The Spectral Expansion for a Nonself-adjoint Hill Operator with a Locally Integrable Potential

By O. Veliev, Melda Duman. 2002

64 citations recorded by the source corpus.

Abstract We construct the spectral expansion for the one-dimensional Schrodinger operator L = − d 2 dx 2 + q(x), − ∞ x ∞, in L 2 ( − ∞, ∞), where q ( x ) is a 1-periodic, Lebesgue integrable on [0,1], and complex-valued potential. We obtain the asymptotic formulas for the eigenfunctions and eigenvalues of the operator L t , for t ≠ 0, π, generated by this operation in L 2 [0, 1] and the t -periodic boundary conditions. Using it, we prove that the eigenfunctions and associated functions of L t form a Riesz basis in L 2 [0, 1] for t ≠ 0, π. Then we find the spectral expansion for the operator L .

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/99bc4607600d3ff735675d848be10ef2bc785d29

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

A criterion for Hill operators to be spectral operators of scalar type (2006, background): “For spectral expansion theorems in connection with finite interval problems and limit circle-type situations we refer to [17], [46], [75], and the literature cited therein.”
On the riesz basisness of the root functions of the nonself-adjoint sturm-liouville operator (2005, background): “These relations are obvious for q(x) E L2(O, 1). For q(x) E L1(0,1), see Lemma 1 of [ 6 ].”
Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators (2011, background): “h guarantee basisness of {uk}, – with or without additional assumptions about the structure or smoothness of a potential v – have been given by A. Makin [37, 38, 39, 40], A. Shkalikov [52], O. Veliev [59, 60, 61, 1], P. Djakov and B. Mityagin [4, 8, 12, 13, 14]. 2. In our papers [24, 3, 4, 9, 14] we”