On a class of nonself-adjoint multidimensional periodic Schrödinger operators

Abstract

We investigate the Schrödinger operator $L_{(q)}$ in $L_{(2)}(\mathbb{R}^d)\;(d\geq1)$ with the complex-valued potential q thatis periodic with respect to a lattice ?. Besides, it is assumed that the Fourier coefficients $q_\gamma$ of q with respect to theorthogonal system $\{\;e^{i<yx>}:\;\gamma\in\Gamma\;\}$ vanish if ? belongs to a half-space, where ? is the lattice dual to ?. We provethat the Bloch eigenvalues are $ \left|y+t\right|^2$ for $y\in\Gamma$ where t is a quasimomentum and find explicit formulas for theBloch functions. Moreover, we investigate the multiplicity of the Bloch eigenvalue and consider necessary and sufficientconditions on the potential which provide some root functions to be eigenfunctions. Besides, in case d = 1 we investigatein detail the root functions of the periodic and antiperiodic boundary value problems.