Logarithmic Stability for Coefficients Inverse Problem of Coupled Schr\"{o}dinger Equations

TL;DR

This paper establishes a logarithmic stability estimate for an inverse problem involving coupled Schrödinger equations, using Carleman estimates and Fourier-Bros-Iagolnitzer transform techniques.

Contribution

It introduces a novel logarithmic stability result for the inverse coefficients problem in coupled Schrödinger equations, employing advanced Carleman estimates and Fourier analysis methods.

Findings

Logarithmic stability estimate derived for the inverse problem.

Effective use of Carleman estimates for coupled Schrödinger equations.

Application of Fourier-Bros-Iagolnitzer transform in stability analysis.

Abstract

In this paper, we study an inverse coefficients problem for two coupled Schr\"{o}dinger equations with an observation of one component of the solution. The observation is done in a nonempty open subset of the domain where the equations hold. A logarithmic type stability result is obtained. The main method is based on the Carleman estimate for coupled Schr\"{o}dinger equations and coupled heatn equations, and the Fourier-Bros-Iagolnitzer transform.

Equations

$$\left\{\begin{array}[]{ll}\displaystyle i\partial_{t}y_{1}+\Delta y_{1}+a_{11}(x)y_{1}+a_{12}(x)y_{2}=0&\text{ in }\Omega\times(0,T),\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle i\partial_{t}y_{2}+\Delta y_{2}+a_{21}(x)y_{1}+a_{22}(x)y_{2}=0&\text{ in }\Omega\times(0,T),\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle y_{1}=0,y_{2}=0&\text{ on }\Gamma\times(0,T),\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle y_{1}(x,0)=y_{10},\;y_{2}(x,0)=y_{20}&\text{ in }\Omega.\end{array}\right.$$

$$||(a_{11},a_{22})||\le C||y_1{|}_{\omega }||$$

$$\{\begin{array}{l}\nabla \varphi \ne 0\text{ in }\overline{\Omega \backslash {\omega }_0},\\ \text{vskip}6.0ptplus2.0ptminus2.0pt\\ \frac{\partial \varphi }{\partial \nu }\le 0\text{ on }\partial \Omega ,\\ \text{vskip}6.0ptplus2.0ptminus2.0pt\\ |\nabla \varphi (x)\cdot \xi {|}^2+\sum _{i,j=1}^3({\partial }_j{\partial }_j\varphi (x)){\xi }_j{\xi }_j>0\text{mbox}in\overline{\Omega \backslash {\omega }_0},\forall \xi =({\xi }_1,{\xi }_2,{\xi }_3)\in {\mathbb{R}}^3,\\ \text{vskip}6.0ptplus2.0ptminus2.0pt\\ \varphi (x)>\frac{2}{3}||\varphi |{|}_{L^{\infty }(\Omega )},\forall x\in \Omega .\end{array}$$

$$\frac{{\Gamma }_0\text{buildrel}△}{=\{x\in \Gamma |(x-x_0)\cdot \nu (x)\ge 0\}.}$$

$$\frac{{\omega }_0=O_{\delta }\text{buildrel}△}{=\{x\in \Omega |\mathrm{dist}(x,{\Gamma }_0)<\delta \}.}$$

$$\{\begin{array}{l}\nabla \varphi \ne 0\text{ in }\overline{\Omega \backslash {\omega }_1},\\ \text{vskip}6.0ptplus2.0ptminus2.0pt\\ |\nabla \varphi (x)\cdot \xi {|}^2+\sum _{i,j=1}^3({\partial }_j{\partial }_j\varphi (x)){\xi }_j{\xi }_j>0\text{mbox}in\overline{\Omega \backslash {\omega }_1},\forall \xi =({\xi }_1,{\xi }_2,{\xi }_3)\in {\mathbb{R}}^3.\end{array}$$

$$\mathcal{H}=C^1(0,T;H^2(\Omega ))\cap C^2(0,T;H^1(\Omega )),$$

$$||z|{|}_{\mathcal{H}}^2=||z|{|}_{C^1(0,T;H^2(\Omega ))}^2+||z|{|}_{C^2(0,T;H^1(\Omega ))}^2,\forall z\in \mathcal{H}.$$

