Ambarzumyan Type Theorem For a Matrix Valued Quadratic Sturm-Liouville Problem

Emrah Yilmaz and Hikmet Koyunbakan

1 Introduction

It is well known that Ambarzumyan’s theorem[Ambarzumyan(1929)]is about the boundary value problem

with the real potential functionq∈L2[0,π].It was proved that ifλn=n2,n≥0 is the spectral set of(1),thenq(x)=0 on(0,π).As an historical viewpoint,this is known as the first result in inverse spectral theory associated with Sturm-Liouville operators.Ambarzumyan’s theorem was extended to the second order differential systems of two dimensions in[Chakravarty and Acharyya(1988)],to Sturm-Liouville differential systems of any dimension in[Chern and Shen(1997)],to the Sturm-Liouville equation(which is concerned only with Neumann boundary conditions)with general boundary conditions by imposing an additional condition on the potential function in[Chern,Law and Wang(2001)],and to the multi-dimensional Dirac operator in[C.F.Yang and X.P.Yang(2009)].In addition,some different results of Ambarzumyan’s theorem have been obtained by many authors[Carlson and Pvovarchik(2007);Horvath(2001);C.F.Yang and X.P.Yang(2011);Shen(2007);C.F.Yang,Huang and X.P.Yang(2010)].

Ambarzumyan’s theorem was extended to the following boundary value problem by imposing to a condition onp

with the homogeneous Neumann boundary conditions

Before giving the main results,we want to mention some physical properties of quadratic equation.The problem of describing the interactions between colliding particles is of fundamental interest in physics.It is interesting in collisions of two spinles particles,and it is supposed that thes?wave scattering matrix and thes?wave binding energies are exactly known from collision experiments.For a radial static potentialV(E,x)ands?wave,the Schr?dinger equation is written as

This paper is organized as follows;Section 2 is devoted to the some known results of matrix quadratic pencil.Section 3 is about some uniqueness theorems and proofs.

2 Matrix Differential Equations

For simplicity,Aijdenotes entry of a matrixAat thei?throw andj?thcolumn andIdis ad×didentity matrix and 0dis ad×dzero matrix.

We are interested in the eigenvalue problem

whereP(x)=diag[p1(x),p2(x),...,pd(x)]andQ(x)ared×drealsymmetric matrixvalued functions,and thosed×dmatricesA,B,CandDsatisfy the following conditions

In this study,we consider the special case of the problem(4),(5)asA=C=0dandB=D=Id.Namely,we introduce the following matrix differential equation

is singular.In case ofC=0dandD=Id,the eigenvalues of the problem(9)-(11)are zeros ofW(μ)=Y0(π,μ)=0d.

In order to describeW(μ)explicitly,we must know how to express the solutionY(x,μ).The solutionY(x,μ)of(9)-(10)can be expressed as[Yang(2012)]

whereA(x,t)andB(x,t)are symmetric matrix-valued functionswhose entries have continuous partial derivatives up to order two respect toxandt.Now,we will give following results that are crucial to obtain our main results.It is pointed out these lemmas were given by[Yang(2012)].

Lemma 2.1.[Yang(2012)]Let A and B be as in(12).Then,A and B satisfy following conditions

Also,it is well known that[Yang(2012)]the eigenvalues of the problem(9)-(11)are

3 Main Theorems

In this section,some uniqueness theorems are given for the problem(9)-(11).It is shown that an explicit formula of eigenvalues can determine the functions bothQ(x)andP(x)be zero by imposing a condition onP(x).Our method is based on[Chern and Shen(1997);Yang(2012)].

Consider a second matrix quadratic pencil of Schr?dinger problem

and similarly for the problem(19)-(21),we can write

Theorem 3.2.Let P(x)=diag[p1(x),p2(x),...,pd(x)]andQ(x)are two d×d real symmetric matrix-valued functions,and α(π)=0.If{0}∪?mj:j=1,2,...?is a subset of the spectrum of the d?dimensional problem

where0is the first eigenvalue of(25),mjis a strictly ascending infinite sequence of positive integers,and0and mjare multiplicity of n,then P(x)=Q(x)=0dalmost everywhere on(0,π).

Proof:Suppose for the(25)Neumann problem,then we have infinitely many eigenvalues of the formmj,mjare positive integers,j=1,2,...and eachmjis of multiplicityn.Then,we get

We have from(28)and Riemann Lebesque lemma thatA(π,π)=0d.Then,by integration of

By using the reality of 0 being the ground state of the eigenvalue problem(25),we may findd?linearly independent constant vectors corresponding to the same eigenvalue 0 by the variational principle and denote them by?j,j=1,2,...,d.Since they should satisfy the equation

ThusQ(x)=0d.Ifwe consider(29)and diagonally property ofP,we getP(x)=0d.This completes the proof.

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