On eigenvalue accumulation for non-self-adjoint magnetic operators

Abstract

In this work, we use regularized determinant approach to study the discrete spectrum generated by relatively compact non-self-adjoint perturbations of the magnetic Schr\"odinger operator $(-i\nabla - \textbf{\textup{A}})^{2} - b$ in dimension $3$ with constant magnetic field of strength $b>0$. The situation near the Landau levels $2bq$, $q \in \mathbb{N}$, is more interesting since they play the role of thresholds of the spectrum of the free operator. First, we obtain sharp upper bounds on the number of the complex eigenvalues near the Landau levels. Under appropriate hypothesis, we then prove the presence of an infinite number of complex eigenvalues near each Landau level $2bq$, $q \in \mathbb{N}$, and the existence of sectors free of complex eigenvalues. We also prove that the eigenvalues are localized in certain sectors adjoining the Landau levels. In particular, we provide an adequate answer to the open problem from [34] about the existence of complex eigenvalues accumulating near the Landau levels. Furthermore, we prove that the Landau levels are the only possible accumulation points of the complex eigenvalues.