Morgan type uncertainty principle and unique continuation properties for abstract Schr\"odinger equations

TL;DR

This paper establishes Morgan type uncertainty principles and unique continuation properties for abstract Schrödinger equations with time-dependent potentials in Hilbert spaces, broadening understanding of their behavior in various physical systems.

Contribution

It introduces new uncertainty principles and unique continuation results for abstract Schrödinger equations with operators in Hilbert spaces, applicable to many physical models.

Findings

Derived Morgan type uncertainty principles for abstract Schrödinger equations.

Proved unique continuation properties for a wide class of Schrödinger systems.

Applicable to various physical systems with time-dependent potentials.

Abstract

In this paper, Morgan type uncertainty principle and unique continuation properties of abstract Schr\"odinger equations with time dependent potentials in vector-valued classes are obtained. The equation involves a possible linear operators considered in the Hilbert spaces. So, by choosing the corresponding spaces H and operators we derived unique continuation properties for numerous classes of Schr\"odinger type equations and its systems which occur in a wide variety of physical systems

Equations

$$i{\partial }_{t}u+\Delta u+Au+V\left(x,t\right)u=0,x\in R^n,t\in \left[0,1\right],$$

$$i{\partial }_{t}u+\Delta u+Au=F\left(u,\bar{u}\right),x\in R^n,t\in \left[0,1\right],$$

$${\partial }_{t}u=i\left[\Delta u+Lu+V\left(x,t\right)u\right],x\in R^n,y\in G,t\in \left[0,1\right],$$

$$D\left(A\right)=\left\{u\in W^{2,2}\left(0,1\right),B_{j}u=Au\left(j\right)=0\text{, }j=0,1\right\},$$

$$Au=a{u}^{\left(2\right)}+b{u}^{\left(1\right)},$$

$$i{\partial }_{t}u+\Delta u+a\frac{{\partial }^{2}u}{\partial y^2}+b\frac{\partial u}{\partial y}=F\left(u,\bar{u}\right),$$

$$u=u\left(x,y,t\right)\text{, }x\in R^n,y\in G,t\in \left[0,1\right],$$

$$a\left(j\right)u_{yy}\left(x,j,t\right)+b\left(j\right)u_y\left(x,j,t\right)=0\text{, }j=0,1\text{, for a.e. }x\in R^n,t\in \left(0,1\right).$$

$${\partial }_t{u}_m=i\left[\Delta u_m+\sum _{j=1}^N\left(a_{mj}+b_{mj}\left(x,t\right)\right)u_j\right],x\in R^n,t\in \left(0,1\right).$$

$${\Vert f\Vert }_{{}_{p,\gamma }}={\Vert f\Vert }_{L_{p,\gamma }\left(\Omega ;E\right)}={\left(\int {\Vert f\left(x\right)\Vert }_E^p\gamma \left(x\right)dx\right)}^{\frac{1}{p}},1\le p<\infty ,$$

$${\Vert f\Vert }_{L_{\infty ,\gamma }\left(\Omega ;E\right)}=ess\underset{x\in \Omega }{\sup }{\Vert f\left(x\right)\Vert }_E\gamma \left(x\right),p=\infty .$$

$${\Vert f\Vert }_{L_x^{p_1}{L}_y^{p_2}\left(\Omega ;E\right)}={\left({\left(\underset{{\Omega }_1}{\int }{\Vert f\left(x,t\right)\Vert }_E^{p_2}dt\right)}^{\frac{p_1}{p_2}}dx\right)}^{\frac{1}{p_1}}.$$

$${\left(f,g\right)}_{L^2\left(\Omega ;H\right)}=\underset{\Omega }{\int }{\left(f\left(x\right),g\left(x\right)\right)}_{H}dx.$$

$${\Vert u\Vert }_{C\left(\Omega ;E\right)}=\underset{x\in \Omega }{\sup }{\Vert u\left(x\right)\Vert }_E.$$

$${\Vert u\Vert }_{C^m\left(\Omega ;E\right)}=\underset{0\le |\alpha |\le m}{\max }\underset{x\in \Omega }{\sup }{\Vert D^{\alpha }u\left(x\right)\Vert }_E.$$

$$O_R=\left\{x\in R^n,\left|x\right|<R\right\}\text{, }R>0.$$

$${\Vert u\Vert }_{W^{m,p}\left(\Omega ;E\right)}={\Vert u\Vert }_{L^p\left(\Omega ;E\right)}+\sum _{k=1}^n{\Vert \frac{{\partial }^{m}u}{\partial x_k^m}\Vert }_{L^p\left(\Omega ;E\right)}<\infty .$$

$${\Vert u\Vert }_{W^{m,p}\left(\Omega ;E_{0,}E\right)}={\Vert u\Vert }_{L^p\left(\Omega ;E_0\right)}+\sum _{k=1}^n{\Vert \frac{{\partial }^{m}u}{\partial x_k^m}\Vert }_{L^p\left(\Omega ;E\right)}<\infty .$$

$$\left[A,B\right]=AB-BA$$

$$X=L^2\left(R^n;H\right)\text{, }{Y}^k=W^{2,k}\left(R^n;H\right)\text{, }k\in \mathbb{N}.$$

$$\underset{R^n}{\int }\underset{R^n}{\int }{\Vert f\left(x\right)\Vert }_H{\Vert \hat{f}\left(\xi \right)\Vert }_H{e}^{|x.\zeta |}dxd\xi <\infty \text{.}$$

$$f\left(x\right)\in L^1\left(R^n;H\right)\cap X,p\in \left(1,2\right),\frac{1}{p}+\frac{1}{q}=1,\alpha ,\beta >0$$

$$\underset{R^n}{\int }{\Vert f\left(x\right)\Vert }_H{e}^{\frac{{\alpha }^p{|x|}^p}{p}}dx+\underset{R^n}{\int }{\Vert \hat{f}\left(\xi \right)\Vert }_H{e}^{\frac{{\beta }^q{|\xi |}^q}{q}}d\xi <\infty \text{, }\alpha \beta >1.$$

$$u_0\left(.\right)\in L^1\left(R^n;H\right)\cap X$$

$$\underset{R^n}{\int }{\Vert u_0\left(x\right)\Vert }_H{e}^{\frac{{\alpha }^p{|x|}^p}{p}}dx+\underset{R^n}{\int }{\Vert e^{it\left(\Delta +A\right)}{u}_0\left(x\right)\Vert }_H{e}^{{\frac{{\beta }^q|\xi |}{q{\left(2t\right)}^q}}^q}dx<\infty \text{, }\alpha \beta >1.$$

