Unique continuation properties for Schroedinger operators in Hilbert spaces

TL;DR

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

Contribution

It introduces a general framework for unique continuation in Hilbert space valued Schrödinger equations, applicable to diverse physical models.

Findings

Derived Morgan type uncertainty principles.

Established unique continuation properties for abstract Schrödinger equations.

Applicable to a wide range of physical systems.

Abstract

Here, the Morgan type uncertainty principle and unique continuation properties of abstract Schredinger equations with time dependent potentials are obtained in Hilbert space valued function classes. The equations include linear operator in abstract Hilbert spaces H dependent on space variables. So, by selecting appropriate spaces H and operators, we derive unique continuation properties for numerous classes of Schr\"odinger type equations and its systems, which occur in a wide variety of physical systems.

Equations

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

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

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

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

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

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

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

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

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

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

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

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

$$\Vert u\Vert ={\Vert u\Vert }_H={\left(u,u\right)}_H^{\frac{1}{2}}={\left(u,u\right)}^{\frac{1}{2}}\text{ for }u\in H.$$

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

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

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

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

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

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

$${\Vert u\Vert }_{H\left(A\right)}=\Vert Au\Vert +\Vert u\Vert .$$

$$\left(Au,u\right)\ge C_A{\Vert u\Vert }^2.$$

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

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

$$\sum _{k=1}^n{\left(x_k\left[A\frac{\partial f}{\partial x_k}-\frac{\partial A}{\partial x_k}f\right],f\right)}_X\ge 0,$$

$$\sum _{k=1}^n{\left(x_k\left[A\frac{\partial f}{\partial x_k}+\frac{\partial A}{\partial x_k}f\right],f\right)}_X\ge 0$$

$$\mathrm{Im}{\left(\left(A+V\right)\upsilon ,\upsilon \right)}_H\ge C_0\mu \left(x,t\right){\Vert \upsilon \left(x,t\right)\Vert }^2,$$

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

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

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

$$\underset{R^n}{\int }{\Vert u\left(x,0\right)\Vert }^2{e}^{2a_0{|x|}^p}dx<\infty ,\text{ for }p\in \left(1,2\right),$$

Full text

Unique continuation properties for Schredinger operators in Hilbert spaces

Veli Shakhmurov

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

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

Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Azerbaijan, AZ1141, Baku, F. Agaev, 9,

E-mail: veli.sahmurov@gmail.com

Abstract

Here, the Morgan type uncertainty principle and unique continuation properties of abstract Schredinger equations with time dependent potentials are obtained in Hilbert space valued function classes. The equations include linear operator in abstract Hilbert spaces $H$ dependent on space variables. So, by selecting appropriate spaces $H$ and operators, we derive unique continuation properties for numerous classes of Schrödinger type equations and its systems, which occur in a wide variety of physical systems.

**Key Word:    Schredinger 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 $A=A\left(x\right)$ is a linear operator, $V\left(x,t\right)$ is a given potential operator function in a Hilbert space $H$, the 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 $A$, 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 is then applied to show that two regular solutions $u_{1}$ and $u_{2}$ of non-linear Schredinger equation

[TABLE]

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 Schredinger equations have been studied by $\left[\text{5-8}\right],$ $\left[\text{21-23}\right]$ and the references therein. In contrast to the mentioned above results we will study the unique continuation properties of abstract Schredinger equations with operator potentials. Abstract differential equations were studied e.g. in $\left[1\right]$, $\left[10\right],$ $\left[13-19\right]$, $\left[25\text{, }26\right].$ Since the Hilbert space $H$ is arbitrary and $A$, $V$ are possible linear operators, by choosing $H$, $A$ and $V$ we can obtain numerous classes of Schrödinger type equations and its systems which occur in different applications. If we choose the abstract space $H$ a concrete Hilbert space, for example $H=L^{2}\left(G\right)$, $A=L,$ $V=q\left(x,t\right)$, where $G$ is a domain in $R^{m}$ with smooth boundary, $L$ is a elliptic operator with respect to the variable $y\in G$ and $q$ is a complex valued function, then we obtain the unique continuation properties of the following Schrödinger equation

