Let´s consider differential operator Sturm-Liouville of the second order:
(1)
with not separable boundary conditions of the first type (see [1]):
(2)
where 
, and it is supposed that potential q(x) - summable function on the segment [0; π]:
(3)
almost everywhere on [0; p].
Theorem. Asymptotics of the eigenvalues of the differential operator (1)-(2) with a condition (3) has the following kind:
(4)
and for this
(5)
(6)
The theorem is proved by methods of the chapter 5 of the monograph [2].
References
-
Sadovnichiy V.A., Sultanayev Ya.T., Akhtyamov A.M. Inverse problems of Sturm-Liouville with not separable boundary conditions. - M.: Publishing House of Moscow University, 2009. - 184 p.
-
Mitrokhin S.I. Spectral theory of operators: smooth, discontinuous, summable coefficients. - M.: INTUIT, 2009. - 364 p.
The work is submitted to the Scientific International Conference «Research on the priority of higher education on-directions of science and technology», on board the cruise ship MSC Musica, June, 10-17, 2012, came to the editorial office оn 03.05.2012.