$${\Vert V\Vert }_{L^{\infty }\left(R^n\times \left(0,1\right);L\left(H\right)\right)}\le C$$

$$\underset{R\to \infty }{\lim }{\Vert V\Vert }_{L_t^1{L}_x^{\infty }\left(L\left(H\right)\right)}=0,$$

$$L_t^1{L}_x^{\infty }\left(L\left(H\right)\right)=L^1\left(0,1;L^{\infty }\left(R^n/O_R\right);L\left(H\right)\right).$$

$$\sigma \left(t\right)=\frac{1}{\alpha \left(1-t\right)+\beta t}.$$

$$\underset{t\in \left[0,1\right]}{\sup }{\Vert e^{{|x|}^2{\sigma }^2\left(t\right)}{V}_2\left(.,t\right)\Vert }_B<\infty .$$

Full text

Morgan type uncertainty principle and unique continuation properties for abstract Schrödinger equations

Veli Shakhmurov

Department of Mechanical Engineering, Okan University, Akfirat, Tuzla 34959 Istanbul, Turkey,

E-mail: veli.sahmurov@okan.edu.tr

Abstract

In this paper, Morgan type uncertainty principle and unique continuation properties of abstract Schrödinger equations with time dependent potentials in vector-valued $L^{2}$ classes are obtained. The equation involves a possible linear operators considered in the Hilbert space $H.$ So, by choosing the corresponding spaces $H$ and operators we derived unique continuation properties for numerous classes of Schrödinger type equations and its systems which occur in a wide variety of physical systems.

**Key Word:    Schrödinger equations, Positive operators, **Semigroups of operators, Unique continuation, Morgan type uncertainty principle

**AMS 2010: 35Q41, 35K15, 47B25, 47Dxx, 46E40 **

1. Introduction, definitions

In this paper, the unique continuation properties of the abstract Schrödinger equations

[TABLE]

are studied, where $A$ is a linear operator, $V\left(x,t\right)$ is a given potential operator function in a Hilbert space $H$, subscript $t$ indicates the partial derivative with respect to $t$, $\Delta$ denotes the Laplace operator in $R^{n}$ and $u=$ $u(x,t)$ is the $H$-valued unknown function. The goal is to obtain sufficient conditions on the opertator $A$, the potential $V$ and the behavior of the solution $u$ at two different times, $t_{0}=0$ and $t_{1}=1$ which guarantee that $u\equiv 0$ in $R^{n}\times[0,1].$

This linear result was then applied to show that two regular solutions $u_{1}$ and $u_{2}$ of non-linear Schrödinger equations

[TABLE]

and for general non-linearities $F$, must agree in $R^{n}\times[0,1]$, when $u_{1}-u_{2}$ and its gradient decay faster than any quadratic exponential at times [math] and $1$.

Unique continuation properties for Schrödinger equations studied e.g in $\left[\text{6-8}\right],$ $\left[\text{21-23}\right]$ and the referances therein. In contrast to the mentioned above results we will study the unique continuation properties of abstract Schrödinger equations with operator potentials. Abstract differential equations studied e.g. in $\left[1\right]$, $\left[5\right]$, $\left[10\text{, }11\right],\left[13-19\right]$, $\left[25\text{, }26\right].$ Since the Hilbert space $H$ is arbitrary and $A$ is a possible linear operator, by choosing $H$ and $A$ we can obtain numerous classes of Schrödinger type equations and its systems which occur in a wide variety of physical systems. If we choose the abstract space $H$ a concrete Hilbert space, for example $H=L^{2}\left(G\right)$, $A=L,$ where $G$ is a domin in $R^{m}$ with sufficientli smooth boundary and $L$ is elliptic operator, then we obtain the unique continuation properties of followinng Schrödinger equation

[TABLE]

where $L$ is an elliptic operator with respect to variable $y\in G\subset R^{m}$ and $u=u\left(x,y,t\right)$.

Moreover, let $H=L^{2}\left(0,1\right)$ and $A$ to be differential operator with generalized Wentzell-Robin boundary condition defined by

[TABLE]

[TABLE]

where $a$ is positive and $b$ is a real-valued functions on $\left(0,1\right)$. Then, we get the unique continuation properties of the Wentzell-Robin type boundary value problem (BVP) for the following Schrödinger type equation

[TABLE]

[TABLE]

[TABLE]

Note that, the regularity properties of Wentzell-Robin type BVP for elliptic equations were studied e.g. in $\left[\text{11, 12 }\right]$ and the references therein. Moreover, if put $H=l_{2}$ and choose $A$ and $V\left(x,t\right)$ as a infinite matrices $\left[a_{mj}\right]$, $b_{mj}\left(x,t\right)$ $m,$ $j=1,2,...,\infty$ respectively, then we obtain the unique continuation properties of the following system of Schrödinger equation

[TABLE]

Let $E$ be a Banach space and $\gamma=\gamma\left(x\right)$ be a positive measurable function on a domain $\Omega\subset R^{n}.$Here, $L_{p,\gamma}\left(\Omega;E\right)$ denotes the space of strongly measurable $E$-valued functions that are defined on $\Omega$ with the norm

[TABLE]

[TABLE]

For $\gamma\left(x\right)\equiv 1$ the space $L_{p,\gamma}\left(\Omega;E\right)$ will be denoted by $L_{p}=L_{p}\left(\Omega;E\right)$ for $p\in\left[1,\infty\right].$

Let $\mathbf{p=}\left(p_{1},p_{2}\right)$ and $\Omega=\Omega_{1}\times\Omega_{2}$, where $\Omega_{k}\in R^{n_{k}}$. $L_{x}^{p_{1}}L_{t}^{p_{2}}\left(\Omega;E\right)$ will denote the space of all $E$-valed $\mathbf{p}$-summable functions with mixed norm, i.e., the space of all measurable functions $f$ defined on $G$ equiped with norm

[TABLE]

For $p=2$ and $H$ Hilbert space we get Hilbert space of $H$-valued functions with inner product of two elements $f$, $g\in L^{2}\left(\Omega;H\right)$:

[TABLE]

Let $C\left(\Omega;E\right)$ denote the space of $E-$valued, bounded uniformly continious functions on $\Omega$ with norm

[TABLE]

$C^{m}\left(\Omega;E\right)$ will denote the space of $E$-valued bounded uniformly strongly continuous and $m$-times continuously differentiable functions on $\Omega$ with norm

[TABLE]

$C_{0}^{\infty}\left(\Omega;E\right)-$denotes the space of $E$-valued infinity many differentiable finite functions.

