This article is cited in 2 papers

On the integral characteristic function of the Sturm-Liouville problem

D. V. Valovik

Penza State University, Penza, Russia

Abstract: We introduce a function whose zeros, and only these zeros, are eigenvalues of the corresponding Sturm-Liouville problem. The boundary conditions of the problem depend continuously on the spectral parameter. Therefore, it makes sense to call the function thus constructed a characteristic function of the Sturm-Liouville problem (however, it is not a characteristic function in the ordinary sense). An investigation of the function thus obtained enables us to prove the solvability of the problem in question, to find the asymptotic behaviour of the eigenvalues, to obtain comparison theorems, and to introduce an indexing of the eigenvalues and the zeros of eigenfunctions in a natural way.
Bibliography: 31 titles.

Keywords: Sturm-Liouville problem, integral characteristic function, asymptotic behaviour of eigenvalues, comparison theorem, Riccati equation.

UDC: 517.927.25

MSC: Primary 34B24; Secondary 34C10, 34L20

Received: 17.02.2019 and 20.04.2020

DOI: 10.4213/sm9235

 English version: Sbornik: Mathematics, 2020, 211:11, 1539–1550

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