{1}{2}\kappa u\\
-\frac{1}{2}e^{i\pi/4}\biggl[\frac{\kappa^2}{4}+\partial_s^2\biggl]e^{i\Phi}\partial_t^{-1/2}e^{-i\Phi}u=0.
\end{multline}
$\ABC_{3b}$:
\begin{multline}
\partial_{{n}}u+e^{-i\pi/4}e^{i\Phi}\partial_t^{1/2}e^{-i\Phi}u+\frac{1}{2}\kappa u\\
-\frac{1}{2}e^{i\pi/4}\biggl[\frac{\kappa^2}{4}+\partial_s^2\biggl]e^{i\Phi}\partial_t^{-1/2}e^{-i\Phi}u\\
+\frac{i}{2}\biggl[\frac{\kappa^3+\partial_s^2\kappa}{4}-\frac{\partial_{{n}}\phi}{2}+\partial_s(\kappa\partial_s)\biggl]e^{i\Phi}\partial^{-1}_te^{-i\Phi}u=0.
\end{multline}
%The special case of the circular boundary can be obtained by taking the curvature $\kappa$ to be constant, so that the $\ABC_{3b}$ simplifies to\\
%$\ABC_{3b}$:
%\begin{multline}
%\partial_{\vv{n}}u+e^{-i\pi/4}e^{i\Phi}\partial_t^{1/2}e^{-i\Phi}u+\frac{1}{2}\kappa u-\frac{1}{2}e^{i\pi/4}\biggl[\frac{\kappa^2}{4}+\partial_s^2\biggl]e^{i\Phi}\partial_t^{-1/2}e^{-i\Phi}u\\
%+\frac{i}{2}\biggl[\frac{\kappa^3}{4}-\frac{\partial_{\vv{n}}\phi}{2}+\kappa\partial^2_s\biggl]e^{i\Phi}\partial^{-1}_te^{-i\Phi}u=0.
%\end{multline}

%======================================================================================
\subsubsection{Stability and uniqueness}
In order to study the stability property of the solution of the IVP defined by
\eqref{eq:2D-NL} with boundary condition $\ABC_{1a}$ and $\ABC_{2a}$,
respectively, we start with the relation:
\begin{equation}\label{eq:Energy-estimate-NL}
||u(\vv{x},T)||^2_{L^2(\Omega_i)}-||u_0(\vv{x})||^2_{L^2(\Omega_i)}
    =2\Re\int_0^T dt\int_{\Gamma}(u^*i\nabla u)\cdot
d\vbs{\varsigma}.
\end{equation}
Realization of the boundary operators require an appropriate Fourier
representation with respect to the tangential variable, $s$. To this end, one
may either employ the Fourier series representation by extending the field 
periodically for all $s\in\field{R}$ or Fourier transform representation by 
extending the field to all $s\in\field{R}$ by setting it zero outside $\Gamma$. 
The result for the first oder ABCs, $\ABC_{1a}$, can be obtained by observing
that the boundary integral
\begin{equation}
    \mathcal{I}_1 = -e^{i\pi/4}\int_0^Tdt\int_{\Gamma}
    \overline{(ue^{-i\Phi})}\sqrt{\partial_t-i\partial^2_s}(u e^{-i\Phi})ds,
\end{equation}
satisfies $\Re\mathcal{I}_1\leq0$ so that
\begin{equation}
||u(\vv{x},T)||^2_{L^2(\Omega_i)}\leq||u_0(\vv{x})||^2_{L^2(\Omega_i)}.
\end{equation}
For the second order ABCs, $\ABC_{2a}$, the energy estimate cannot be
obtained for a general potential function. A special case of interest is when 
$\partial_s\phi=0$ so that $\ABC_{2a}$ is given by
\begin{multline}
\partial_{{n}}u+e^{-i\pi/4}e^{i\Phi}(\partial_t-i\partial_s^2)^{1/2}e^{-i\Phi}u+\frac{1}{2}\kappa u\\
+\frac{1}{2}i\kappa e^{i\Phi}(\partial_t-i\partial_s^2)^{-1}\partial_s^2(e^{-i\Phi}u)=0.
\end{multline}
Define the boundary integrals
\begin{equation}
\begin{split}
    &\mathcal{I}_2 = -\frac{i}{2}\int_0^Tdt\int_{\Gamma}\kappa(s)|u|^2ds,\\
    &\mathcal{I}_3 = \frac{1}{2}\int_0^Tdt\int_{\Gamma}
    \kappa(s)\overline{(ue^{-i\Phi})}(\partial_t-i\partial^2_s)^{-1}\partial_s^2(u
    e^{-i\Phi})ds.