Let

[TABLE]

Let $\mathbb{N}$ denote the set of all natural numbers, $\mathbb{C}$ denote the set of all complex numbers.

Let $\Omega$ be a domain in $R^{n}$ and $m$ be a positive integer. $W^{m,p}\left(\Omega;E\right)$ denotes the space of all functions $u\in L^{p}\left(\Omega;E\right)$ that have the generalized derivatives $\frac{\partial^{m}u}{\partial x_{k}^{m}}\in L^{p}\left(\Omega;E\right),$ $1\leq p\leq\infty$ with the norm

[TABLE]

Let $E_{0}$ and $E$ be two Banach spaces and $E_{0}$ is continuously and densely embedded into $E$. Here, $W^{m,p}\left(\Omega;E_{0},E\right)$ denote the space $W^{m,p}\left(\Omega;E\right)\cap$ $L^{p}\left(\Omega;E\right)$ equipped with norm

[TABLE]

Let $E_{1}$ and $E_{2}$ be two Banach spaces. $L\left(E_{1},E_{2}\right)$ will denote the space of all bounded linear operators from $E_{1}$ to $E_{2}.$ For $E_{1}=E_{2}=E$ it will be denoted by $L\left(E\right).$

A linear operator $A$ is said to be positive in a Banach space $E$ with bound $M>0$ if $D\left(A\right)$ is dense on $E$ and $\left\|\left(A+sI\right)^{-1}\right\|_{L\left(E\right)}\leq M\left(1+\left|s\right|\right)^{-1}$ for any $s\in$ $\left(-\infty,0\right),$ where $I$ is the identity operator in $E.$

Let $\left[A,B\right]$ be a commutator operator, i.e.

[TABLE]

for linear operators $A$ and $B.$

Sometimes we use one and the same symbol $C$ without distinction in order to denote positive constants which may differ from each other even in a single context. When we want to specify the dependence of such a constant on a parameter, say $\alpha$, we write $C_{\alpha}$.

2. Main results for absract Scrödinger equation

First of all, we generalize the result G. W. Morgan (see e.g $\left[7\right]$) about Morgan type uncertainty principle for Fourier transform. Let

[TABLE]

**Lemma 2.1. Let ** $f\left(x\right)\in L^{1}\left(R^{n};H\right)\cap X$ and

[TABLE]

Then $f\left(x\right)\equiv 0.$

In particular, using Young’s inequality this implies:

Result 2.1. Let

[TABLE]

and

[TABLE]

Then $f\left(x\right)\equiv 0.$

The Morgan type uncertainty principle, in terms of the solution of the free Schrödinger equation will be as:

Let

[TABLE]

and for some $t\neq 0$

[TABLE]

Then $u_{0}\left(x\right)\equiv 0.$

**Condition 1. **Assume $A$ is a pozitive operator in Hilbert space $H$ and $iA$ generates a semigrop $U\left(t\right)=e^{iAt}$.  Suppose

[TABLE]

and

[TABLE]

where

[TABLE]

Here,

[TABLE]

In $\left[\text{17, Theorem 1}\right]$ we proved the following result:

**Theorem A1. **Assume the Condition 1 holds or $V\left(x,t\right)=V_{1}\left(x\right)+V_{2}\left(x,t\right)$, where $V_{1}\left(x\right)\in L\left(H\right)$ for $x\in R^{n}$ and

[TABLE]

Suppose $\alpha$, $\beta>0$ and $\alpha\beta<4$ such that any solution $u\in C\left(\left[0,1\right];X\right)$ of $\left(1.1\right)$ satisfay

[TABLE]

Then $u\left(x,t\right)\equiv 0.$

Our main result in this paper is the following:

**Theorem 1. **Assume the Condition 1 holds and there exist constants $a_{0},$ $a_{1},$ $a_{2}>0$ such that for any $k\in\mathbb{Z}^{+}$ a solution $u\in C\left(\left[0,1\right];X\right)$ of $\left(1.1\right)$ satisfy

[TABLE]

[TABLE]

Moreover, there exists $M_{p}>0$ such that

[TABLE]

Then $u\left(x,t\right)\equiv 0.$

**Corollary 1. **Assume the Condition 1 holds and

[TABLE]

There exist positive constants $\alpha$, $\beta$ such that a solution $u\in C\left(\left[0,1\right];X\right)$ of $\left(1.1\right)$ satisfy

[TABLE]

with

[TABLE]

and there exists $N_{p}>0$ such that

[TABLE]

Then $u\left(x,t\right)\equiv 0.$

As a direct consequence of Corollary 1 we have the following result regarding the uniqueness of solutions for nonlinear equation $\left(1.2\right)$

**Theorem 2. **Assume the Condition 1 holds and $u_{1},$ $u_{2}\in C\left(\left[0,1\right];Y^{2,k}\right)$ strong solutions of $(1.2)$ with $k\in\mathbb{Z}^{+},$ $k>\frac{n}{2}.$ Suppose $F:H\times H\rightarrow H,$ $F\in C^{k}$, $F\left(0\right)=\partial_{u}F\left(0\right)=\partial_{\bar{u}}F\left(0\right)=0$ and there exist positive constants $\alpha$, $\beta$ such that

[TABLE]

with

[TABLE]

and there exists $N_{p}>0$ such that

[TABLE]

Then $u_{1}\left(x,t\right)\equiv u_{2}\left(x,t\right).$

Corollary 2. Assume the Condition 1 holds and there exist positive constants $\alpha$ and $\beta$ such that a solution $u\in C\left(\left[0,1\right];X\right)$ of $\left(1.1\right)$ satisfy

[TABLE]

for $j=1,2,...,n$ and $p\in\left(1,2\right)$, $\frac{1}{p}+\frac{1}{q}=1.$ Moreover, there exists $N_{p}>0$ such that

[TABLE]

Then $u\left(x,t\right)\equiv 0.$

**Remark 2.1. **The Theorem 3 still holds, with different constant $N_{p}>0$, if one replaces the hypothesis $(2.8)$ by

[TABLE]

for $j=1,2,...,n.$

Next, we shall extend the method used in the proof Theorem 3 to study the blow up phenomenon of solutions of nonlinear Schrödinger equations

[TABLE]

where $A$ is a linear operatorin a Hilbert space $H.$

Let $u\left(x,t\right)$ be a solution of the equation $\left(2.12\right).$ Then it can be shown that the function

[TABLE]

is a solution of the focussing $L^{2}$-critical solution of abstract Schrodinger equation

[TABLE]

which blows up at time $t=1,$ where $U\left(x,t\right)$ is a fundamental solution of the Schrödinger equation

[TABLE]

i.e.