[TABLE]

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

[TABLE]

[TABLE]

where $a$ and $b$ are real valued functions on $\ R^{n}\times\left[0,1\right]$. Then, we obtain the unique continuation properties of the Wentzell-Robin type boundary value problem (BVP) for the nonlinear Schredinger type equation

[TABLE]

[TABLE]

[TABLE]

Note that, the regularity properties of Wentzell-Robin type BVP for elliptic equations have been 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 infinite matrices $\left[a_{mj}\right]$, $\left[b_{mj}\left(x,t\right)\right],$ $m,$ $j=1,2,...,\infty$ respectively, then we obtain the unique continuation properties of the following system of Schredinger 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 $G$ equipped with norm

[TABLE]

Let $H$ be a Hilbert space and

[TABLE]

For $p=2$ and $E=H$ is a Hilbert space, $L^{p}\left(\Omega;E\right)$ is to be a Hilbert space with inner product

[TABLE]

for $f$, $g\in L^{2}\left(\Omega;H\right)$.

Let $C\left(\Omega;E\right)$ denote the space of $E-$valued, bounded uniformly continuous 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]

Moreover, $C_{0}^{\infty}\left(\Omega;E\right)-$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 let $m$ 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 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 suppose $E_{0}$ is continuously and densely embedded into $E$. Here, $W^{m,p}\left(\Omega;E_{0},E\right)$ denotes 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. Let $L\left(E_{1},E_{2}\right)$ 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 $L\left(E\right).$

Let $A$ is a symmetric operator in a Hilbert sapace $H$ with domain $D\left(A\right).$ Here, $H\left(A\right)$ denotes the domain of $A$ equipped with graphical norm, i.e.

[TABLE]

The symmetric operator $A$ is positive defined if $D\left(A\right)$ is dense in $H$ and there exists a positive constant $C_{A}$ depend only on $A$ such that

[TABLE]

It is known that see e.g. $\left[\text{24, \S 1.15.1}\right]$) there exist fractional powers $A^{\theta}$ of the positive defined operator $A.$

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 $C_{\alpha}$.

2.1. Main results for absract Scrödinger equation

Consider the problem $\left(1.1\right)$.  Here,

[TABLE]

Definition 2.1. A function $u\in L^{\infty}\left(0,T;H\left(A\right)\right)$ is called a local weak solution to $\left(1.1\right)$ on $\left(0,T\right)$ if $u$ belongs to $L^{\infty}\left(0,T;H\left(A\right)\right)\cap W^{1,2}\left(0,T;H\right)$ and satisfies $\left(1.1\right)$ in the sense of $L^{\infty}\left(0,T;H\left(A\right)\right).$ In particular, if $\left(0,T\right)$ coincides with $\mathbb{R}$, then $u$ is called a global weak solution to $\left(1.1\right).$

If the solution of $\left(1.1\right)$ belongs to $C\left(\left[0,T\right];H\left(A\right)\right)\cap W^{2,2}\left(0,T;H\right),$ then its called a stronge solution.

Condition 1. Assume: (1) $A=A\left(x\right)$ is a symmetric operator in Hilbert space $H$ with independent on $x$ domain $D\left(A\right)$ that is dense on $H;$

(2) $\frac{\partial A}{\partial x_{k}}$ are symmetric operators in Hilbert space $H$ with independent on $x$ domain $D\left(\frac{\partial A}{\partial x_{k}}\right)=D\left(A\right)$ and

[TABLE]

for each $f\in L^{\infty}\left(0,T;Y^{1}\left(A\right)\right)$;

(3)

[TABLE]

for each $f\in L^{\infty}\left(0,T;Y^{1}\left(A\right)\right)$;

(4) $iA$ generates a Schrödinger grop and $V\left(x,t\right)\in L\left(H\right)$ for $\left(x,t\right)\in R^{n}\times\left[0,1\right];$

