An Extension of The First Eigen-type Ambarzumyan theorem

TL;DR

This paper extends the first eigenvalue-type Ambarzumyan theorem to arbitrary self-adjoint Sturm-Liouville operators, contributing to inverse spectral theory and broadening the theorem's applicability.

Contribution

It generalizes Ambarzumyan's theorem for a wider class of Sturm-Liouville operators, enhancing inverse spectral analysis methods.

Findings

Extended Ambarzumyan's theorem to arbitrary self-adjoint Sturm-Liouville operators

Provided new insights into inverse spectral theory

Broadened the applicability of eigenvalue-based spectral characterizations

Abstract

An extension of the first eigenvalue-type Ambarzumyan's theorem are provided for the arbitrary self-adjoint Sturm-Liouville differential operators. The result makes a contribution to the P\"oschel-Trubowitz inverse spectral theory as well.

Equations

$$-y^{\prime \prime }(x)+q(x)y(x)=\lambda y(x),y^{\prime }(0)=y^{\prime }(1)=0$$

$$-y^{\prime \prime }(x)+q(x)y(x)=\lambda y(x)$$

$${\lambda }_1-{\tilde{\lambda }}_1=(\hat{q}{\tilde{y}}_1,{\tilde{y}}_1),$$

$${\lambda }_1-{\tilde{\lambda }}_1=ess\inf \hat{q}\text{or}{\lambda }_1-{\tilde{\lambda }}_1=ess\sup \hat{q},$$

$${\lambda }_n-{\tilde{\lambda }}_n=(\hat{q}{\tilde{y}}_n,{\tilde{y}}_n)$$

$${\lambda }_n-{\tilde{\lambda }}_n=ess\inf \hat{q}\text{or}{\lambda }_n-{\tilde{\lambda }}_n=ess\sup \hat{q},$$

$$((\hat{q}+{\tilde{\lambda }}_n-{\lambda }_n){\tilde{y}}_n,{\tilde{y}}_n)=0.$$

$$-y^{\prime \prime }(x)+q(x)y(x)=\lambda y(x),y(0)=y(1)=0.$$

$${\lambda }_n-(n\pi {)}^2=2{\int }_0^{1}q(x){\sin }^{2}n\pi xdx$$

$${\lambda }_n-(n\pi {)}^2=ess\inf q\text{or}{\lambda }_n-(n\pi {)}^2=ess\sup q.$$

$$0=2{\int }_0^{1}q(x){\sin }^{2}n\pi xdx={\int }_0^{1}q(x)dx-{\int }_0^{1}q(x)\cos 2n\pi xdx.$$

$${\int }_0^{1}q(x)\cos 2n\pi xdx=0$$

$$a_n={\int }_0^{1}q(x)\cos 2n\pi xdx={\int }_0^{1}e(u)\cos 2n\pi xdx=0.$$

$$2{\int }_0^{1}q(x){\sin }^{2}n\pi xdx\ne 0,$$

$$a_n={\int }_0^{1}q(x)\cos 2n\pi xdx\ne 0.$$

Full text

1 Introduction

In $1929$, Ambarzumyan [1] proved that if $\{(n\pi)^{2}:n\in\mathbb{N}\cup\{0\}\}$ is the spectrum of the boundary value problem

[TABLE]

with real potential $q\in L^{1}(0,1)$, then $q=0$ a.e. Clearly, if $q=0$ a.e., then the eigenvalues $\lambda_{n}=(n\pi)^{2}$, $n\in\mathbb{N}\cup\{0\}$.

We note that in Ambarzumyan’s theorem the whole spectrum is specified. But then Freiling and Yurko [2] proved that it is enough to specify only the first eigenvalue. More precisely, they proved the following first eigenvalue-type of Ambarzumyan theorem:

Theorem 1.1**.**

If $\lambda_{0}=\int_{0}^{1}q(x)\,dx$, then $q(x)=\lambda_{0}$ a.e.

Consider the boundary value problems $L(q)$ generated in the space $L^{2}(0,1)$ by the following differential equation

[TABLE]

with arbitrary self-adjoint boundary conditions, where $q\in L^{1}(0,1)$ is a real-valued function. The operator $L(q)$ is self-adjoint and its spectrum is discrete, real and bounded from below. We suppose that the eigenvalues of the operator $L(q)$ consist of the sequence $\{\lambda_{n}\}_{n\geq 1}$ $(\lambda_{n}\leq\lambda_{n+1},\lim_{n\rightarrow\infty}\lambda_{n}=+\infty)$ (counting with multiplicities) and $\{y_{n}\}_{n\geq 1}$ denotes the corresponding normalized eigenfunctions of the operator $L(q)$.

Let us consider another known and fixed operator $\tilde{L}:=L(\tilde{q})$ of the same domain with $L:=L(q)$ but with different potential $\tilde{q}\in L^{1}(0,1)$. Denote by $\{\tilde{\lambda}_{n}\}_{n\geq 1}$ and $\{\tilde{y}_{n}\}_{n\geq 1}$ eigenvalues and normalized eigenfunctions of the operator $\tilde{L}$, respectively.

In [3], Yurko provided the following generalization of Theorem 1.1 on wide classes of self-adjoint differential operators. Here $(.,.)$ denotes inner product in $L^{2}(0,1)$ and $\hat{q}:=q-\tilde{q}$.

