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On characteristic matrices and eigenfunction expansions of two singular point symmetric systems
Author(s) -
Mogilevskii Vadim
Publication year - 2015
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201300242
Subject(s) - mathematics , eigenfunction , boundary value problem , mathematical analysis , boundary (topology) , self adjoint operator , type (biology) , hamiltonian system , hamiltonian (control theory) , pure mathematics , eigenvalues and eigenvectors , hilbert space , ecology , mathematical optimization , physics , quantum mechanics , biology
We study general (not necessarily Hamiltonian) first‐order symmetric systems J y ′ − B ( t ) y = Δ ( t ) f ( t )on an interval I = ( a , b ) with both singular endpoints a and b . For such a system we give a criterion of existence and description of self‐adjoint separated boundary conditions. We prove the Titchmarsh type formula for the characteristic matrix Ω ( λ ) of the self‐adjoint linear relation inL Δ 2 ( I )generated by separated boundary conditions as well as by a certain type of mixed boundary conditions. This formula enables one to express Ω ( λ ) in terms of the m ‐functions at the endpoints of I . By using the Titchmarsh type formula we parametrize all spectral functions corresponding to boundary problems with the mentioned boundary conditions immediately in terms of self‐adjoint boundary parameters at the end points a and b .