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Asymptotic analysis of non-self-adjoint Hill operators

100%

Veliev O.

Open Mathematics

2013

tom 11

nr 12

2234-2256

We obtain uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L t (q) with a potential q ∈ L 1[0,1] and t-periodic boundary conditions, t ∈ (−π, π]. Using these formulas, we find sufficient conditions on the potential q such that the number of spectral singularities in the spectrum of the Hill operator L(q) in L 2(−∞,∞) is finite. Then we prove that the operator L(q) has no spectral singularities at infinity and it is an asymptotically spectral operator provided that the potential q satisfies sufficient conditions.

Decisiveness of the spectral gaps of periodic Schrodinger operators on the Dumbbell-like metric graph

100%

Niikuni H.

Opuscula Mathematica

2015

tom Vol. 35, no. 2

199--234

In this paper, we consider periodic Schrodinger operators on the dumbbell-like metric graph, which is a periodic graph consisting of lines and rings. Let one line and two rings be in the basic period. We see the relationship between the structure of graph and the band-gap spectrum.

Quantum graph spectra of a graphyne structure

75%

Do N. T. , Kuchment P.

2013

tom 2

107-123

We study the dispersion relations and spectra of invariant Schrödinger operators on a graphyne structure (lithographite). In particular, description of different parts of the spectrum, band-gap structure, and Dirac points are provided.