Theorem 1.2**.**

Let

[TABLE]

where $\tilde{y}_{1}$ is a normalized eigenfunction of $\tilde{L}$ related to the first eigenvalue $\tilde{\lambda}_{1}$. Then $q(x)=\tilde{q}(x)+\lambda_{1}-\tilde{\lambda}_{1}$ a.e. on $(0,1)$.

Note that the proof of this theorem is based on the well-known variational principle of the smallest eigenvalue.

Recently Ashrafyan [4] proved the following another generalization of the first eigenvalue-type of Ambarzumyan theorem for Sturm-Liouville problems with arbitrary self-adjoint boundary conditions.

Theorem 1.3**.**

Let

[TABLE]

where $\tilde{\lambda}_{1}$ is the first eigenvalue $\tilde{L}$. Then $q(x)=\tilde{q}(x)+\lambda_{1}-\tilde{\lambda}_{1}$ a.e. on $(0,1)$.

As in the proof of Theorem 1.2, the above uniqueness theorem is also provided by using the property of the smallest eigenvalue. That is, the proof of Theorem 1.3 is based on the Sturm oscillation theorem that the first eigenfunction has no zeros on interval $(0,1)$.

2 Main results

The main result of this paper is as follows. Note that, to obtain the following extension theorem, it is enough to have information about the arbitrary eigenvalue $\lambda_{n}$ instead of the first eigenvalue only.

Theorem 2.1**.**

Let, for some $n$,

[TABLE]

and

[TABLE]

where $\tilde{y}_{n}$ is a normalized eigenfunction of $\tilde{L}$ related to the eigenvalue $\tilde{\lambda}_{n}$. Then $q(x)=\tilde{q}(x)+\lambda_{n}-\tilde{\lambda}_{n}$ a.e. on $(0,1)$.

Proof.

It follows from the first assumption that

[TABLE]

Since $\tilde{y}_{n}$ is a normalized eigenfunctions of the operator $\tilde{L}$ corresponding to the eigenvalue $\tilde{\lambda}_{n}$, we obtain that $\tilde{y}_{n}$ has at most finitely many isolated zero points in $(0,1)$. Therefore, the measure of the set of zero points is [math]. Hence, using this and $(\hat{q}+\tilde{\lambda}_{n}-\lambda_{n})\tilde{y}_{n}^{2}\in L^{1}(0,1)$, we get $\hat{q}=\lambda_{n}-\tilde{\lambda}_{n}$ a.e. on $(0,1)$. ∎

From Theorem 2.1, one can easily verify the following assertion.

Theorem 2.2**.**

Let the assumptions of Theorem 2.1 be valid and let $\int_{0}^{1}q(x)dx=\int_{0}^{1}\tilde{q}(x)dx$. Then $q(x)=\tilde{q}(x)$ a.e. on $(0,1)$.

2.1 Example

Let us give an example to illustrate Theorem 2.1.

Consider the Dirichlet boundary value problem

[TABLE]

Let $\tilde{q}(x)\equiv 0$. Then $\tilde{\lambda}_{n}=(n\pi)^{2}$, $\tilde{y}_{n}=\sqrt{2}\sin n\pi x$ for some $n\geq 1$ and Theorem 2.1 implies the following assertion.

Corollary 2.3**.**

Let, for some $n$,

[TABLE]

and

[TABLE]

Then $q(x)=\lambda_{n}-(n\pi)^{2}$ a.e. on $(0,1)$.

And, Theorem 2.2 implies the following assertion. The assertion makes a contribution to the Pöschel-Trubowitz inverse spectral theory.

Corollary 2.4**.**

Let the assumptions of Corollary 2.3 be valid and let $\int_{0}^{1}q(x)dx=0$. Then $q(x)=0$ a.e. on $(0,1)$.

Remark 1*.*

In [5], Pöschel and Trubowitz showed that, for the Dirichlet problem, there is an infinite dimensional set of $L^{2}(0,1)$ potentials with the same Dirichlet zero spectrum $\sigma:=\{(n\pi)^{2}:n\in\mathbb{N}\}$ as $q=0$ a.e. That is, if the spectrum is $\sigma$, then the potential is not necessarily zero. Thus Ambarzumyan’s theorem is not valid.

Arguing as in the paper [6] (see p. 337), from Pöschel and Trubowitz [5], an isospectral $L^{2}(0,1)$ potential can be written in the form $q=u+e(u)$, where $u$ is odd part and $e$ is even part which can be uniquely expressed as a function of its odd part ($e=e(u)$).

Let $q$ be an isospectral potential with the Dirichlet zero spectrum $\sigma=\{(n\pi)^{2}:n\in\mathbb{N}\}$. Then $\int_{0}^{1}q(x)dx=0$ and from (5), we get for some $n$

[TABLE]

Hence, for some $n$,

[TABLE]

which implies that the n-th even Fourier coefficient $a_{n}$ has the following equalities

[TABLE]

Then $a_{n}=0$ for all $n$. Similarly, all the odd Fourier coefficients vanish. Therefore, the even part of an isospectral potential does not vanish. Consequently, from Corollary 2.4, for the isospectral potential $q$ (i.e. $q\neq 0$), we get, for all $n$,

[TABLE]

that is, we get from (6), for all $n$,

[TABLE]

Thus, the present paper supplements the Pöschel-Trubowitz inverse spectral theory.