$$a_{21}\ge a_0\text{ or }-a_{21}\ge a_0\text{ in }\omega \times (0,T).$$

$$\begin{array}[]{ll}\displaystyle\mathcal{A}(\tilde{\omega},M)\negthinspace\negthinspace\negthinspace&\displaystyle\buildrel\triangle\over{=}\{(a_{11},a_{22})\in L^{\infty}(\Omega;{\mathbb{R}})^{2}|\,||a_{jj}||_{L^{\infty}(\Omega)}\leq M,a_{jj}(x)=\varpi_{j}(x)\text{ on }\tilde{\omega},\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr&\displaystyle\quad\;\;\mbox{the equation \eqref{eq1} has a unique solution $(y_{1},y_{2})\in\mathcal{H}$ satisfying }\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr&\displaystyle\quad\;\;||y_{j}||_{\mathcal{H}}\leq{\bf C}(M)\mbox{ for some constant ${\bf C}(M)$ depending on }M,\;j=1,2\}.\end{array}$$

$$\{\begin{array}{l}{y}_{j0}(x)\in \mathbb{R}\text{ or }i{y}_{j0}(x)\in \mathbb{R}\text{ a.e. in }\Omega ,\\ \text{vskip}6.0ptplus2.0ptminus2.0pt\\ |y_{j0}(x)|\ge r>0\text{ a.e. in }\Omega ,\\ \text{vskip}6.0ptplus2.0ptminus2.0pt\\ y_j(a_{11},a_{22})\in H^1(0,T;L^{\infty }(\Omega )).\end{array}$$

$$\begin{array}[]{ll}\displaystyle|(a_{11}-\tilde{a}_{11},a_{22}-\tilde{a}_{22})|_{L^{2}(\Omega)}\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle\leq C\Big{(}\big{|}\ln||y_{1}(a_{11},a_{22})-\tilde{y}_{1}(\tilde{a}_{11},\tilde{a}_{22})||_{H^{1}(0,T;H^{1}(\omega))}\big{|}^{-1}\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle\qquad+||y_{1}(a_{11},a_{22})-\tilde{y}_{1}(\tilde{a}_{11},\tilde{a}_{22})||_{H^{1}(0,T;H^{1}(\omega))}\Big{)},\end{array}$$

$$\sum _{j=1}^2||a_{jj}-{\tilde{a}}_{jj}|{|}_{L^2(\Omega )}^2\le C\sum _{j=1}^2||y_j(a_{11},a_{22})-{\tilde{y}}_j({\tilde{a}}_{11},{\tilde{a}}_{22})|{|}_{H^1(0,T;H^1({\omega }_1))}^2.$$

$$\hat{\phi }(x,t)\triangleq \frac{e^{\eta \varphi (x)}}{(T+t)(T-t)},\hat{\alpha }(x,t)\triangleq \frac{e^{2\eta ||\varphi |{|}_{L^{\infty }(\Omega )}}-e^{\eta \varphi (x)}}{(T+t)(T-t)},\forall (x,t)\in \Omega \times (0,T),$$

$$z_j=y_j(a_{11},a_{22})-{\tilde{y}}_j({\tilde{a}}_{11},{\tilde{a}}_{22}),f_j(x)=a_{jj}(x)-{\tilde{a}}_{jj}(x),R_j(x,t)={\tilde{y}}_j(x,t).$$

$$\left\{\begin{array}[]{ll}\displaystyle i\partial_{t}z_{1}+\Delta z_{1}+a_{11}z_{1}+a_{12}z_{2}=f_{1}(x)R_{1}(x,t)&\text{ in }\Omega\times(0,T),\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle i\partial_{t}z_{2}+\Delta z_{2}+a_{21}z_{1}+a_{22}z_{2}=f_{2}(x)R_{2}(x,t)&\text{ in }\Omega\times(0,T),\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle z_{1}(x,0)=z_{2}(x,0)=0&\text{ in }\Omega,\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle z_{1}=z_{2}=0&\text{ on }\Gamma\times(0,T).\end{array}\right.$$

$$(z_1(x,t),z_2(x,t))=(\overline{z_1(x,-t)},\overline{z_2(x,-t)})\text{mbox}fort\in (-T,0).$$

$$(R_1(x,t),R_2(x,t))=(\overline{R_1(x,-t)},\overline{R_2(x,-t)})\text{mbox}fort\in (-T,0).$$

$$(R_1(x,t),R_2(x,t))=(-\overline{R_1(x,-t)},-\overline{R_2(x,-t)})\text{mbox}fort\in (-T,0).$$