\end{split}
\end{equation}
It follows that $\Re\mathcal{I}_2=0$. Define the symbol 
$\sigma_{P}=(1/2)\kappa(s)(i\xi+i\zeta^2)^{-1}(-\zeta^2)$ so that the
symbol for the adjoint operator $\sigma_{P^{\dagger}}$ works out to be
\begin{equation}
    \sigma_{P^{\dagger}}\sim\frac{-i}{2}\sum_{k\in\field{N}}\frac{1}{k!i^k}(\partial_s^k\kappa)
    \partial^k_{\zeta}\biggl(\frac{\zeta^2}{\xi+\zeta^2}\biggl).
\end{equation}
Let $2Q=P+P^{\dagger}$ so that 
\begin{equation}
\begin{split}
2\sigma_{Q}
&\sim\frac{i}{2}\frac{\kappa\zeta^2}{\xi+\zeta^2}+\frac{-i}{2}
\sum_{k\in\field{N}}\frac{1}{k!i^k}(\partial_s^k\kappa)\partial^k_{\zeta}\biggl(\frac{\zeta^2}{\xi+\zeta^2}\biggl)\\
&=-\frac{1}{2}(\partial_s\kappa)\partial_{\zeta}\biggl(\frac{\zeta^2}{\xi+\zeta^2}\biggl)+\ldots,
\end{split}
\end{equation}
and putting $\psi=u e^{-i\Phi}$, we have
\begin{multline}
\Re\mathcal{I}_3=\frac{1}{(2\pi)^2}\iiint d\zeta d\xi ds\\
\times\fourier_{t}\fourier_{s}[\psi(s,t)]
\sigma_{Q}(s,\zeta,\xi)\fourier_t[\psi(s,t)]e^{-i\zeta s}.
\end{multline}
From the asymptotic expansion, to the zeroth order, we have
$\Re\mathcal{I}_3\approx 0$ (for constant curvature boundary, this result is
exact). The energy estimate can thus be approximately established for the
special case when $\partial_s\phi=0$. 


For the ABCs obtained in the high-frequency approximation, it is evident that
the first order ABC, $\ABC_{1b}$, satisfies the energy estimate. For the
higher-order ABCs, one requires an additional conditions, $\partial_s\phi=0$ in
order to obtain the energy estimate. In $\ABC_{2b}$, the term with the factor
$\kappa/2$ can be ignored as it leads to a purely imaginary quantity. 
Putting $\psi=ue^{i\Phi}$, consider the following boundary integrals
\begin{equation}
\begin{split}
&\mathcal{I}_1 =
-\int_0^Tdt\int_{\Gamma}\psi^*(s,t)
[e^{i\pi/4}\partial_t^{1/2}+\frac{1}{2}e^{-i\pi/4}\partial^{-1/2}_t\partial_s^2]\psi(s,t)ds,\\
&\mathcal{I}_2 =
-\frac{1}{8}e^{-i\pi/4}\int_0^Tdt\int_{\Gamma}\psi^*(s,t)\kappa^2\partial^{-1/2}_t\psi(s,t)ds.
\end{split}
\end{equation}
The first integral can be dealt with in the manner done before. The second
integral can be dealt with in the manner described in the appendix. Defining 
$\sigma_P=-e^{-i\pi/4}\kappa^2(i\xi)^{-1/2}$ so that
$\sigma_{P^{\dagger}}=-e^{i\pi/4}\kappa^2(-i\xi)^{-1/2}$. Putting
$2Q=P+P^{\dagger}$, we have $\sigma_Q=-\kappa^2\cos[\pi/4+\pi\sgn(\xi)/4]$.
The boundary integral can now be written as 
\begin{multline}
\Re\mathcal{I}_2=-\frac{1}{16\pi}\iint d\xi ds\\
\times[\kappa(s)]^2|\xi|^{-1/2}\cos[\pi(1+\sgn(\xi))/4]|\fourier_t[\psi(s,t)]|^2,
\end{multline}
so that $\Re\mathcal{I}_2\leq0$.

For $\ABC_{3b}$, we define the following boundary integrals
\begin{equation}
\begin{split}
    &\mathcal{I}_3 =
    \frac{1}{8}\int_0^Tdt\int_{\Gamma}\psi^*(s,t)
    [(\kappa^3+\partial_s^2\kappa)\partial_t^{-1}]\psi(s,t)ds,\\
    &\mathcal{I}_4 =
    \frac{1}{2}\int_0^Tdt\int_{\Gamma}\psi^*(s,t)
    [(\partial_s\kappa\partial_s)\partial_t^{-1}]\psi(s,t)ds,\\
    &\mathcal{I}_5 =
    -\frac{1}{4}\int_0^Tdt\int_{\Gamma}\psi^*(s,t)
    [(\partial_n\phi)\partial_t^{-1}]\psi(s,t)ds.