(5) there is a constant $C_{0}>0$ so that

[TABLE]

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

(6)

[TABLE]

where

[TABLE]

Let $A=A\left(x\right)$ is a symmetric operator in Hilbert space $H$ with independent on $x$ domain $D\left(A\right)$ and

[TABLE]

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 $u\in C\left(\left[0,1\right];X\left(A\right)\right)$ 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 Condition1 holds and

[TABLE]

There exist positive constants $\alpha$, $\beta$ such that a solution $u\in C\left(\left[0,1\right];X\left(A\right)\right)$ 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 Corollary1 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 $u_{1},$ $u_{2}\in C\left(\left[0,1\right];Y^{2,k}\left(A\right)\right)$ 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 hold and there exist positive constants $\alpha$ and $\beta$ such that a solution $u\in C\left(\left[0,1\right];X\left(A\right)\right)$ 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 2 still holds, with different constant $N_{p}>0$, if one replaces the hypothesis $(2.6)$ by

[TABLE]

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

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

[TABLE]

where $A$ is a linear operator in a Hilbert space $H.$

Let $u\left(x,t\right)$ be a solution of the equation $\left(2.10\right).$ Then it can be shown that the function

[TABLE]

is a solution of the focussing $L^{2}$-critical solution of abstract Schredinger 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]

By using the above result we wil prove the following main result

**Theorem 3. **Assume the Condition 1 holds and there exist positive constants $b_{0}$ and $\theta$ such that a solution $u\in C\left(\left[-1,1\right];X\left(A\right)\right)$ of $\left(2.10\right)$ satisfied $\left\|F\left(u,\bar{u}\right)\right\|\leq b_{0}\left\|u\right\|^{\theta}$ for $\left\|u\right\|>1$. Suppose

[TABLE]

and

[TABLE]

where,

[TABLE]

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

2.2. Some auxiliary results

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

**Lemma 2.0. 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 Schredinger equation will be as:

Let

[TABLE]

and

[TABLE]

for some $t\neq 0$. Then $u_{0}\left(x\right)\equiv 0.$

Let $L^{2}\left(H\right)=L^{2}\left(\left(0,1\right)\times R^{n};H\right).$

Consider the abstract Schredinger equation

[TABLE]

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

Let $S=S\left(t\right)$ be a symmetric, $K=K\left(t\right)$ be a skew-symmetric operators in $H$,  $f=f\left(.,t\right)\in X$ for $t\in\left[0,1\right]$ and

[TABLE]

[TABLE]

Let

[TABLE]

**Lemma 2.1.  **Let the Condition 1 holds. Assume that $u\in L^{\infty}\left(\left[0,1\right];X\left(A\right)\right)\cap L^{2}\left(0,1;Y^{1}\right)$ is a solution of $\left(2.16\right)$ for $a\in\mathbb{R}_{+}$ and $b\in\mathbb{R}.$ Then,

[TABLE]

[TABLE]

uniformly in $t\in\left[0,1\right],$ when $\gamma\geq 0,$ $0\leq T\leq 1$ and

[TABLE]

**Proof. **Set $\upsilon$ $=e^{\varphi}u$, where $\varphi$ is a real-valued function to be chosen later. The function $\upsilon=\upsilon\left(x,t\right)$ verifies

[TABLE]

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

[TABLE]

[TABLE]

here

[TABLE]

Formally,

[TABLE]

for $t\geq 0.$ Again a formal integration by parts gives that

[TABLE]

[TABLE]

By Cauchy-Schwarz’s inequality and in view of assumption on $A,$ $V$ we get

[TABLE]

[TABLE]

when

[TABLE]

Let

[TABLE]

where $\theta_{\rho}$ is a radial mollifier and

[TABLE]

Then by reasoning as in the end of proof $\left[\text{8, Lemma 1}\right],$ from $\left(2.19\right)$ we obtain that the estimate holds

[TABLE]

[TABLE]

holds uniformly in $\rho$, $r$ and $t\in\left[0,1\right].$ Lemma 2.1 follows after letting $\rho$ tend to zero and $r$ to infinity.