$$\left\{\begin{array}[]{ll}\displaystyle i\partial_{t}u_{1}+\Delta u_{1}+a_{11}u_{1}+a_{12}u_{2}=f_{1}(x)\partial_{t}R_{1}(x,t)&\text{ in }\Omega\times(0,T),\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle i\partial_{t}u_{2}+\Delta u_{2}+a_{21}u_{1}+a_{22}u_{2}=f_{2}(x)\partial_{t}R_{2}(x,t)&\text{ in }\Omega\times(0,T),\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle u_{1}(x,0)=-if_{1}(x)R_{1}(x,0),\quad u_{2}(x,0)=-if_{2}(x)R_{2}(x,0)&\text{ in }\Omega,\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle u_{1}=u_{2}=0&\text{ on }\Gamma\times(0,T).\end{array}\right.$$

$$||(u_1,u_2)|{|}_{[L^2(-T,T;H^2(\Omega )){]}^2}^2+||({\partial }_t{u}_1,{\partial }_t{u}_2)|{|}_{[L^2(-T,T;H_0^1(\Omega )){]}^2}^2\le \mathbf{C}.$$

$$\{\begin{array}{l}\frac{M_{j1}\text{buildrel}△}{=i(2\tau \nabla \hat{\alpha }\cdot \nabla {\hat{u}}_j+\tau \Delta \hat{\alpha }{\hat{u}}_j)+\tau {\partial }_t\hat{\alpha }{\hat{u}}_j,}\\ \text{vskip}6.0ptplus2.0ptminus2.0pt\\ \frac{M_{j2}\text{buildrel}△}{={\partial }_t{\hat{u}}_j+i(\Delta {\hat{u}}_j+{\tau }^2|\nabla \hat{\alpha }{|}^2{\hat{u}}_j).}\end{array}$$

$$\begin{array}[]{ll}\displaystyle\int_{-T}^{T}\int_{\Omega}e^{-2\tau\hat{\alpha}}\tau^{3}\eta^{4}\hat{\varphi}^{3}(|u_{1}|^{2}+|u_{2}|^{2})dxdt+\int_{-T}^{T}\int_{\Omega}\sum_{j=1}^{2}|M_{j2}|^{2}dxdt\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle\leq C\Big{\{}\int_{-T}^{T}\int_{\omega_{1}}e^{-2\tau\hat{\alpha}}\big{[}\tau^{3}\eta^{4}\hat{\varphi}^{3}\big{(}|u_{1}|^{2}+|u_{2}|^{2}\big{)}+\tau\eta^{2}\hat{\varphi}\big{(}|\nabla u_{1}|^{2}+|\nabla u_{2}|^{2}\big{)}\big{]}dxdt\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle\qquad\;+\int_{-T}^{T}\int_{\Omega}e^{-2\tau\hat{\alpha}}\big{(}|f_{1}(x)\partial_{t}R_{1}(x,t)|^{2}+|f_{2}(x)\partial_{t}R_{2}(x,t)|^{2}\big{)}dxdt\Big{\}}.\end{array}$$

$$J=-{\int }_0^T{\int }_{\Omega }{e}^{-\tau \hat{\alpha }}{M}_{12}{\bar{u}}_{1}dxdt-{\int }_0^T{\int }_{\Omega }{e}^{-\tau \hat{\alpha }}{M}_{22}{\bar{u}}_{2}dxdt.$$

$$\begin{array}[]{ll}\displaystyle{\mathop{\rm Re}\,}(J)&\negthinspace\negthinspace\negthinspace\displaystyle=-{\mathop{\rm Re}\,}\Big{[}\int_{0}^{T}\int_{\Omega}\partial_{t}\hat{u}_{1}\bar{\hat{u}}_{1}dxdt+i\int_{0}^{T}\int_{\Omega}\big{(}-|\nabla\hat{u}_{1}|^{2}+\tau^{2}|\nabla\hat{\alpha}|^{2}|\hat{u}_{1}|^{2}\big{)}dxdt\Big{]}\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr&\displaystyle\quad-{\mathop{\rm Re}\,}\Big{[}\int_{0}^{T}\int_{\Omega}\partial_{t}\hat{u}_{2}\bar{\hat{u}}_{2}dxdt+i\int_{0}^{T}\int_{\Omega}\big{(}-|\nabla\hat{u}_{2}|^{2}+\tau^{2}|\nabla\hat{\alpha}|^{2}|\hat{u}_{2}|^{2}\big{)}dxdt\Big{]}\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr&\displaystyle\negthinspace\negthinspace\negthinspace=\frac{1}{2}\int_{\Omega}\big{(}|\hat{u}_{1}(x,0)|^{2}+|\hat{u}_{2}(x,0)|^{2}\big{)}dx\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr&\displaystyle\negthinspace\negthinspace\negthinspace=\frac{1}{2}\int_{\Omega}e^{-2\tau\hat{\alpha}(x,0)}\big{(}|f_{1}(x)|^{2}|R_{1}(x,0)|^{2}+|f_{2}(x)|^{2}|R_{2}(x,0)|^{2}\big{)}dxdt.\end{array}$$

