10 research outputs found
Isospectral Mathieu-Hill Operators
In this paper we prove that the spectrum of the Mathieu-Hill Operators with
potentials ae^{-i2{\pi}x}+be^{i2{\pi}x} and ce^{-i2{\pi}x}+de^{i2{\pi}x} are
the same if and only if ab=cd, where a,b,c and d are complex numbers. This
result implies some corollaries about the extension of Harrell-Avron-Simon
formula. Moreover, we find explicit formulas for the eigenvalues and
eigenfunctions of the t-periodic boundary value problem for the Hill operator
with Gasymov's potential
Bethe-Sommerfeld conjecture for periodic operators with strong perturbations
We consider a periodic self-adjoint pseudo-differential operator
, , in which satisfies the following conditions:
(i) the symbol of is smooth in \bx, and (ii) the perturbation has
order less than . Under these assumptions, we prove that the spectrum of
contains a half-line. This, in particular implies the Bethe-Sommerfeld
Conjecture for the Schr\"odinger operator with a periodic magnetic potential in
all dimensions.Comment: 61 page
On the Schrödinger operator with a periodic PT-symmetric matrix potential
In this article, we obtain asymptotic formulas for the Bloch eigenvalues of the operator L generated by a system of Schrödinger equations with periodic PT-symmetric complex-valued coefficients. Then, using these formulas, we classify the spectrum ?(L) of L and find a condition on the coefficients for which ?(L) contains all half line [H, ?) for some H. © 2021 Author(s)