**Lemma 2.2.  Let $S=S\left(x,t\right)$ be a symmetric operator, $K=K\left(x,t\right)$ be skew-symmetric in Hilbert space $H$ with independent on $x$ domain $D\left(S\right)$ and $D\left(K\right),$ respectively. **Assume $G\left(.,t\right)\in L^{2}\left(R^{n}\right)$ is a positive funtion and $f(x,t)$ is a $H-$valued reasonable function. Then,

[TABLE]

[TABLE]

and

[TABLE]

Moreover, if

[TABLE]

for $x\in R^{N}$, $t\in\left[0,1\right]$ and

[TABLE]

Then $Q\left(t\right)$ is logarithmically convex in $[0,1]$ and there is a constant $M$ such that

[TABLE]

Proof. It is clear that

[TABLE]

Also,

[TABLE]

[TABLE]

Then by reasoning as in $\left[\text{8, Lemma 2}\right]$ we obtain the assertion.

Consider the following abstract Schredinger equation

[TABLE]

Let

[TABLE]

[TABLE]

**Lemma 2.3.  **Let the Condition 1 holds. Assume that $a$, $\gamma>0,$ $b\in\mathbb{R}.$ Let $u\in L^{\infty}\left(0,1;X\left(A\right)\right)\cap L^{2}\left(0,1;Y^{1}\right)$ be solution of the equation $\left(2.22\right)$ and

[TABLE]

for $x\in R^{n},$ $t\in\left[0,1\right]$ and $u\in D\left(A\right)$, where $\eta$ is a positive function in $L^{1}\left(0,T;L^{\infty}\left(R^{n}\right)\right)$. Moreover, suppose

[TABLE]

and

[TABLE]

Then, $e^{\gamma\left|x\right|^{2}}u\left(.,t\right)$ is logarithmically convex in $[0,1]$ and there is a constant $N$ such that

[TABLE]

where

[TABLE]

when $0\leq t\leq 1.$

**Proof. **Let $f=e^{\gamma\varphi}u,$ where $\varphi$ is a real-valued function to be chosen. The function $f\left(x,t\right)$ verifies

[TABLE]

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

[TABLE]

where

[TABLE]

A calculation shows that

[TABLE]

[TABLE]

If we put $\varphi=\left|x\right|^{2}$, then $\left(2.25\right)$ reduce the following

[TABLE]

Moreover by assumtion (2) of Condition 1,

[TABLE]

[TABLE]

This identity, the condition on $A=A\left(x\right),$ $V=V\left(x,t\right)$ and $\left(2.24\right)$ imply that

[TABLE]

[TABLE]

If we knew that the quantities and calculations involved in the proof of Lemma 2.2. ( this fact are derived ın a similar way as in $\left[\text{8, Lemma 2}\right]$) were finite and correct, when $f=e^{\gamma\left|x\right|^{2}}u$, we would have the logarithmic convexity of $Q(t)=\left\|e^{\gamma\left|x\right|^{2}}u\left(.,t\right)\right\|_{X}$ and get $(2.23)$ from Lemma 2.2. To justify the validity of the previous arguments now we consider the function $\varphi_{a}\left(x\right)$ constructed in $\left[\text{8, Lemma 3}\right]$, i.e.,

[TABLE]

and replace $\varphi=\left|x\right|^{2}$ by $\varphi_{a,\rho}=\theta_{\rho}\ast\varphi_{a},$ where $a,$ $\rho\in\left(0,1\right)$ and $\theta\in C_{0}^{\infty}\left(R^{n}\right)$ is a radial function. From the proof $\left[\text{8, Lemma 3}\right]$ we get that $\varphi_{a,\rho}$ is convex function and

[TABLE]