$${\mathop{\rm Re}\,}(J)\geq\frac{r^{2}}{2}\int_{\Omega}e^{-2\tau\hat{\alpha}(x,0)}\big{(}|f_{1}(x)|^{2}+|f_{2}(x)|^{2}\big{)}dxdt.$$

$$\begin{array}[]{ll}\displaystyle|J|&\negthinspace\negthinspace\negthinspace\displaystyle\leq\Big{(}\int_{0}^{T}\int_{\Omega}e^{-2\tau\hat{\alpha}}|u_{1}|^{2}dxdt\Big{)}^{\frac{1}{2}}\Big{(}\int_{0}^{T}\int_{\Omega}|M_{12}|^{2}dxdt\Big{)}^{\frac{1}{2}}\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr&\displaystyle\quad+\Big{(}\int_{0}^{T}\int_{\Omega}e^{-2\tau\hat{\alpha}}|u_{2}|^{2}dxdt\Big{)}^{\frac{1}{2}}\Big{(}\int_{0}^{T}\int_{\Omega}|M_{22}|^{2}dxdt\Big{)}^{\frac{1}{2}}\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr&\displaystyle\negthinspace\negthinspace\negthinspace\leq\tau^{\frac{3}{2}}\eta^{2}\int_{0}^{T}\!\int_{\Omega}\!e^{-2\tau\hat{\alpha}}\big{(}|u_{1}|^{2}\!+\!|u_{2}|^{2}\big{)}dxdt\!+\!\tau^{-\frac{3}{2}}\eta^{-2}\!\int_{0}^{T}\!\int_{\Omega}\!\big{(}|M_{12}|^{2}\!+\!|M_{22}|^{2}\big{)}dxdt.\end{array}$$

$$\begin{array}[]{ll}\displaystyle\quad\int_{-T}^{T}\int_{\Omega}e^{-2\tau\hat{\alpha}}\big{(}|f_{1}(x)\partial_{t}R_{1}(x,t)|^{2}+|f_{2}(x)\partial_{t}R_{2}(x,t)|^{2}\big{)}dxdt\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle\leq C\int_{\Omega}e^{-2\tau\hat{\alpha}(x,0)}\big{(}|f_{1}(x)|^{2}+|f_{2}(x)|^{2}\big{)}dx.\end{array}$$

$$\begin{array}[]{ll}\displaystyle\quad\frac{r^{2}}{2}\int_{\Omega}e^{-2\tau\hat{\alpha}(x,0)}\big{(}|f_{1}(x)|^{2}+|f_{2}(x)|^{2}\big{)}dxdt\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle\leq C\tau^{-\frac{3}{2}}\eta^{-2}\Big{\{}\int_{-T}^{T}\int_{\omega_{1}}e^{-2\tau\hat{\alpha}}\big{[}\tau^{3}\eta^{4}\hat{\varphi}^{3}\big{(}|u_{1}|^{2}+|u_{2}|^{2}\big{)}+\tau\eta^{2}\hat{\varphi}\big{(}|\nabla u_{1}|^{2}+|\nabla u_{2}|^{2}\big{)}\big{]}dxdt\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle\qquad\qquad\qquad+\int_{\Omega}e^{-2\tau\hat{\alpha}(x,0)}\big{(}|f_{1}(x)|^{2}+|f_{2}(x)|^{2}\big{)}dx\Big{\}}.\end{array}$$

$$\begin{array}[]{ll}\displaystyle\quad\int_{\Omega}e^{-2\tau\hat{\alpha}(x,0)}\big{(}|f_{1}(x)|^{2}+|f_{2}(x)|^{2}\big{)}dxdt\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle\leq C\tau^{-\frac{3}{2}}\eta^{-2}\int_{-T}^{T}\int_{\omega_{1}}e^{-2\tau\hat{\alpha}}\big{[}\tau^{3}\eta^{4}\hat{\varphi}^{3}\big{(}|u_{1}|^{2}+|u_{2}|^{2}\big{)}+\tau\eta^{2}\hat{\varphi}\big{(}|\nabla u_{1}|^{2}+|\nabla u_{2}|^{2}\big{)}\big{]}dxdt.\end{array}$$