\end{split}
\end{equation}
Defining $\sigma_P=(\kappa^3+\partial_s^2\kappa)(i\xi)^{-1}$ so that
\[
\sigma_Q=(\kappa^3+\partial_s^2\kappa)|\xi|^{-1}\cos[\pi\sgn(\xi)/4].
\] 
From here it follows that $\mathcal{I}_3$ is purely imaginary. Carrying out an
integration by parts in $\mathcal{I}_4$ with respect to $s$, we have
\begin{equation}
\begin{split}
\mathcal{I}_4
&=-\frac{1}{2}\int_0^Tdt\int_{\Gamma}[\partial_s\psi^*(s,t)]\kappa\partial_t^{-1}[\partial_s\psi(s,t)]ds\\
&=-\frac{1}{2\pi}\int_0^Td\xi\int_{\Gamma}\kappa(i\xi)^{-1}|\fourier_t[\partial_s\psi(s,t)]|^2ds.
\end{split}
\end{equation}
Therefore, $\mathcal{I}_4$ is purely imaginary. The discussion of the last
integral is more involved. Define the symbol 
$\sigma_{P}=-(\partial_n\phi)(i\xi)^{-1}$ so that the
symbol for the adjoint operator $\sigma_{P^{\dagger}}$ works out to be
\begin{equation}
    \sigma_{P^{\dagger}}\sim-\sum_{k\in\field{N}}\frac{1}{k!i^k}(\partial_t^k\partial_n\phi)
    \partial^k_{\xi}(-i\xi)^{-1}.
\end{equation}
Now
\begin{equation}
\begin{split}
2\sigma_{Q}
&\sim-(\partial_n\phi)(i\xi)^{-1}-\sum_{k\in\field{N}}\frac{1}{k!i^k}(\partial_t^k\partial_n\phi)\partial^k_{\xi}(-i\xi)^{-1}\\
&=-(\partial_t\partial_n\phi)(\xi)^{-2}+\ldots.
\end{split}
\end{equation}
Therefore, the contribution from this last integral can be ignored on account
of the lower order leading term in the above asymptotic expansion. Thus the
energy estimate
\begin{equation}
||u(\vv{x},T)||^2_{L^2(\Omega_i)}\leq||u_0(\vv{x})||^2_{L^2(\Omega_i)}.
\end{equation}
can be established for the ABC-family~\footnote{Only the second
    and third order ABCs require $\partial_s\phi=0$ in order to establish the
energy estimate.}: $\ABC_{jb},j=1,2,3$.

\subsection{Domains with straight boundary: infinite strip }
The ABCs for the straight boundary can be written by setting the curvature $\kappa$ 
to zero.\\
$\ABC_{1a}$:
\begin{equation}
\partial_{{n}}u+e^{-i\pi/4}e^{i\Phi}(\partial_t-i\partial_s^2)^{1/2}e^{-i\Phi}u=0.
\end{equation}
$\ABC_{2a}$:
\begin{multline}
\partial_{{n}}u+e^{-i\pi/4}e^{i\Phi}(\partial_t-i\partial_s^2)^{1/2}e^{-i\Phi}u\\
+e^{-i\pi/4}\Phi_se^{i\Phi}(\partial_t-i\partial_s^2)^{-1/2}\partial_s(e^{-i\Phi}u)=0.
\end{multline}
Along the direction in which the strip extends to infinity, we impose the periodic 
boundary condition. Note that an exact representation of the boundary operators is 
possible in this case which is based on the observations made in earlier sections. 
The energy estimate for the first order ABCs is easy to obtain; however, for the 
second order ABCs this result is not available. The high-frequency ABCs work out to be\\
$\ABC_{1b}$:
\begin{equation}
\partial_{{n}}u+e^{-i\pi/4}e^{i\Phi}\partial_t^{1/2}e^{-i\Phi}u=0.
\end{equation}
$\ABC_{2b}$:
\begin{multline}
\partial_{{n}}u+e^{-i\pi/4}e^{i\Phi}\partial_t^{1/2}e^{-i\Phi}u\\
-\frac{1}{2}e^{i\pi/4}\partial_s^2\biggl(e^{i\Phi}\partial_t^{-1/2}e^{-i\Phi}u\biggl)=0.
