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Spectral theory of Sturm–Liouville difference operators

✍ Scribed by Guoliang Shi; Hongyou Wu

Publisher
Elsevier Science
Year
2009
Tongue
English
Weight
241 KB
Volume
430
Category
Article
ISSN
0024-3795

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✦ Synopsis

We present several classes of explicit self-adjoint Sturm-Liouville difference operators with either a non-Hermitian leading coefficient function, or a non-Hermitian potential function, or a non-definite weight function, or a non-self-adjoint boundary condition. These examples are obtained using a general procedure for constructing difference operators realizing discrete Sturm-Liouville problems, and the minimum conditions for such difference operators to be self-adjoint with respect to a natural quadratic form. It is shown that a discrete Sturm-Liouville problem admits a difference operator realization if and only if it does not have all complex numbers as eigenvalues. Spectral properties of self-adjoint Sturm-Liouville difference operators are studied. In particular, several eigenvalue comparison results are proved.

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