Full text

Keywords: logarithmic stability, coefficients inverse problem, coupled Schrödinger equations, Carleman estimate

1 Introduction

Let $T>0$ and $\Omega\subset{\mathbb{R}}^{3}$ be a nonempty bounded domain with smooth boundary and let $i=\sqrt{-1}$. Consider the following coupled Schrödinger equations:

[TABLE]

System (1) is a useful model for describing molecular multiphoton transitions induced by a laser (e.g.[1, 12]), where $a_{11}(x)$ and $a_{22}(x)$ are field-free molecular electronic potentials, and $a_{12}(x)$ and $a_{21}(x)$ are radiation-molecule interactions. In physical models, usually, the radiation-molecule interactions can be deduced a priori while the field-free molecular electronic potentials should be determined a posteriori.

Let $\omega$ be a nonempty open subset of $\Omega$. In this paper, we study the following inverse problems:

Problem (IP) Can one recover the field-free molecular electronic potentials $(a_{11},a_{22})$ from suitable observation of $y_{1}$ on $[0,T]\times\omega$?

Here the word “recover” means two issues: One is that the observation determines the potentials uniquely. The other is to find an algorithm to compute the potentials efficiently.

A stability estimate

[TABLE]

with suitable norms under suitable boundedness conditions is not only important theoretically but also essential for the second issue: it can guarantee the convergence of the numerical algorithm for computing $(a_{11},a_{22})$.

Inequalities in the type of (2) for Schrödinger equations were studied extensively (e.g. [5, 3, 4, 2, 8, 7, 10, 14, 17, 19]). Roughly speaking, the existing works fall into two categories: one is Lipschitz type stability when the observation domain fulfills some geometrically condition (e.g. [3, 2, 8, 7, 10, 14, 17, 19]), while the other is logarithmic type stability when the observation domain is a general nonempty open subset of the domain or its boundary (e.g. [5, 4]). For the latter case, some a priori knowledge about the potential on a suitable subdomain should be known (see [5]).

A main method for establishing the Lipschitz type stability is based on Carleman estimate. On the other hand, the key method for proving the logarithmic type stability is a combination of the Carleman estimate and the Fourier-Bros-Iagolnitzer (F.B.I.) transformation. For readers who are not familiar with the F.B.I. transform, we refer them to [9] for an introduction and to [18] for the application of F.B.I. transform to establish observability estimate for Schrödinger equations.

To the best of our knowledge, although there are several interesting works concerning inverse problem for a parabolic system with two components by measurements of one component, for [6] as an example, there is no work on the inverse coefficients problem for the coupled Schrödinger equations with an observation on one component of the solution. Due to the essential difference between these two equations, we have to argue independently of [6] in the case of parabolic systems. In this paper, we will study this problem by the Carleman estimate for Schrödinger equation, coupled heat equations and F.B.I. transform. Although we borrow some idea in [5] to prove our main result, since we study the inverse problem for couple Schrödinger equations with a single observation on one component of the solution, we cannot simply mimic the method in [5] to obtain the desired logarithmic type stability. Some technical obstacles should be overcome, as is seen in the proof.

The rest of this paper is organized as follows. Section 2 is devoted to presenting the main result while section 3 is devoted to the proof of the main result.

2 Statement of the main result

Let $\omega_{0}$ be an open subset of $\Omega$ such that there exists a function $\phi\in C^{4}(\overline{\Omega})$ satisfying

[TABLE]

Here $\nu=\nu(x)$ denotes the outward normal vector of $\Omega$.

There are plenty of choices of $\omega_{0}$ satisfying the above condition. A typical example can be constructed as follows.

Let $x_{0}\in{\mathbb{R}}^{3}\setminus\overline{\Omega}$ and

[TABLE]

Let $\delta>0$. Put

[TABLE]

Let $\tilde{\psi}(\cdot)\in C^{4}(\overline{\Omega})$ be a nonnegative function such that $\tilde{\psi}(x)=|x-x_{0}|^{2}$ for $x\in\overline{\Omega\backslash\omega_{0}}$ and $\tilde{\psi}(x)>0$ for $x\in O_{\delta/2}$ and $\tilde{\psi}=0$ on $\Gamma_{0}$. Then $\psi(x)=\tilde{\psi}(x)+2|\tilde{\psi}|_{L^{\infty}(\Omega)}$ is the desired function.

More examples of such kind of $\omega_{0}$ and $\psi$ can be found in [17].

