Essentially isospectral transformations and their applications

Abstract

We define and study the properties of Darboux-type transformations between Sturm-Liouville problems with boundary conditions containing rational Herglotz-Nevanlinna functions of the eigenvalue parameter (including the Dirichlet boundary conditions). Using these transformations, we obtain various direct and inverse spectral results for these problems in a unified manner, such as asymptotics of eigenvalues and norming constants, oscillation of eigenfunctions, regularized trace formulas, and inverse uniqueness and existence theorems.

Harmonic oscillator with a step and its isospectral properties

“…Isospectral deformation of Hamiltonian is of importance, for it is certainly related to the construction of exactly solvable potentials [13,14,[27][28][29][30]. It has extensively been explored by many authors (For recent works, see, e.g., [31][32][33][34]). Some isospectral deformations produce a deformed Hamiltonian that shares many, if not all, eigenvalues with the original Hamiltonian, while other isospectral deformations construct a deformed Hamiltonian whose energy spectrum is entirely identical to the original one's.…”

Asymptotic Analysis of Sturm-Liouville Problem With Dirichlet and Nonlocal Two-Point Boundary Conditions

“…Asymptotic formulas for eigenvalues and eigenfunctions for Sturm-Liouville equation (1.1) with local BCs are investigated in the classical books [17,18,38]. These results were generalized for tasks with retarded argument [3,21,24,27] and for some other local BCs [1,20] and Sturm-Liouville Problem (SLP) with eigenparameter in BCs [7,9,11,12,13]. Asymptotical analysis of eigenvalues and eigenfunctions of SLPs with periodic BCs was obtained in [2,6,10].…”

Inverse problems for discontinuous Dirac operator with eigenparameter dependent boundary and transmission conditions

“…Besides all this, in recent years, Sturm-Liouville operators with eigenparameter dependent discontinuity conditions were studied in [25]. And also, complete solutions of various direct and inverse spectral problems for Sturm-Liouville operators with boundary conditions containing rational Herglotz-Nevanlinna functions of the eigenvalue parameter were provided in [26] and [27], and were subsequently extended to distributional [28] and Bessel-type potentials [29,33]. On the other hand, inverse problems for Dirac and Sturm-Liouville operators with eigenparameter dependent nonseparated boundary conditions were solved in [30,31], respectively.…”