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Study of the asymptotic eigenvalue distribution and trace formula of a second order operator-differential equation. (English) Zbl 1268.34049
Summary: The purpose of this article is to show some spectral properties of the Bessel operator equation, with spectral parameter-dependent boundary condition. This problem arises upon separation of variables in heat or wave equations, when one of the boundary conditions contains a partial derivative with respect to time. To illustrate the problem and the proof in detail, as a first step, the discreteness of the spectrum of the corresponding operator is proved. Then, the nature of the eigenvalue distribution is established. Finally, based on these results, a regularized trace formula for the eigenvalues is obtained.

MSC:
34B09 Boundary eigenvalue problems for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34L05 General spectral theory of ordinary differential operators
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A10 Spectrum, resolvent
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References:
[1] doi:10.1007/BF01177870 · Zbl 0246.47058 · doi:10.1007/BF01177870
[2] doi:10.1016/j.amc.2006.08.066 · Zbl 1123.34308 · doi:10.1016/j.amc.2006.08.066
[3] doi:10.1215/S0012-7094-60-02758-7 · Zbl 0095.09502 · doi:10.1215/S0012-7094-60-02758-7
[4] doi:10.1016/0022-1236(88)90031-6 · Zbl 0649.47012 · doi:10.1016/0022-1236(88)90031-6
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