Clearly, if (3) holds, then there exists $\omega_{1}\subset\subset\omega_{0}$ such that

[TABLE]

Let $\tilde{\omega}\in\Omega$ be a neighborhood of $\omega_{1}$ such that $\omega_{1}\subset\subset\tilde{\omega}$ and $\partial\tilde{\omega}$ is $C^{2}$. Set

[TABLE]

where $H^{k}(\Omega)$ is the usual Sobolev space. The Banach space $\mathcal{H}$ is equipped with its natural norm

[TABLE]

Let $\omega\subset\omega_{1}\subset\Omega$ be an arbitrary nonempty open subset. Suppose that $\{a_{jk}\}_{j,k=1}^{2}\subset L^{\infty}(Q)$ and we can choose a constant $a_{0}>0$ such that

[TABLE]

Remark 2.1**.**

(7) means that the coupling between $y_{1}$ and $y_{2}$ does not degenerate. More precisely, $y_{1}$ can effect $y_{2}$ adequately. Without (7), one cannot obtain information of $y_{2}$ from $y_{1}$.

Let us now define the admissible set of unknown coefficients. Fix a constant $M>0$ and two functions $\varpi_{1},\varpi_{2}\in L^{\infty}(\tilde{\omega};{\mathbb{R}})$. Let $\mathcal{A}(\tilde{\omega},M)$ be the set of pairs of real-valued functions $(a_{11},a_{22})$ such that

[TABLE]

Remark 2.2**.**

There are mainly two restrictions on a element in $\mathcal{A}(\tilde{\omega},M)$. The first one is that there is a priori bound $M$. This is reasonable since in a physical model, one can assume to know some preliminary upper bound on unknown potentials. The second one is that we know the value of $(a_{11}(x),a_{22}(x))$ for $x\in\tilde{\omega}$. This is technically restrictive but is acceptable because we may be able to directly measure potentials near the boundary. Furthermore we note that compared with [5], we need less information on unknown potentials.

In what follows, in order to emphasize the dependence of the solution to (1) on the unknown potentials, we write $(y_{1}(a_{11},a_{22}),y_{2}(a_{11},a_{22}))$ for the solution to (1).

We choose the initial data $(y_{10},y_{20})$ which satisfy all conditions ensuring that $\mathcal{A}(\tilde{\omega},M)$ is nonempty. Also, for $j=1,2$, they fulfill

[TABLE]

Remark 2.3**.**

Condition (9) means that we have to choose initial data suiatably, and is a technical restriction. Similarly to Appendix B in [5], we can verify that such $(y_{10},y_{20})$ exists.

The main result of this paper is stated as follows.

Theorem 2.1**.**

There exists a constant $C>0$ such that

[TABLE]

for all $(a_{11},a_{22}),(\tilde{a}_{11},\tilde{a}_{22})\in{\cal A}(\tilde{\omega},M)$.

Remark 2.4**.**

One can consider the problem that all the coefficients $\{a_{jk}\}_{j,k=1}^{2}$ are unknown. In this case, the following three conditions are needed: (1) the unknown coefficient $a_{21}$ must be nonzero in a nonempty open subset $\omega$; (2) the functions $a_{11}$ and $a_{12}$, $a_{21}$ and $a_{22}$ must be linearity independence, respectively; (3) two times of observations with different suitable chosen initial data of $y_{1}$ are required. As pointed in Remark 2.1, condition (1) can not be removed since we only observe a single component of the solutions. Condition (2) is reasonable since what we can observe is only the linear combination of the coefficients. Condition (3) can not be deleted because for each observation we only observe the linear combinations to get the coefficients from these combinations and we need observe the system twice.

Remark 2.5**.**

From the proof of Theorem 2.1, one can see that it can be generalized to a system coupled by more than two Schrödinger equations with an observation on some components of the solution. In this paper, to present the key idea in a simple way, we do not pursue the full technical generality.

3 Proof of Theorem 2.1

Before giving the proof, we present a preliminary result.

Lemma 3.1**.**

For all $(a_{11},a_{22}),(\tilde{a}_{11},\tilde{a}_{22})\in{\cal A}(\tilde{\omega},M)$,

[TABLE]

In order to obtain the Lipschitz stabilty in (11), the subdomain $\omega_{1}$ can not be arbitrarily small and must satisfy (4). Lemma 3.1 should be a known result. However, since we failed to find an exact reference, we provide it here for the sake of completeness and readers’ convenience.

Proof of Lemma 3.1.

Let $\phi\in C^{4}(\overline{\Omega})$ be the function satisfying (3) and (4). Set

[TABLE]

where $\eta$ denotes some positive number which can be specified later.