[TABLE]

Assume that the functon $\upsilon$ defıned by $\left(2\text{.}13\right)$ satisfies the following estimate

[TABLE]

where

[TABLE]

The following result will be get:

**Theorem 3. **Assume the Condition 1 holds and there exist positive constants $b_{0}$ and $\theta$ such that a solution $u\in C\left(\left[-1,1\right];X\right)$ of $\left(2.12\right)$ satisfied

[TABLE]

Suppose

[TABLE]

and that $\left(2.15\right)$ holds with $Q\left(.\right)$ satisfies $\left(2.16\right).$ If $p>p\left(\theta\right)=\frac{2\left(\theta n-2\right)}{\left(\theta n-1\right)},$ then $a\equiv 0.$

3. Some properties of solutions of abstract Schredinger equations

Let

[TABLE]

[TABLE]

We recall the following lemma (see $\left[\text{17, Lemma 3.1}\right]$).

Let

[TABLE]

[TABLE]

**Lemma A1. **Assume $a>0,$ $b\in\mathbb{R}$, $A$ is a symmetric operator in $H.$ Moreover, there is a constant $C_{0}>0$ so that

[TABLE]

for $x\in R^{n},$ $t\in\left[0,T\right],$ $\gamma\geq 0$, $T\in\left[0,1\right]$ and $\upsilon\in H\left(A\right)$, where $\mu$ is a positive function in $L^{1}\left(0,T;L^{\infty}\left(R^{n}\right)\right)$.

Then the solution $u$ of $\left(3.0\right)$ belonging to $L^{\infty}\left(0,1;X\right)\cap L^{2}\left(0,1;Y^{1}\right)$ satisfies the following estimate

[TABLE]

where

[TABLE]

Let $u=u\left(x,s\right)$ be a solution of the equation

[TABLE]

and $a+ib\neq 0$, $\gamma\in\mathbb{R}$, $\alpha$, $\beta\in\mathbb{R}_{+}$. Set

[TABLE]

Then, $\tilde{u}\left(x,t\right)$ verifies the equation

[TABLE]

with

[TABLE]

[TABLE]

Moreover,

[TABLE]

when $s=\beta t\sigma\left(t\right)$.

**Remark 3.1. **Let $\beta=\beta\left(k\right).$ By assumption we have

[TABLE]

[TABLE]

Thus, for $\gamma=\gamma(k)$ $\in[0,\infty)$ to be chosen later, one has

[TABLE]

[TABLE]

Let we take

[TABLE]

i.e.

[TABLE]

Let

[TABLE]

From $\left(3.2\right)$, using energy estimates it follows

[TABLE]

where

[TABLE]

Consider the following problem

[TABLE]

[TABLE]

where $A$ is a linear operator, $V\left(x,t\right)$ is a given potential operator function in a Hilbert space $H$ and $F$ is a $H$-valued function.

Let as define operator valued integral operators in $L^{p}\left(\Omega;E\right)$. Let $k$: $R^{n}\backslash\left\{0\right\}\rightarrow L\left(E\right).$ We say $k\left(x\right)$ is a $L\left(E\right)$-valued Calderon-Zygmund kernel ($C-Z$ kernel) if $k\in C^{\infty}\left(R^{n}\backslash\left\{0\right\},L\left(E\right)\right),$ $k$ is homogenous of degree $-n,$ $\mathop{\displaystyle\int}\limits_{B}k\left(x\right)d\sigma=0,$ where

[TABLE]

For $f\in L^{p}\left(\Omega;E\right),$ $p\in\left(1,\infty\right),$ $a\in L^{\infty}\left(R^{n}\right),$ and $x\in\Omega$ we set the Calderon-Zygmund operator

[TABLE]

and commutator operator

[TABLE]

By using Calderón’s first commutator estimates $\left[4\right],$ convolution operators on abstract functions $\left[2\right]$ and abstract commutaror theorem in $\left[20\right]$ we obtain the following result:

Theorem A${}_{2}.$ Assume $k(.)$ is $L\left(E\right)$-valued $C-Z$ kernel that have locally integrable first-order derivatives in $\left|x\right|>0$, and

[TABLE]

Let $a(.)$ have first-order derivatives in $L^{r}\left(R^{n}\right)$,$1<r\leq\infty$. Then for $p$, $q\in\left(1,\infty\right),$ $q^{-1}=p^{-1}+r^{-1}$ the following estimates hold

[TABLE]

[TABLE]

for $f\in C_{0}^{\infty}\left(R^{n};E\right),$ where the constant $C>0$ is independent of $f.$

Let

[TABLE]

By following $\left[\text{4, Lemma 2.1}\right]$ let us show the following

Lemma 3.1. Assume the Condition 1 holds and there exists $\varepsilon_{0}>0$ such that

[TABLE]

Moreover, suppose $u\in C\left(\left[0,1\right];X\right)$ is a stronge solution of $\left(3.10\right)$ with

[TABLE]

for $\gamma\left(x\right)=e^{2\lambda.x}$   and for some $\lambda\in R^{n}.$ Then there exists a positive constant $M_{0}=M_{0}\left(n,A,H\right)$ independent of $\lambda$ such that

[TABLE]

**Proof. **First, we consider the case, when $\gamma\left(x\right)=\beta\left(x\right)=e^{2\beta x_{1}}.$ Without loss of generality we shall assume $\beta>0.$ Let $\varphi_{n}\in C^{\infty}\left(\mathbb{R}\right)$ such that $\varphi_{n}\left(\tau\right)=1$, $\tau\leq n$ and $\varphi_{n}\left(\tau\right)=0$ for $\tau\geq 10n$ with $0\leq\varphi_{n}\leq 1,$ $\left|\varphi_{n}^{\left(j\right)}\left(\tau\right)\right|\leq C_{j}\tau n^{-j}.$ Let

