Spectral properties of the Sturm–Liouville operator with a spectral parameter quadratically included in the boundary condition

L. I. Mammadova   ; I. M. Nabiev

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 30 (2020) no. 2, pp. 237-248

Résumé

The article considers the Sturm–Liouville operator with a real quadratically integrable potential. Boundary conditions are non-separated. One of these boundary conditions includes the quadratic function of the spectral parameter. Some spectral properties of the operator are studied. It is proves that eigenvalues are real and non-zero and there are no associated functions to the eigenfunctions. An asymptotic formula for the spectrum of the operator is derived, and a representation of the characteristic function as an infinite product is obtained. The results of the paper play an important role in solving inverse problems of spectral analysis for differential operators.

Détail

Keywords: Sturm-Liouville operator, non-separated boundary conditions, eigenvalues, infinite product.

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