In this paper, the classical Ambarzumyan’s theorem for the regular Sturm-Liouville problem is extended to energy-dependent Sturm-Liouville problem with a spectral parameter depending on the boundary condition
Quadratic pencil of Sturm-Liouville operators; uniqueness theorem; Ambarzumyan’s theorem; inverse problem; asymptotic of eigenvalues
1. Introduction
In 1929, Ambarzumyan investigated the Sturm-Liouville operator with Neumann boundary conditions, and proved that if its spectrum consists of zero and infinitely many other square integers, then the potential is zero. From a historical viewpoint, the work of Ambarzumyan [1] was the first paper in the theory of inverse spectral problems associated with Sturm-Liouville operators. Ambarzumyan’s theorem was generalized in many directions [2-6]. In [7], Ambarzumyan's theorem was extended to the quadratic pencil of the Sturm–Liouville operators with spectral parameter contained in the boundary conditions by adding an additional condition for the potential. Boundary value problems with spectral parameter in boundary conditions have received much attention in the recent research literature [8-10]. Various physical applications of such problems were found in [11].
In this paper we prove Ambarzumyan's theorem for the quadratic pencil of the Sturm–Liouville operators with spectral parameter contained in the boundary conditions without any additional condition for the potential.
Quadratic pencil of the Sturm–Liouville operators arises in various models of quantum and classical mechanics. For instance, to this form can be reduced the corresponding evolution equations (such as the Klein–Gordon equation [12, 13]) that are used to model interactions between colliding relativistic spinless particles. Another typical example is related to vibrations of mechanical systems in viscous media, see [14].
We consider the boundary-value problem
where 
is a spectral parameter and the functions 
and 
are real.
We denote by 
the spectrum of the problem (1). It is well known [15] that the sequence 
satisfies the classical asymptotic form
where 
and 
.
Let 
be spectrum of the problem
The main results of this paper are as follows.
Theorem. If 
then, 
.
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