\end{multline}
$\ABC_{3b}$:
\begin{multline}
\partial_{{n}}u+e^{-i\pi/4}e^{i\Phi}\partial_t^{1/2}e^{-i\Phi}u\\
-\frac{1}{2}e^{i\pi/4}\partial_s^2\biggl(e^{i\Phi}\partial_t^{-1/2}e^{-i\Phi}u\biggl)
-\frac{i\partial_{{n}}\phi}{4}e^{i\Phi}\partial^{-1}_te^{-i\Phi}u=0.
\end{multline}
The energy estimate for the general case is not available for such ABCs.

\subsection{Rectangular domains: corner conditions}
Continuing with the rectangular domain as defined in Section
\ref{sec:rect-domain}, let us consider the possibility of extending the results
obtained in Section \ref{sec:HF-rect} to the nonlinear/variable potential case. 
We confine our attention to the high-frequency ABCs only. We demonstrate the
posibility of constructing the corner condition at the corner defined by 
$\Gamma_r\cap\Gamma_t$. Consider the $\ABC_{2b}$ given by
\begin{multline}
\partial_{x_1}u+e^{-i\pi/4}e^{i\Phi}\partial_t^{1/2}e^{-i\Phi}u\\
-\frac{1}{2}e^{i\pi/4}\partial_{x_2}^2\biggl(e^{i\Phi}\partial_t^{-1/2}e^{-i\Phi}u\biggl)
=0,\,\,\vv{x}\in\Gamma_r,
\end{multline}
\begin{multline}
\partial_{x_2}u+e^{-i\pi/4}e^{i\Phi}\partial_t^{1/2}e^{-i\Phi}u\\
-\frac{1}{2}e^{i\pi/4}\partial_{x_1}^2\biggl(e^{i\Phi}\partial_t^{-1/2}e^{-i\Phi}u\biggl)
=0,\,\,\vv{x}\in\Gamma_t.
\end{multline}
Let us consider the weak formulation of the IVP defined by
\begin{equation}\label{eq:IVP-NL-estimate}
\begin{split}
&i\partial_tu+\triangle u+\phi u=0,\,\,(\vv{x},t)\in\Omega_i\times\field{R}_+,\\
&u(\vv{x},0)=u_0(\vv{x}),\,\,\vv{x}\in\Omega_i,\,\,\text{with}\,\,\supp\,u_0(\vv{x})\subset\Omega_i.
\end{split}\end{equation}
Consider the test function $\psi(\vv{x})\in H^1(\Omega_i)$ so that
\begin{multline}
\int_{\Omega_i}(i\partial_tu+\nabla^2u+\phi u)\psi d^2\vv{x}\\
=\int_{\Omega_i}[i\partial_tu-(\nabla u)\cdot(\nabla\psi)+\phi u]d^2\vv{x}
+\int_{\Gamma}\psi(\nabla u)\cdot d\vbs{\varsigma}.
\end{multline}
Let us consider the top and right boundaries. The boundary integrals are given by
\begin{widetext}
\begin{equation}
\begin{split}
\mathcal{I}_R+\mathcal{I}_T
&= \int_{\Gamma_r}\psi(\partial_{{x_1}}u) dx_2
    +\int_{\Gamma_t}\psi(\partial_{{x_2}}u) dx_1\\
&=-e^{-i\pi/4}\int_{\Gamma_r\cup\Gamma_l}\psi e^{i\Phi}\partial^{1/2}_te^{-i\Phi}u
    +\frac{1}{2}e^{i\pi/4}\biggl[\int_{\Gamma_r}\psi\partial^2_{{x_2}}(e^{i\Phi}\partial^{-1/2}_te^{-i\Phi}u) dx_2
    +\int_{\Gamma_t}\psi\partial^2_{{x_1}}(e^{i\Phi}\partial^{-1/2}_te^{-i\Phi}u)dx_1\biggl]\\
&=-e^{-i\pi/4}\int_{\Gamma_r\cup\Gamma_l}\psi e^{i\Phi}\partial^{1/2}_te^{-i\Phi}u
   +\frac{1}{2}e^{i\pi/4}\biggl[\psi\partial_{x_2}(e^{i\Phi}\partial^{-1/2}_te^{-i\Phi}u)\biggl|_{x_2=x_b}^{x_t}
-\int_{\Gamma_r}(\partial_{x_2}\psi)\partial_{{x_2}}(e^{i\Phi}\partial^{-1/2}_te^{-i\Phi}u) dx_2\\
&+\psi\partial_{x_1}(e^{i\Phi}\partial^{-1/2}_te^{-i\Phi}u)\biggl|_{x_1=x_l}^{x_r}
-\int_{\Gamma_t}(\partial_{x_1}\psi)\partial_{{x_1}}(e^{-i\Phi}\partial^{-1/2}_te^{-i\Phi}u)dx_1\biggl].