[TABLE]

so that $\theta_{n}\in C^{\infty}\left(\mathbb{R}\right)$ nondecreasing with $\theta_{n}(\tau)=\beta\tau$ for $\tau<n,$ $\theta_{n}(\tau)=C_{n}\beta$ for $\tau>10n$ and

[TABLE]

Let $\phi_{n}\left(\tau\right)=\exp\left(2\theta_{n}(\tau)\right)$ so that $\phi_{n}\left(\tau\right)\leq\exp\left(2\beta\tau\right)$ and $\phi_{n}\left(\tau\right)\rightarrow\exp\left(2\beta\tau\right)$ for $n\rightarrow\infty.$ Let $u\left(x,t\right)$ be a solution of the equation $\left(3.10\right)$, then one gets the equation $\upsilon_{n}\left(x,t\right)=\phi_{n}\left(x_{1}\right)u\left(x,t\right)$ satisfies the following

[TABLE]

where

[TABLE]

Now, we consider a new function

[TABLE]

Then from $\left(3.13\right)$ we get

[TABLE]

where

[TABLE]

[TABLE]

when

[TABLE]

[TABLE]

It is clear to see that

[TABLE]

[TABLE]

[TABLE]

Consider the smooth function $\eta\in C^{\infty}\left(R^{n}\right)$ such that $\eta\left(x\right)=1$ for $\left|x\right|\leq\frac{1}{2}$ and $\eta\left(x\right)=0$ for $\left|x\right|\geq 1,$ with $0\leq\eta\left(x\right)\leq 1.$ Moreover, let

[TABLE]

Define the multipliers operators

[TABLE]

Then by applying the equation $\left(3.10\right)$ for example to $P_{\varepsilon}P_{+}w_{n}$ we get

[TABLE]

[TABLE]

From $\left(3.16\right)$ we obtain

[TABLE]

[TABLE]

[TABLE]

for all $\upsilon\in C_{0}^{\infty}\left(R^{n};H\right)$, where $\left(u,\upsilon\right)$ denotes scalar product of $u\left(x,t\right)$ and $\upsilon\left(x\right)$ in $H$ for all $x\in R^{n}$ and $t\in\left[0,1\right].$ Taking the complex conjugate of the above, we get the equation

[TABLE]

[TABLE]

[TABLE]

here $\bar{K}\left(u\right)$ denotes the complex conjugate of $u.$

Multiplying $\left(3.17\right)$ and $\left(3.18\right)$ by $\bar{K}\left(\left(P_{\varepsilon}P_{+}w_{n},\upsilon\right)\right)$ and $-\left(P_{\varepsilon}P_{+}w_{n},\upsilon\right)$ respectively, and adding the result, we obtain

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By taking the imaginary part in $\left(3.19\right)$ we get

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Since for all $n\in\mathbb{Z}^{+}$, $w_{n}\left(.\right)\in X$, $\tilde{F}_{n}\left(.,t\right)\in X$ and for a.e. $t\in\left[0,1\right]$ by integrating both sides of $\left(3.20\right)$ on $R^{n}$ we get

[TABLE]

It is clear to see that

[TABLE]

Then applying the Cauchy-Schwartz and Holder inequalites for a.e. $t\in\left[0,1\right]$ we obtain

[TABLE]

[TABLE]

[TABLE]

Moreover, again applying the Cauchy-Schwartz and Holder inequalites due to symmetricity of the operator $A,$ for a.e. $t\in\left[0,1\right]$ we get

[TABLE]

where, the constant $C$ in $\left(3.21\right)-\left(3.24\right)$ is intependent of $\upsilon\in C_{0}^{\infty}\left(R^{n};H\right),$ $\varepsilon\in\left(0,\left.1\right]\right.$ and $n\in\mathbb{Z}^{+}.$

Since $C_{0}^{\infty}\left(R^{n};H\right)$ is dense in $X,$ from $\left(3.24\right)$-$\left(3.27\right)$ in view of operator theory in Hilbert spaces, we obtain the following

[TABLE]

[TABLE]

[TABLE]

[TABLE]

For bounding the last two terms in $\left(3.20\right)$ we will use the abstract version of Calderón’s first commutator estimates $\left[4\right].$ Really by Cauchy-Schvartz inequality and in view of Theorem A1we get

[TABLE]

[TABLE]

Also, from the calculus of pseudodifferential operators with operator coefficients (see e.g. $\left[5\right]$ ) and the inequality $(3.15)$, we have

[TABLE]

[TABLE]

where the constant $C$ in $\left(3.26\right)-(3.29)$ is independent of $\varepsilon\in[0,1]$ and $n.$

We remark that estimates $(3.26)-(3.29)$ also hold with $b\left(x_{1}\right)$ replacing $a(x_{1})$. Since $C_{0}^{\infty}\left(R^{n};H\right)$ is dense in $X,$ from $\left(3.26\right)$-$\left(3.29\right)$ in view of operator theory in Hilbert spaces, we obtain

[TABLE]

[TABLE]

and the same estimates $\left(3.30\right)$ with $b(x_{1}$) replacing $a(x_{1})$.

By reasoning as in $\left[\text{4, Lemma 2.1}\right]$ (claim 1 and 2 ) from $\left(3.30\right)$ we obtain

[TABLE]

[TABLE]

Now, the estimates $\left(3.25\right)$ and $\left(3.31\right)$ implay the assertion.