Put then, $f_{a,\rho}=e^{\gamma\varphi_{a,\rho}}u$ for $u\in X$ and $Q_{a,\rho}\left(t\right)=$ $\left\|f_{a,\rho}\left(.,t\right)\right\|_{X}^{2}$ in Lemma 2.2. The decay bound in Lemma 2.1 and the interior regularity for solutions of $(2.23)$ can now be used qualitatively to make sure that the quantities or calculations involved in the proof of $\left[\text{8, Lemma 2}\right]$ ( with replicing $L^{2}$ by $X)$ are finite and correct for $f_{a,\rho}$. In this case, $f_{a,\rho}$ verifies

[TABLE]

with symmetric and skew-symmetric operators $S^{a,\rho}$ and $K^{a,\rho}$ given by $(2.24)$ with $\varphi$ replaced by $\varphi_{a,\rho}$. The formula $\left(2\text{.}25\right)$, the convexity of $\varphi_{a,\rho}$, the bounds $(2.26)$ and $(2.27)$ imply that the inequalities

[TABLE]

[TABLE]

hold and $M_{2}\left(a,\rho\right)\leq e^{C\left(n\right)\rho^{2}}M_{2},$ when $a,\rho\in\left(0,1\right).$ Particularly, $Q_{a,\rho}$ is logarithmically convex in $[0,1]$ and

[TABLE]

Then, $(2.23)$ follows after taking first the limit, when a tends to zero in $(2.28)$ and then, when $\rho$ tends to zero.

3. Some properties of solutions of the abstract Schredinger equations

Let

[TABLE]

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

[TABLE]

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

[TABLE]

and commutator operator

[TABLE]

[TABLE]

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

Theorem A${}_{2}.$ Assume $k(.)$ is $L\left(H\right)-$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 $f\in C_{0}^{\infty}\left(R^{n};H\right),$ where the constant $C>0$ is independent of $f.$

Let $X_{\gamma}$ denote the weithed Lebesque space $L_{\gamma}^{2}\left(R^{n}\text{:}H\right)$ with $\gamma\left(x\right)=e^{2\lambda.x}.$ By following $\left[\text{5, Lemma 2.1}\right]$ let us show:

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

[TABLE]

Moreover, suppose $u\in C\left(\left[0,1\right];X\left(A\right)\right)$ 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]

Then by reasoning as in $\left[\text{5, Lemma 2.1}\right]$ and by using the properties of symmetric operators $A$ and $V$, we obtain

[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 $\left(3.16\right)$ 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 inequalities due to symmetricity of the operator $A,$ for a.e. $t\in\left[0,1\right]$ we get

[TABLE]

where, the constant $C$ in $\left(3.18\right)-\left(3.20\right)$ is independent 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 $\left(3.17\right)$-$\left(3.20\right)$ in view of operator theory in Hilbert spaces, we obtain the following

[TABLE]

[TABLE]

[TABLE]

[TABLE]

For bounding the last two terms in $\left(3.16\right)$ 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 A2 we 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 $\left(3.22\right)-(3.25)$ is independent of $\varepsilon\in[0,1]$ and $n.$ We remark that estimates $(3.22)-(3.25)$ 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 $\left(3.22\right)$-$\left(3.25\right)$ in view of operator theory in Hilbert spaces, we obtain

[TABLE]

[TABLE]

and the same estimates $\left(3.26\right)$ with $b(x_{1}$) replacing $a(x_{1})$. By reasoning as in $\left[\text{6, Lemma 2.1}\right]$ (claim 1 and 2 ) from $\left(3.26\right)$ we obtain

[TABLE]

[TABLE]

Now, the estimates $\left(3.21\right)$ and $\left(3.27\right)$ 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 $\mathop{\displaystyle\int}\limits_{0}^{1}\left\|V\left(.,s\right)\right\|_{L^{\infty}\left(\left|y\right|>\varkappa;H\right)}ds<\varepsilon_{0}$ 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 Lemma 2.3 we have

[TABLE]

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

[TABLE]

Let

[TABLE]

A calculation shows that,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By Lemma 2.2

[TABLE]

[TABLE]

so,

[TABLE]

