We construct the spectral expansion for the one-dimensional Schrödinger operatorwhere 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. 2002 Elsevier Science
We obtain uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L ( ) with a potential ∈ L 1 [0 1] and -periodic boundary conditions, ∈ (−π π]. Using these formulas, we find sufficient conditions on the potential such that the number of spectral singularities in the spectrum of the Hill operator L( ) in L 2 (−∞ ∞) is finite. Then we prove that the operator L( ) has no spectral singularities at infinity and it is an asymptotically spectral operator provided that the potential satisfies sufficient conditions.
MSC:34L05, 34L20
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