4. Proof of Theorem 1.

We will apply Lemma 3.1 to a solution of the equation $(3.2)$. Since $0<\alpha<\beta=$ $\beta(k)$ for $k>k_{0}$ it follows that $\alpha\leq\sigma\left(t\right)\leq\beta$ for any $t\in[0,1]$. Therefore if $y=\sqrt{\alpha\beta}x\sigma\left(t\right)$, then from $\left(3.8\right)$ we get

[TABLE]

Thus,

[TABLE]

and so,

[TABLE]

Also, for $s=\beta t\sigma\left(t\right)$ it is clear that

[TABLE]

Therefore,

[TABLE]

and from $\left(4.1\right)$ we get

[TABLE]

where

[TABLE]

So, if

[TABLE]

then,

[TABLE]

and we can applay Lemma 3.1 to the equation $\left(3.2\right)$ with

[TABLE]

to get the following estimate

[TABLE]

[TABLE]

where $M$ a positive constant defined in Remark 2.1 and

[TABLE]

From $\left(4.6\right)$ we have

[TABLE]

[TABLE]

and multiply the above inequality by $e^{\left|\lambda\right|/q}\left|\lambda\right|^{n\left(q-2\right)/2}$, integrate in $\lambda$ and in $x$, use Fubini theorem and the following formula

[TABLE]

proven in $\left[\text{7, Appendx}\right]$ to obtain

[TABLE]

[TABLE]

Hence, the esimates $\left(3.6\right)$, $\left(3.8\right)$, $\left(3.9\right),$ $\left(4.3\right)$ and $\left(4.8\right)$ imply

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Next, we shall obtain bounds for the $\nabla\tilde{u}$. Let $\tilde{\gamma}=\frac{\gamma}{2}$ and $\varphi$ be a strictly convex complex valued function on compact sets of $R^{n}$, radial such that (see $[7]$)

[TABLE]

[TABLE]

[TABLE]

Let us consider the equation

[TABLE]

where $F\left(x,t\right)=\tilde{V}\upsilon,$ $A$ is a symmetric operator in $H$ and $\tilde{V}$ is a operator in $H$ defined by $\left(3.3\right).$

Let

[TABLE]

where $\upsilon$ is a solution of $\left(4.10\right)$. Then, by reasoning as in $\left[\text{17, Lemma 3.3}\right]$ we have

[TABLE]

where $S$, $K$ are symmetric and skew-symmetric operator, respectively given by

[TABLE]

Let

[TABLE]

A calculation shows that,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By $\left[\text{17, Lemma 3.2}\right]$

[TABLE]

[TABLE]

so,

[TABLE]

Multiplying $(4.15)$ by $t(1-t)$ and integrating in $t$ we obtain

[TABLE]

This computation can be justified by parabolic regularization using the fact that we already know the decay estimate for the solution of $\left(4.10\right)$. Hence, combining $(3.8)$, $(4.3)$ and $(4.9$) it follows that

[TABLE]

[TABLE]

[TABLE]

It is clear to see that

[TABLE]

So, by using the properties of $\varphi$ we get

[TABLE]

From here, we can conclude that

[TABLE]

[TABLE]

for $k\geq k_{0}(M_{0})$ sufficiently large, where

[TABLE]

For proving Theorem 1 first, we deduce the following estimate

[TABLE]

for $R$ sufficiently large, $\nu_{1}$, $\nu_{2}\in\left(0,1\right)$, $\nu_{1}<\nu_{2}$ and $\nu=\left(\nu_{1},\nu_{2}\right)$. From $\left(3.1\right)$ by using the change of variables $s=\beta t\sigma\left(t\right)$ and $y=\sqrt{\alpha\beta}x\sigma\left(t\right)$ we get

[TABLE]

[TABLE]

for $k>M_{0},$ $s\nu_{1}>\frac{1}{2}$ and $R_{0}=R\left(ka_{0}^{-1}\right)^{\frac{1}{2p}}.$ Thus, taking

[TABLE]

with $\omega=\omega(u)$ a constant to be determined, it follows that

[TABLE]

where the interval $I=I_{k}=[s\nu_{1},s\nu_{2}]$ satisfies $I\subset[1/2,1]$ for $k$ sufficiently large. Moreover, given $\varepsilon>0$ there exists $k_{0}(\varepsilon)>0$ such that for any $k\geq k_{0}$ one has that $I_{k}\subset[1-\varepsilon,1]$. By hypothesis on $u(x,t)$, i.e. the continuity of $\left\|u(\text{\textperiodcentered},s)\right\|_{X}$ at $s=1$, it follows that there exists $\omega>1$ and $K_{0}=K_{0}(u)$ such that for any $k\geq K_{0}$ and for any $s\in I_{k}$

[TABLE]

which yields the desired result. Next, we deduce the following estimate

[TABLE]

for $R$ sufficiently large, $\nu_{1},$ $\nu_{2}\in\left(0,1\right)$, $\mu_{1}=\frac{\left(\nu_{2}-\nu_{1}\right)}{8}$, $\mu_{2}=1-\mu_{1},$ $\mu_{1}<\mu_{2}$ and $\mu=\left(\mu_{1},\mu_{2}\right)$.

Indeed, from $\left(3.9\right)$ and $\left(4.17\right)$ we obtain

[TABLE]

[TABLE]

[TABLE]

Hence, from $\left(4.22\right)$ we get $\left(4.21\right)$ for $k\geq k_{0}(C_{0})$ sufficiently large.

Let $Y=L^{2}\left(R^{n}\times\left[0,1\right];H\right)$. By reasoning as in $\left[6\text{, Lemma 3.1}\right]$ we obtain

Lemma 4.1. Let $A$ be a pozitive operator in the Hilbert space $H$ and $iA$ generates a semigrop $U\left(t\right)=e^{iAt}$. Assume that $R>0$ and $\varphi$ : $[0,1]\rightarrow\mathbb{R}$ is a smooth function. Then, there exists $C=C(n,\varphi,H,A)>0$ such that, the inequality

[TABLE]

holds, for $\varkappa\geq CR^{2}$ and $g\in C_{0}^{\infty}\left(R^{n+1};H\right)$ with support contained in the set

[TABLE]

**Proof. **Let $f=e^{\alpha\left|\psi\left(x,t\right)\right|^{2}}g.$ Then, by acts of Schredinger operator $\left(i\partial_{t}+\Delta+A\right)$ to $f\in X$ we get

[TABLE]

where

[TABLE]

[TABLE]

Hence,

[TABLE]

and

[TABLE]

[TABLE]

A calculation shows that

[TABLE]

and

[TABLE]

[TABLE]

[TABLE]

Hence, using the hypothesis on the support on $g$ and the Cauchy–Schwarz inequality, the absolute value of the last two terms in $(4.23)$ can be bounded by a fraction of the first two terms on the right-hand side of $(4.23)$, when

$\alpha>CR^{2}$ for some large $C$ depending on $\left\|\varphi^{\prime}\right\|_{\infty}+\left\|\varphi^{\prime\prime}\right\|_{\infty}.$ This yields the assertion.