\end{split}
\end{equation}
\end{widetext}
From here it is evident that the corner condition must provide the value of the following expression at 
$\Gamma_r\cap\Gamma_t$:
\begin{multline}
\partial_{x_1}(e^{i\Phi}\partial_t^{-1/2}e^{-i\Phi}u)+\partial_{x_2}(e^{i\Phi}\partial_t^{-1/2}e^{-i\Phi}u)\\
=i(\Phi_{x_1}+\Phi_{x_2})e^{i\Phi}\partial_t^{-1/2}e^{-i\Phi}u-ie^{i\Phi}\partial_t^{-1/2}[(\Phi_{x_1}+\Phi_{x_2})e^{-i\Phi}u]\\
+e^{i\Phi}\partial_t^{-1/2}e^{-i\Phi}(\partial_{x_1}u+\partial_{x_2}u).
\end{multline}
For the linear case with variable potential, the last term in the above 
equation must be computed from a corner condition. However, for the nonlinear 
case, $\Phi$ is supposed to be dependent on some known field while our 
final intent is to restore its nonlinear dependence. In this light, the 
above equation cannot be made free of the derivatives of $u$ at the corner
as the equation is nonlinear. However, we may derive a condition at the corner
which can be combined with the ABCs and eventually be solved by an iterative 
scheme. Putting $u=\psi e^{i\Phi}$ in the evolution equation 
\eqref{eq:IVP-NL-estimate}, we have 
\begin{equation}
    i\partial_t\psi+\triangle\psi+2i(\nabla\Phi)\cdot(\nabla\psi)+(e^{-i\Phi}\triangle
    e^{i\Phi})\psi=0.
\end{equation}
Carrying out the operation $e^{i\Phi}\partial_t^{-1/2}$ on both side of the
equation above, we have
\begin{widetext}
\begin{equation}
\begin{split}
&ie^{i\Phi}\partial_t^{1/2}\psi+e^{i\Phi}\partial_t^{-1/2}\triangle\psi
+2e^{i\Phi}\partial_t^{-1/2}(i\nabla\Phi)\cdot(\nabla\psi)+e^{i\Phi}\partial_t^{-1/2}(e^{-i\Phi}\triangle
e^{i\Phi})\psi=0,\\
&ie^{i\Phi}\partial_t^{1/2}\psi+e^{i\Phi}\partial_t^{-1/2}\triangle\psi
+2e^{i\Phi}(i\nabla\Phi)\cdot\partial_t^{-1/2}(\nabla\psi)+(\triangle
e^{i\Phi})\partial_t^{-1/2}\psi+\ldots=0,\\
\end{split}
\end{equation}
\end{widetext}
which simplifies to
\begin{equation}
ie^{i\Phi}\partial_t^{1/2}e^{-i\Phi}u
+\triangle\biggl(e^{i\Phi}\partial_t^{-1/2}e^{-i\Phi}u\biggl)
=0\,\,\mod(\partial_t^{-3/2}e^{-i\Phi}u).
\end{equation}
This equation combined with the ABCs gives the following corner condition
\begin{equation}
\text{CC}_1:\,\,\partial_{x_1}u
+\partial_{x_2}u+\frac{3}{2}e^{-i\pi/4}e^{i\Phi}\partial_t^{1/2}e^{-i\Phi}u=0,
\end{equation}
or, equivalently,
\begin{equation}
\text{CC}_1:\,\,e^{i\Phi}\partial_t^{-1/2}e^{-i\Phi}(\partial_{x_1}u+\partial_{x_2}u)+\frac{3}{2}e^{-i\pi/4}u=0.
\end{equation}
Similarly, for $\ABC_{3b}$ one has the following corner condition
\begin{multline}
\text{CC}_2:\,\,\partial_{x_1}u+\partial_{x_2}u+\frac{3}{2}e^{-i\pi/4}e^{i\Phi}\partial_t^{1/2}e^{-i\Phi}u\\
-i\frac{\partial_{x_1}\phi+\partial_{x_2}\phi}{4}e^{i\Phi}\partial^{-1}_te^{-i\Phi}u
=0\,\,\mod(\partial^{-5/2}_te^{-i\Phi}u).
