Degenerate boundary conditions on a geometric graph

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Abstract


The boundary conditions of the Sturm-Liouville problem defined on a star-shaped geometric graph of three edges are studied. It is shown that if the lengths of the edges are different, then the Sturm-Liouville problem does not have degenerate boundary conditions. If the lengths of the edges and the potentials are the same, then the characteristic determinant of the Sturm-Liouville problem can not be equal to a constant different from zero. But the set of Sturm-Liouville problems for which the characteristic determinant is identically equal to zero is an infinite (continuum). In this way, in contrast to the Sturm-Liouville problem defined on an interval, the set of boundary-value problems on a star-shaped graph whose spectrum completely fills the entire plane is much richer. In the particular case when the minor A124 for matrix of coefficients is nonzero, it does not consist of two problems, as in the case of the Sturm-Liouville problem given on an interval, but of 18 classes, each containing two to four arbitrary constants.


About the authors

V. A. Sadovnichy

Lomonosov Moscow State University

Author for correspondence.
Email: rector@msu.ru

Russian Federation, 1, Leninskie gory, Moscow, 119991

Academician of the RAS

Ya. T. Sultanaev

Ufa Branch of the Russian Academy of Sciences; Bashkir State Pedagogical University

Email: akhtyamovam@mail.ru

Russian Federation, 71, Prospect Oktyabrya, Ufa, 450054; 3A, October Revolution street, Republic of Bashkortostan, Ufa, 450000

A. M. Akhtyamov

Ufa Branch of the Russian Academy of Sciences; Bashkiria State University, Ufa

Email: akhtyamovam@mail.ru

Russian Federation, 71, Prospect Oktyabrya, Ufa, 450054; 32, Validy Str., Ufa, 450076

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