For $j=1,2$, let

[TABLE]

Then $(z_{1},z_{2})\in[C([0,T];H^{1}_{0}(\Omega))]^{2}$ is the solution of the following system:

[TABLE]

Take the even-conjugate extensions of $(z_{1},z_{2})$ to the interval $(-T,T)$, i.e., set

[TABLE]

If $(R_{1}(x,0),R_{2}(x,0))\in{\mathbb{R}}^{2}$ for a.e. $x\in\Omega$, then we set

[TABLE]

If $(iR_{1}(x,0),iR_{2}(x,0))\in{\mathbb{R}}^{2}$ for a.e. $x\in\Omega$, then we set

[TABLE]

In such context, we have that $(R_{1},R_{2})\in H^{1}(-T,T;L^{\infty}(\Omega))^{2}$, and $(z_{1},z_{2})$ solves the system (13) in $\Omega\times(-T,T)$.

Assume $(u_{1},u_{2})=(\partial_{t}z_{1},\partial_{t}z_{2})$. We have

[TABLE]

It follows from (6), (13) and (14) that $(u_{1},u_{2})\in[C^{1}([-T,T];H_{0}^{1}(\Omega))\cap C([-T,T];$ $H^{2}(\Omega))]^{2}$. Further, there exists a constant ${\bf C}={\bf C}(M,T)>0$ such that

[TABLE]

For $j=1,2$ and $\tau>0$, let $\hat{u}_{j}=e^{-\tau\hat{\alpha}}u_{j}$ and

[TABLE]

By Proposition 3.1 in [17], we know that there exist $\tau_{0}>0$ and $\eta_{0}(\tau_{0})>0$ such that for all $\tau\geq\tau_{0}$ and $\eta\geq\eta_{0}(s_{0})$, it holds that

[TABLE]

Put

[TABLE]

Then

[TABLE]

This, together with the conditions on $R_{1}(x,0)$ and $R_{2}(x,0)$, implies that

[TABLE]

On the other hand, it follows from (18) that

[TABLE]

From the choice of $\hat{\alpha}$, we find that

[TABLE]

This, together with (17), (19) and (20), implies that

[TABLE]

Thus, there is an $\tau_{1}>0$ such that for all $\tau\geq\max\{\tau_{0},\tau_{1}\}$ and $\eta\geq\eta_{0}(\tau_{0})$,

[TABLE]

This concludes (11) and completes the proof of Lemma 3.1. ∎

Next, in order to keep the self-containment, we give a brief introduction to F.B.I. transformation here. Let

[TABLE]

Then

[TABLE]

For every $\lambda\geq 1$, define

[TABLE]

Then,

[TABLE]

Let $s,l_{0}\in\mathbb{R}$, the F.B.I. transformation $\mathcal{F}_{\lambda}$ for $u\in\mathcal{S}(\mathbb{R}^{n+1})$ is defined as follows:

[TABLE]

Now we are in a position to prove Theorem 2.1.

Proof of Theorem 2.1.

The proof is long. We divide it into four steps.

Step 1. In this step, we introduce an equation on $(-T,T)\times\tilde{\omega}$.

Recall that $\omega$ is an arbitrary fixed nonempty subset of $\tilde{\omega}$ such that $\overline{\omega}\subset\tilde{\omega}$. By [11, Lemma 1.1], there exists a function $\psi\in C^{2}(\overline{\tilde{\omega}})$ such that

[TABLE]

We can conclude from (24) that there exist a constant $\beta>0$ and $\omega_{2}\subset\subset\tilde{\omega}$ such that

[TABLE]

and that

[TABLE]

It follows from the last condition in (24) that the maximum value of $\psi$ can only be attained in $\omega$, i.e., there exists a point $x_{0}\in\omega$ such that

[TABLE]

Let $\chi\in C^{\infty}_{0}(\tilde{\omega})$ be a cut-off function, which satisfies $0\leq\chi\leq 1$ and

[TABLE]

where $\omega_{3}$ is a subset of $\tilde{\omega}$ such that $\omega_{2}\subset\subset\omega_{3}$.

Let $(w_{1},w_{2})=(\chi u_{1},\chi u_{2})$. Then by (8) and (14), we have that

[TABLE]

By (15), there exists ${\bf C}={\bf C}(M,T)>0$ such that

[TABLE]

Step 2. In this step, we introduce a system of parabolic equations related to (29) and a Carleman estimate to the parabolic system.