Now, from $\left(3.3\right)$ we have

[TABLE]

Then from $\left(4.20\right)$ we get

[TABLE]

Define

[TABLE]

Let $\nu_{1}$, $\nu_{2}\in\left(0,1\right)$, $\nu_{1}<\nu_{2}$ and $\nu_{2}<2\nu_{1}$. We choose $\varphi\in C^{\infty}\left(\left[0,1\right]\right)$ and $\theta$, $\theta_{R}\in C_{0}^{\infty}\left(R^{n}\right)$ satisfying

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Let

[TABLE]

where $\tilde{u}\left(x,t\right)$ is a solution of $\left(3.2\right)$ when $\tilde{V}=\tilde{F}\equiv 0$. It is clear to see that

[TABLE]

Hence,

[TABLE]

Moreover, from $\left(4.26\right)$ also we get that

[TABLE]

so

[TABLE]

Then, for $\xi=\psi\left(x,t\right)$ we have

[TABLE]

[TABLE]

Note that,

[TABLE]

and

[TABLE]

Now applying Lemma 4.1 choosing $\varkappa=d_{n}^{2}R^{2},$ $d_{n}^{2}\geq\left\|\varphi^{\prime}\right\|_{\infty}+\left\|\varphi^{\prime\prime}\right\|_{\infty}$ it follows that

[TABLE]

[TABLE]

[TABLE]

Since

[TABLE]

$D_{1}$ can be absorbed in the left hand side of $\left(4.27\right).$ Moreover, $\left|\psi\left(x,t\right)\right|\leq 4$ on the support of $B_{1},$ thus

[TABLE]

Let

[TABLE]

Then $R_{\mu}^{n}\subset$supp $B_{2}$, and $1\leq\left|\psi\left(x,t\right)\right|\leq 2$, so

[TABLE]

By using $\left(4.18\right)$ and $\left(4.22\right)$ we have

[TABLE]

[TABLE]

Puting $\varkappa=d_{n}R^{2}=2a_{1}k^{\frac{1}{2-p}}$ it follows from $\left(4.28\right)$ that, if $\left\|u\left(.,0\right)\right\|_{X}\neq 0$ then

[TABLE]

for $k\geq k_{0}(C_{\mu})$ sufficiently large.

Now, by $\left(4.22\right)$ we get

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The estimates $\left(4.28\right)-\left(4.30\right)$ imply

[TABLE]

where

[TABLE]

Hence, if $42a_{1}<\sqrt{a_{1}^{p}a_{0}}\left(\frac{2}{d_{n}}\right)^{\frac{p}{2}}$ by letting $k$ tends to infinity it follows from $\left(4.31\right)$ that $\left\|u\left(.,0\right)\right\|_{X}=0$, which gives $u\left(x,t\right)\equiv 0$.

**Proof of Corollary 1. **Since

[TABLE]

one has that

[TABLE]

Then, by reasoning as in $\left[\text{7, Corollary 1}\right]$ we obtain the assertion.

**Proof of Theorem 2. **Indeed, just applying Corollary 1 with

[TABLE]

and

[TABLE]

we obtain the assertion of Theorem 2.

**5. Proof of Theorem 3. **

First, we deduce the corresponding upper bounds. Assume

[TABLE]

Fix $\bar{t}$ near $1,$ and let

[TABLE]

which satisfies the equation $\left(2.12\right)$ with

[TABLE]

where $A$ is a linear operator, $V\left(x,t\right)$ is a given potential operator function in a Hilbert space $H.$

From $\left(5.1\right)$ we get

[TABLE]

where

[TABLE]

For $V\left(x,t\right)=F\left(u,\bar{u}\right)$, by hypothesis

[TABLE]

By using Appell transformation if we suppose that $\upsilon\left(y,s\right)$ is a solution of

[TABLE]

$\alpha$ and $\beta$ are positive, then

[TABLE]

verifies the equation

[TABLE]

with $\tilde{V}\left(x,t\right),$ $\tilde{F}\left(x,t\right)$ defined by $\left(3.3\right)$, $\left(3.4\right)$ and

[TABLE]

[TABLE]

It follows from expressions $\left(3.6\right)$ and $\left(3.8\right)$ that

[TABLE]

Next, we shall estimate

[TABLE]

where

[TABLE]

Thus,

[TABLE]

with

[TABLE]

Hence,

[TABLE]

and

[TABLE]

[TABLE]

where

[TABLE]

To apply Lemma 3.1 we need

[TABLE]

for some $R,$ i.e.,

[TABLE]

where

[TABLE]

Let

[TABLE]

By using $\left(5.5\right)-\left(5.10\right),$ by virtue of Lemma 3.1 and $\left(4.7\right)$ we deduced

[TABLE]

[TABLE]

where

[TABLE]

Next, using the same argument given in section $4$, $(4.10)-(4.22)$, one finds that

[TABLE]

Now we turn to the lower bounds estimates. Since they are similar to those given in detail in section 3, we obtain that the estimate $(4.24)$ for potential operator function $\tilde{V}\left(x,t\right)$ when

[TABLE]

Finally, we get

[TABLE]

for $p<\frac{p}{2}+\frac{p^{2}}{2\left(2-p\right)}$, i.e. $p>1$ that assumed in conditon of Theorem 4, i.e. we obtain the assertion of Theorem 4.

**Remark 5.1. **Let us consider the case $\theta=4/n$ in Theorem $4$, i.e. $p>4/3$. Then from Theorem 4 we obtain the following result

Result 5.1. Assume the conditions of Theorem 4 are satisfied for $p>4/3$. Then $u\left(x,t\right)\equiv 0.$

**6. Unique continuation properties for the system of Schrödinger equation **

Consider the Cauchy problem for the finite or infinite system of Schrödinger equation

[TABLE]

where $u=\left(u_{1},u_{2},...,u_{N}\right),$ $u_{j}=u_{j}\left(x,t\right),$ $a_{mj}$ are complex numbers and $b_{mj}=b_{mj}\left(x,t\right)$ are complex valued functions. Let $l_{2}=l_{2}\left(N\right)$ and $l_{2}^{s}=l_{2}^{s}\left(N\right)$ (see $\left[\text{23, \S\ 1.18}\right]$). Let $A$ be the operator in $l_{2}\left(N\right)$ defined by

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

Let

[TABLE]

From Theorem 1 we obtain the following result

**Theorem 6.1. **Assume there exist the constants $a_{0},$ $a_{1},$ $a_{2}>0$ such that for any $k\in\mathbb{Z}^{+}$ a solution $u\in C\left(\left[0,1\right];X_{2}\right)$ of $\left(6.1\right)$ satisfy

[TABLE]

[TABLE]

Moreover, there exists $M_{p}>0$ such that

[TABLE]

Then $u\left(x,t\right)\equiv 0.$

Proof. It is easy to see that $A$ is a symmetric operator in $l_{2}$ and other conditions of Theorem 1 are satisfied. Hence, from Teorem 1 we obtain the conculision.

