An Algorithm for Finding the Periodic Potential of the Three-dimensional Schrodinger Operator from the Spectral Invariants (original) (raw)

On a Class of Non-self-adjoint Multidimensional Periodic Schrodinger Operators

TURKISH JOURNAL OF MATHEMATICS

We investigate the multidimensional Schrodinger operator L(q) in L2 R d (d ≥ 2) with complex-valued periodic, with respect to a lattice Ω, potential q when the Fourier coefficients qγ of q with respect to the orthogonal system {e i γx : γ ∈ Γ} vanish if γ belong to a half-space, where Γ is the lattice dual to Ω. We prove that the Bloch eigenvalues are | γ + t | 2 for γ ∈ Γ, where t is a quasimomentum and find explicit formulas for the Bloch functions. It implies that the Fermi surfaces of L(q) and L(0) are the same. The considered set of operators includes a large class of PT symmetric operators used in the PT symmetric quantum theory.

On the Schrödinger operator with a periodic PT-symmetric matrix potential

Journal of Mathematical Physics

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 σ(L) of L and find a condition on the coefficients for which σ(L) contains all half line [H, ∞) for some H.

Analysis of periodic Schrödinger operators: Regularity and approximation of eigenfunctions

Journal of Mathematical Physics, 2008

Let V be a real valued potential that is smooth everywhere on R 3 , except at a periodic, discrete set S of points, where it has singularities of the Coulomb type Z/r. We assume that the potential V is periodic with period lattice L. We study the spectrum of the Schrödinger operator H = −∆ + V acting on the space of Bloch waves with arbitrary, but fixed, wavevector k. Let T := R 3 /L. Let u be an eigenfunction of H with eigenvalue λ and let ǫ > 0 be arbitrarily small. We show that the classical regularity of the eigenfunction u is u ∈ H 5/2−ǫ (T) in the usual Sobolev spaces, and u ∈ K m 3/2−ǫ (T S) in the weighted Sobolev spaces. The regularity index m can be as large as desired, which is crucial for numerical methods. For any choice of the Bloch wavevector k, we also

Perturbation Theory for the Multidimensional Schrodinger Operator with a Periodic Potential

2005

In this paper we obtain asymptotic formulas of arbitrary order for the Bloch eigenvalue and the Bloch function of the periodic Schrodinger operator − ∆ + q(x), of arbitrary dimension, when corresponding quasimomentum lies near a diffraction hyperplane. Moreover, we estimate the measure of the isoenergetic surfaces in the high energy region. Bisides, writing the asymptotic formulas for the Bloch eigenvalue and the Bloch function, when corresponding quasimomentum lies far from the diffraction hyperplanes, obtained in my previous papers in improved and enlarged form, we obtain the complete perturbation theory for the multidimensional Schrodinger operator with a periodic potential. 1

Asymptotic Analysis of the Periodic Schrodinger Operator

arXiv: Mathematical Physics, 2005

In this paper we obtain asymptotic formulas of arbitrary order for the Bloch eigenvalues and Bloch functions of the Schrodinger operator of arbitrary dimension, with periodic, with respect to arbitrary lattice, potential. Moreover, we estimate the measure of the isoenergetic surfaces in the high energy region.

On the Guided States of 3D Biperiodic Schrödinger Operators

Communications in Partial Differential Equations, 2012

We consider the Laplacian operator H0 = −∆ perturbed by a non-positive potential V , which is periodic in two directions, and decays in the remaining one. We are interested in the characterization and decay properties of the guided states, defined as the eigenfunctions of the reduced operators in the Bloch-Floquet-Gelfand transform of H0 + V in the periodic variables. If V is sufficiently small and decreases fast enough in the infinite direction, we prove that, generically, these guided states are characterized by quasi-momenta belonging to some one-dimensional compact real analytic submanifold of the Brillouin zone. Moreover they decay faster than any polynomial function in the infinite direction.