On a class of nonself-adjoint multidimensional periodic Schrödinger operators

Abstract

We investigate the Schrödinger operator $L(q)$ in $L_{2}\left( \mathbb{R}^{d}\right) \ (d\geq1)$ with the complex-valued potential $q$ that is periodic with respect to a lattice $\Omega.$ Besides, it is assumed that the Fourier coefficients $q_{\gamma}$ of $q$ with respect to the orthogonal system $\{e^{i\left\langle \gamma x\right\rangle }:\gamma\in\Gamma\}$ vanish if $\gamma$ belongs to a half-space, where $\Gamma$ is the lattice dual to $\Omega.$ We prove that the Bloch eigenvalues are $\mid\gamma+t\mid^{2}$ for $\gamma\in\Gamma,$ where $t$ is a quasimomentum and find explicit formulas for \ the Bloch functions. Moreover, we investigate the multiplicity of the Bloch eigenvalue and consider necessary and sufficient conditions on the potential which provide some root functions to be eigenfunctions. Besides, in case $d=1$ we investigate in detail the root functions of the periodic and antiperiodic boundary value problems.