**7. Unique continuation properties for nonlinear anisotropic Schrödinger equation **

The regularity property of BVP for elliptic equations were studied e.g. in $\left[\text{1, 2}\right]$. Let $\Omega=R^{n}\times G$, $G\subset R^{d},$ $d\geq 2$ is a bounded domain with $\left(d-1\right)$-dimensional boundary $\partial G$. Let us consider the following problem

[TABLE]

[TABLE]

[TABLE]

where $a_{\alpha},$ $b_{j\beta}$ are the complex valued functions, $\alpha=\left(\alpha_{1},\alpha_{2},...,\alpha_{n}\right)$, $\beta=\left(\beta_{1},\beta_{2},...,\beta_{n}\right),$ $\mu_{i}<2m,$ $K=K\left(x,y,t\right)$ and

[TABLE]

Let

[TABLE]

[TABLE]

[TABLE]

Theorem 7.1. Let the following conditions be satisfied:

(1) $G\in C^{2}$, $a_{\alpha}\in C\left(\bar{G}\right)$ for each $\left|\alpha\right|=2m$ and $a_{\alpha}\in L_{\infty}\left(G\right)$ for each $\left|\alpha\right|<2m$;

(2) $b_{j\beta}\in C^{2m-m_{j}}\left(\partial G\right)$ for each $j$, $\beta$ and $\ m_{j}<2m$, $\sum\limits_{j=1}^{m}b_{j\beta}\left(y^{\prime}\right)\sigma_{j}\neq 0,$ for $\left|\beta\right|=m_{j},$ $y^{{}^{\shortmid}}\in\partial G,$ where $\sigma=\left(\sigma_{1},\sigma_{2},...,\sigma_{n}\right)\in R^{n}$ is a normal to $\partial G$ ;

(3) for $y\in\bar{G}$, $\xi\in R^{n}$, $\lambda\in S\left(\varphi_{0}\right)$ for $0\leq\varphi_{0}<\pi$, $\left|\xi\right|+\left|\lambda\right|\neq 0$ let $\lambda+$ $\sum\limits_{\left|\alpha\right|=2m}a_{\alpha}\left(y\right)\xi^{\alpha}\neq 0$;

(4) for each $y_{0}\in\partial G$ local BVP in local coordinates corresponding to $y_{0}$:

[TABLE]

[TABLE]

has a unique solution $\vartheta\in C_{0}\left(\mathbb{R}_{+}\right)$ for all $h=\left(h_{1},h_{2},...,h_{n}\right)\in\mathbb{C}^{n}$ and for $\xi^{\prime}\in R^{n-1};$

(5) there exist positive constants $b_{0}$ and $\theta$ such that a solution $u\in C\left(\left[-1,1\right];X_{2}\right)$ of $\left(7.1\right)-\left(7.2\right)$ satisfied

[TABLE]

(6) Suppose

[TABLE]

for $t\in\left[-1,1\right]$ and that $\left(2.15\right)$ holds with $Q\left(.\right)$ satisfies $\left(2.16\right)$ for $H=L^{2}\left(G\right).$

If $p>p\left(\theta\right)=\frac{2\left(\theta n-2\right)}{\left(\theta n-1\right)},$ then $a\equiv 0.$

**Proof. **Let us consider operators $A$ and $V\left(x,t\right)$ in $H=L^{2}\left(G\right)$ that are defined by the equalities

[TABLE]

Then the problem $\left(7.1\right)-\left(7.2\right)$ can be rewritten as the problem $\left(2.12\right)$, where $u\left(x\right)=u\left(x,.\right),$ $f\left(x\right)=f\left(x,.\right)$, $x\in R^{n}$ are the functions with values in $H=L^{2}\left(G\right)$. By virtue of $\left[\text{1}\right]$ operator $A+\mu$ is positive in $L^{2}\left(G\right)$ for sufficiently large $\mu>0$. Moreover, in view of (1)-(6) all conditons of Theorem 3 are hold. Then Theorem 3 implies the assertion.

8. The Wentzell-Robin type mixed problem for Schrödinger equations

Consider the problem $\left(1.5\right)-\left(1.6\right)$.  Let

[TABLE]

Suppose $\nu=\left(\nu_{1},\nu_{2},...,\nu_{n}\right)$ are nonnegative real numbers. In this section, from Theorem 1 we obtain the following result:

**Theorem 8.1. ** Suppose the the following conditions are satisfied:

(1) $a$ is positive, $b$ is a real-valued functions on $\left(0,1\right)$. Moreover, $a\left(.\right)\in C\left(0,1\right)$ and

[TABLE]

(2) $u_{1},$ $u_{2}\in C\left(\left[0,1\right];Y^{2,k}\right)$ are strong solutions of $(1.2)$ with $k>\frac{n}{2};$

(3) $F:\mathbb{C}\times\mathbb{C}\rightarrow\mathbb{C},$ $F\in C^{k}$, $F\left(0\right)=\partial_{u}F\left(0\right)=\partial_{\bar{u}}F\left(0\right)=0;$

(4) there exist positive constants $\alpha$ and $\beta$ such that

[TABLE]

with

[TABLE]

(5) there exists $N_{p}>0$ such that

[TABLE]

Then $u_{1}\left(x,t\right)\equiv u_{2}\left(x,t\right).$

Proof. Let $H=L^{2}\left(0,1\right)$ and $A$ is a operator defined by $\left(1.4\right).$ Then the problem $\left(1.5\right)-\left(1.6\right)$ can be rewritten as the problem $\left(1.2\right)$. By virtue of $\left[\text{10, 11}\right]$ the operator $A$ generates analytic semigroup in $L^{2}\left(0,1\right)$. Hence, by virtue of (1)-(5) all conditons of Theorem 1.2 are satisfied. Then Theorem 1.2 implies the assertion.

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