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

[TABLE]

[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]

[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]

[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. Assume the assumptıions (1) and  (3) of Condition 1 are satisfied. Suppose 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 $g\in C_{0}^{\infty}\left(R^{n+1};H\left(A\right)\right)$ 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 $f\in X\left(A\right)$ we get

[TABLE]

where

[TABLE]

[TABLE]

Hence,

[TABLE]

and

[TABLE]

[TABLE]

A calculation shows that

[TABLE]

where

[TABLE]

[TABLE]

Since $A$ is a symmetric operator in $H,$ from $\left(4.23\right)$ we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In view of the hypothesis on $g$ and the Cauchy–Schwarz inequality, the absolute value of the third fourth terms in $(4.23)$ can be bounded by a fraction of the first two terms on the right-hand side of $(4.24)$, when $\alpha>Cr^{2}$ for some large $C$ depending on $\left\|\varphi^{\prime}\right\|_{\infty}+\left\|\varphi^{\prime\prime}\right\|_{\infty}.$ Moreover, by using the assumption on $A=A\left(x\right)$, we get that the last two terms are nonnegative. This yields the assertion. Now, from $\left(3.3\right)$ we have

[TABLE]

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

[TABLE]

Define

[TABLE]

[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 $\bar{t}\in\left(0,1\right)$ near $1,$ and let

[TABLE]

which satisfies the equation $\left(2.10\right)$ 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]

for a $r>0$, 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]

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, i.e. we obtain the assertion of Theorem 3.

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

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

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

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

[TABLE]

where $u=\left(u_{1},u_{2},...,u_{N}\right),$ $u_{j}=u_{j}\left(x,t\right),$ $a_{mj}=a_{mj}\left(x\right)$ are 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. **Suppose $a_{mj}$ are bounded cotinious on $R^{n}$ and $b_{mj}$  are bounded functions on $R^{n}\times\left[0,T\right].$ Assume there exist the constants $a_{0},$ $a_{1},$ $a_{2}>0$ such that for any $k\in\mathbb{Z}^{+}$ a solution $u\in C\left(\left[0,1\right];X_{2}\left(A\right)\right)$ 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 Schredinger equation **

The regularity property of BVPs 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, $\alpha=\left(\alpha_{1},\alpha_{2},...,\alpha_{d}\right)$, $\beta=\left(\beta_{1},\beta_{2},...,\beta_{d}\right),$ $\mu_{i}<2m,$ $K=K\left(x,y,t\right)$ and

[TABLE]

Let

[TABLE]

[TABLE]

[TABLE]

Let

[TABLE]

Theorem 7.1. Let the following conditions be satisfied:

(1) $G\in C^{2}$, $a_{\alpha}\in C\left(\bar{\Omega}\right)$ 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 $\sigma=\left(\sigma_{1},\sigma_{2},...,\sigma_{d}\right)\in R^{d}$ is a normal to $\partial G$ ;

(3) for $y\in\bar{G}$, $\xi\in R^{d}$, $\lambda$ with $\left|\arg\lambda\right|\leq\varphi$ for $0\leq\varphi<\pi$, $\left|\xi\right|+\left|\lambda\right|\neq 0$ 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 $h=\left(h_{1},h_{2},...,h_{d}\right)\in\mathbb{C}^{d}$ and for $\xi^{\prime}\in R^{d-1};$

(5) there exist positive constants $b_{0}$ and $\theta$ such that a solution $u\in C\left(\left[-1,1\right];X_{2}\left(A\right)\right)$ 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 Schredinger 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 following conditions are satisfied:

(1) $a$  and $b$ are a real-valued functions on $\left(0,1\right)$ and $a\left(t\right)>0$. Moreover, $a\left(.\right)$ is bounded continious function on $R^{n}\times\left[0,1\right]$ 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 2 are satisfied. Then Theorem 2 implies the assertion.

Acknowledgements

The author would like to express a gratitude to Dr. Neil. Course for his useful advice in English in preparing of this paper

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