For $j=1,2$, let $\displaystyle W_{j}(x,s)=\int_{\mathbb{R}}F_{\lambda}(l_{0}+is-l)\Phi(l)w_{j}(x,l)dl$. Since

[TABLE]

we follow that

[TABLE]

where for $j=1,2$,

[TABLE]

Let

[TABLE]

where $\eta>0$.

Let $\Phi\in C^{\infty}_{0}(\mathbb{R})$ satisfying the following conditions:

[TABLE]

where $L>0$ will be chosen later.

Take

[TABLE]

Then $l_{0}\in K_{0}$ in (23).

According to Theorem 1.1 in [13], there exist a positive function $\alpha_{0}\in C^{2}(\overline{\tilde{\omega}})$ (only depending on $\tilde{\omega}$ and $\omega$), two positive constants $C_{0}$ (only depending on $\tilde{\omega}$, $\omega$, $\alpha_{0}$ and $M_{0}$) and $\sigma_{0}=\sigma_{0}(\tilde{\omega},\omega,M_{0})$ such that the solution $(W_{1},W_{2})\in[C([-T,T];L^{2}(\tilde{\omega}))\cap L^{2}([-T,T];$ $H^{1}(\tilde{\omega}))]^{2}$ of (31) satisfies that

[TABLE]

where $\gamma(s)=\frac{1}{(T+s)(T-s)}$ and $\sigma\geq\sigma_{0}$.

Step 3. In this step, we estimate all the terms in the right hand side of (33).

Let

[TABLE]

There exists $\delta_{2}>0$ such that

[TABLE]

By the property of F.B.I. transformation, we have that

[TABLE]

From the definition of $F_{j}$, we see that

[TABLE]

Since $\text{supp}\chi^{\prime}\subset\tilde{\omega}\backslash\omega_{1}$ and $G_{j}(x,s)=0$ in $\omega_{1}$, it holds that

[TABLE]

Set

[TABLE]

By (25), we know that there exists $\delta_{1}>0$ such that

[TABLE]

Consequently,

[TABLE]

and

[TABLE]

Substituting (35), (39) and (40) into (33), we obtain that

[TABLE]

In order to reduce the computation complexity of the proof, and without loss of generality, in the following steps, we assume $T=1$. Let $A>1$. By choosing $L=8AT=8A$, we have

[TABLE]

where

[TABLE]

Similarly, we can get that

[TABLE]

Fix $\epsilon\in(0,1)$ such that

[TABLE]

This is equivalent to say that

[TABLE]

Hence,

[TABLE]

Step 4. By Lemma 3.1,

[TABLE]

It follows from Parseval’s identity that

[TABLE]

The first term in the right hand side of (47) reads

[TABLE]

Let

[TABLE]

By applying the Cauchy integral formula, for $\rho\in(0,T-\epsilon)$ and by setting $z=\kappa+\rho e^{i\phi}$, we have that

[TABLE]

Thus,

[TABLE]

Integrating (51) with respect to $x$ over $\omega_{1}$ and with respect to $\kappa$ over $[-\frac{L}{2},\frac{L}{2}]$, we get that

[TABLE]

Substituting (3), (52) into (47) and noting that $T=1$, we find that

[TABLE]

Similarly, we can obtain that

[TABLE]

Let $\displaystyle\sigma=\frac{\lambda^{2}}{2\tau\mu_{3}}$ and $C_{2}=\max\{|\nabla\chi|^{2},|\Delta\chi|^{2}\}$ such that

[TABLE]

From (3), (47), (53) and (54), we have

[TABLE]

Let $\lambda\geq\lambda_{0}$ be such that

[TABLE]

where $C_{3}$ and $C_{4}$ are two constants independent of $\lambda$. Taking

[TABLE]

If $||w_{1}||_{L^{2}(0,T;H^{1}(\omega))}$ is small enough, then

[TABLE]

Otherwise, there exists a constant $m>0$ such that $||w_{1}||_{L^{2}(-T,T;H^{1}(\omega))}\geq m$. Thus, by (30) we have

[TABLE]

∎

acknowledgement

The first author thanks the support of the National Natural Science Foundation of China (No. 11501086), the Fundamental Research Funds for the Central Universities (No. ZYGX2016J137) and the Science Strength Promotion Programme of UESTC. The second author is supported by Grant-in-Aid for Scientific Research (S) 15H05740 and A3 Foresight Program Modeling and Computation of Applied Inverse Problems” of Japan Society for the Promotion of Science, and the ”RUDN University Program 5-100”.