\end{multline}
or, equivalently,
\begin{multline}
\text{CC}_2:\,\,e^{i\Phi}\partial_t^{-1/2}e^{-i\Phi}(\partial_{x_1}u+\partial_{x_2}u)+\frac{3}{2}e^{-i\pi/4}u\\
-i\frac{\partial_{x_1}\phi+\partial_{x_2}\phi}{4}e^{i\Phi}\partial^{-3/2}_te^{-i\Phi}u
=0\,\,\mod(\partial^{-5/2}_te^{-i\Phi}u).
\end{multline}
\subsection{Special case: $\phi=\phi(t)$}
For time-dependent potentials with no spatial variation, the quantity $\Phi$ is
purely time-dependent and the ABCs for different domains types can be simplified
considerably. It must be remarked that the exact form of the transparent boundary condition
is only obtainable for an infinite-strip with periodic boundary condition in
the unbounded direction or rectangular domains. At the appropriate segments of
the boundary, it reads as
\begin{equation}
\partial_{{n}}u+e^{-i\pi/4}e^{i\Phi}(\partial_t-i\partial_s^2)^{1/2}e^{-i\Phi}u=0.
\end{equation}
For rectangular domains, we consider this problem is more detail. At the boundary
$\Gamma_r$, we have
\begin{equation}
    \partial_{x_1}u+e^{-i\pi/4}e^{i\Phi}(\partial_t-i\partial_{x_2}^2)^{1/2}e^{-i\Phi}u=0.
\end{equation}
The auxiliary function, $\varphi(x_1,x_2,t,t')$ as defined in Section
\ref{sec:rect-domain} in the present case is given by
\begin{equation}
    \fourier_{x_2}[\varphi(x_1,x_2,t,t')] =
    e^{-i\zeta^2_2(t-t')-i\Phi(t')}\fourier_{x_2}[u(x_1,x_2,t')],
\end{equation}
so that $\varphi(x_1,x_2,t,t)=u(x_1,x_2,t)e^{-i\Phi(t)}$. Putting 
$\tilde{u}_0(\vbs{\zeta})=\fourier_{x_1}\fourier_{x_2}u_0(\vv{x})$, we have
\begin{equation}
\begin{split}
    &\fourier_{x_1}\fourier_{x_2}u(\vv{x},t')=e^{-i(\zeta_1^2+\zeta_2^2)t\rq{}+i\Phi(t')}
\tilde{u}_0(\vbs{\zeta}),\\
&\varphi(x_1,x_2,t,t')=\frac{1}{(2\pi)^2}\int_{\field{R}^2}e^{i\vbs{\zeta}\cdot\vv{x}-i\zeta_1^2t'-i\zeta_2^2t}
\tilde{u}_0(\vbs{\zeta})d^2\vbs{\zeta}.
\end{split}
\end{equation}
From here is follows that the IVPs satisfied by this auxiliary field is not
different from that described in Section \ref{sec:rect-domain}.

\section{Conclusion}
In this paper we have discussed the formulation of the operator
$(\partial_t-i\triangle_{\Gamma})^{\alpha}, \alpha=1/2,-1/2,-1,\ldots,$ in terms of the
fractional operators. This allows the TBCs/ABCs for the free Schr\"odinger equation 
and general Schr\"odinger equation formulated on various types of computational
domains to be expressed in a natural way. In particular, two families of ABCs
are studied in this paper: first obtained with the standard pseudo-differential approach 
and the second one obtained as the high-frequency approximation of the former. For the rectangular
domains, we have developed various order corner conditions for the family of
ABCs obtained in the high-frequency approximation. Each of these families of 
ABCs (along with corner conditions) are also investigated for stability and
uniqueness of the solution of the resulting initial-boundary value problem. 
The discussion of the time-discrete version of the
TBCs/ABCs is beyond the scope of this paper; this investigation is in progress
and it will be published in a forthcoming paper along with the numerical results.

\appendix
\section{Some properties of pseudo-differential operators\label{app:psido}}
Let us consider the symbol space $\fs{S}^m_M(Y\times\field{R}^n)$ of $M$-quasi
homogeneous symbols \cite{Lascar1977} where $Y$ is an open subset of $\field{R}^n$ and
$M=(\mu_1,\mu_2,\ldots,\mu_n)$ is an n-tuple of numbers $\mu_i>0$. Let $P$ be a pseudo-differential operator with the symbol
$p(y,\zeta)\in\fs{S}_M^m(Y\times\field{R}^n)$ so that
\begin{equation}
\begin{split}
Pu(y)&=\int p(y,\zeta)e^{i\zeta\cdot y}\tilde{u}(\zeta)d^n\zeta\\
&=\iint p(y,\zeta)e^{i\zeta\cdot(y-y')}u(y')dy'd\zeta,
\end{split}
\end{equation}
where $u(y)\in{\fs{C}_0^{\infty}(Y)}$ and
$d\zeta=d\zeta_1 d\zeta_2\ldots,d\zeta_n$, $dy=dy_1 dy_2\ldots,dy_n$. The adjoint of this operator, denoted by
${P}^{\dagger}$, can be defined as
\begin{equation}
P^{\dagger} u(y)=(2\pi)^{-n}\iint p^*(y',\zeta)e^{i\zeta\cdot(y-y')}u(y')dy'd\zeta.
