Essentially isospectral transformations and their applications

Namig Guliyev (1)

1 Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences

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.

Mots clés: boundary conditions dependent on the eigenvalue parameter; asymptotics; oscillation; inverse problems; regularized trace; Darboux transformation; one-dimensional Schrödinger equation

Domaines: Physique mathématique [math-ph]; Théorie spectrale [math.SP]; Analyse fonctionnelle [math.FA]; Analyse classique [math.CA]

Namig Guliyev. Essentially isospectral transformations and their applications. Annali di Matematica Pura ed Applicata, 2020, 199 (4), pp.1621-1648. ⟨10.1007/s10231-019-00934-w⟩. ⟨hal-01579888⟩