\end{equation}
This operator belongs to a more general class of pseudo-differential operators defined by
\begin{equation}
Pu(y)=\iint p(y,y',\zeta)e^{i\zeta\cdot(y-y')}u(y')dy'd\zeta,
\end{equation}
where the symbol $p(y,y',\zeta)\in\fs{C}^{\infty}(Y\times Y\times\field{R}^n)$ 
lies in $\fs{S}_M^m(Y\times Y\times\field{R}^n)$. The adjoint of this operator is defined by the symbol
\begin{equation}
    p^{\dagger}(y,y',\zeta)=p^*(y',y,\zeta).
\end{equation}
Note that the primed variable is to be integrated over. These new operators do not have unique symbols but 
they do admit of a representation in terms of the former kind of operators as 
$P=\OP(\sigma_P(y,\zeta))+R$ where $\sigma_P(y,\zeta)\in\fs{S}_M^m(Y\times\field{R}^n)$ and 
$R$ is an operator with kernel $K_R(y,y')\in\fs{C}^{\infty}(Y\times Y)$ such that
\begin{widetext}
\begin{equation}\label{eq:asymptotic-defn}
\sigma_P(y,\zeta)-\sum_{\alpha\in\field{N}^n,\,{|\alpha|}<N-1}\frac{1}{{\alpha!}i^{|\alpha|}}\partial^{\alpha}_{\zeta}
[\partial^{\alpha}_{y'}p(y,y',\zeta)]_{y'=y}
\in\fs{S}^{m-\mu N}_M(Y\times Y\times\field{R}^n),
\end{equation}
\end{widetext}
where $N>0$ and $\mu$ is smallest element of $M$. The relationship in equation
\eqref{eq:asymptotic-defn} defines the asymptotic expansion of the symbol and we write
\begin{equation}\label{eq:asymptotic-expansion}
\sigma_P(y,\zeta)\sim\sum_{\alpha\in\field{N}^n}
\frac{1}{\alpha!i^{|\alpha|}}\partial^{\alpha}_{\zeta}[\partial^{\alpha}_{y'}p(y,y',\zeta)]_{y'=y}.
\end{equation}
This determines $\sigma_P(y,\zeta)$ only up to a smoothing operator. The formula
expressing $\sigma_{P^{\dagger}}(y,\zeta)$ in terms of $\sigma^*_P(y',\zeta)$ can be obtained by writing
\begin{equation}
p^*(y',y,\zeta)-\sigma_P^*(y',\zeta) \in\fs{S}^{-\infty}_M(Y\times Y\times\field{R}^n),,
\end{equation}
and using the asymptotic expansion \eqref{eq:asymptotic-expansion}
\begin{equation}
\begin{split}
\sigma_{P^{\dagger}}(y,\zeta)&\sim\sum_{\alpha\in\field{N}^n}\frac{1}{\alpha!i^{|\alpha|}}\partial^{\alpha}_{\zeta}[\partial^{\alpha}_{y'}\sigma^*_P(y',\zeta)]_{y'=y},
\end{split}
\end{equation}
where $R$ being a smoothing operator does not show up in the asymptotic
expansion. The adjoint can 
be used to write the Fourier transform of $Pu(y)$ by noticing that
$(P^{\dagger})^{\dagger}=P$, so that
\begin{equation}
    Pu(y)=(2\pi)^{-n}\int\biggl(\int
    \sigma^*_{P^{\dagger}}(y',\zeta)e^{-i\zeta\cdot
    y'}u(y')dy'\biggl)e^{i\zeta\cdot y}d\zeta.
\end{equation}
Consider the inner product defined by $\langle u|v\rangle = \int u^*v dy$. An 
expression of the form $\Re\langle u|P u\rangle$ arises in establishing the
stability of the ABCs. This can be computed by observing
\begin{equation}
2\Re\langle u|P u\rangle=\langle u|P u\rangle+\langle u|P^{\dagger} u\rangle.
\end{equation}
Define $2Q = (P+P^{\dagger})$ and using Plancheral's theorem, we have
\begin{equation}
    \Re\langle u|P u\rangle=(2\pi)^{-n}\iint
    d\zeta dy'\tilde{u}^*(\zeta)\sigma_{Q}(y',\zeta)u(y')e^{-i\zeta\cdot y'}.
\end{equation}
If $\sigma_{Q}(y,\zeta)$ is independent of $y$, then 
\begin{equation}
    \Re\langle u|P u\rangle=(2\pi)^{-n}\int
    d\zeta|\tilde{u}(\zeta)|^2\sigma_{Q}(\zeta).
\end{equation}
The sign of this expression can solely be decided by the sign of
$\sigma_{Q}(\zeta)$. If $\sigma_{Q}(y,\zeta)=\phi(y_1)\sigma(\tilde{\zeta})$
where $\tilde{y}=(y_2,y_3,\ldots,y_n)\in\field{R}^{n-1}$ and
$\tilde{\zeta}=(\zeta_2,\zeta_3,\ldots,\zeta_n)\in\field{R}^{n-1}$ then
\begin{equation}
    \Re\langle u|P u\rangle=(2\pi)^{-(n-1)}\iint
    dy_1d\tilde{\zeta}|\fourier_{\tilde{y}}u(y_1,y)|^2\phi(y_1)\sigma(\tilde{\zeta}).
\end{equation}
On account of the ambiguity in the knowledge of the exact
symbol, the conclusion remains valid only up to an infinitely smoothing
operator (except for the special cases where the symbol is of principle type).
\begin{itemize}
    \item Fractional operators with symbol $p(t,\xi)=\eta(i\xi)^{\alpha}$ (where
        $t\in\field{R}$ with covariable $\xi\in\field{R}\setminus\{0\}$): 
\begin{equation}
\begin{split}
\sigma_{Q}(\xi)&\sim\frac{1}{2}[\eta(i\xi)^{\alpha}+\eta^*(-i\xi)^{\alpha}]\\
               &=|\eta||\xi|^{\alpha}\cos\biggl[\frac{\pi\alpha}{4}\sgn(\xi)+\arg{\eta}\biggl].
\end{split}
\end{equation}
\item Operators with symbol $p(x,t,\zeta,\xi)=\eta(i\xi+i \zeta^2)^{\alpha}$ 
    (where $(x,t)\in\field{R}^2$ with covariables
    $(\zeta,\xi)\in\field{R}^2\setminus\{(\xi,\zeta)\in\field{R}^2:\xi+\zeta^2=0\}$): 
\begin{equation}
\begin{split}
    \sigma_{Q}(\zeta,\xi)&\sim\frac{1}{2}[\eta(i\xi+i\zeta^2)^{\alpha}+\eta^*(-i\xi-i\zeta)^{\alpha}]\\
                         &=|\eta||\xi+\zeta^2|^{\alpha}\cos\biggl[\frac{\pi\alpha}{4}\sgn{(\xi+\zeta^2)}+\arg{\eta}\biggl].
\end{split}
\end{equation}
\item Let $\sigma_P=\eta\phi(x)(i\xi)^{\alpha}$ with real valued function
    $\phi(x)>0,\forall x\in\field{R}$ where the meaning
    of the variables is same as that of the last example. Then
\begin{equation}
\begin{split}
\sigma_{Q}(x,t,\zeta,\xi)&\sim\frac{1}{2}[\eta\phi(x)(i\xi)^{\alpha}+\eta^*\phi(x)(-i\xi)^{\alpha}]\\
&=|\eta|\phi(x)|\xi|^{\alpha}\cos\biggl[\frac{\pi\alpha}{4}\sgn(\xi)+\arg{\eta}\biggl],
\end{split}
\end{equation}
so that
\begin{widetext}
\begin{equation}
    \Re\langle u|P u\rangle=\frac{|\eta|}{2\pi}\iint
    d\xi dx|\fourier_t[{u}(x,t)](x,\xi)|^2\phi(x)|\xi|^{\alpha}\cos\biggl[\frac{\pi\alpha}{4}\sgn(\xi)+\arg{\eta}\biggl].
\end{equation}
\end{widetext}

\end{itemize}
In all the above cases, when $\alpha<1$ and $\arg{\eta}=\pi/4$, it is easy to
show that the sign of $\sigma_{Q}$ remains fixed. This case is treated exactly in the
main body of the paper for the aforementioned operators without resorting to the properties of the
pseudo-differential operators (i.e. the ambiguity resulting from the lack of
knowledge of the exact symbol is circumvented).
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%\end{acknowledgments}


\end{document}
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