Schrödinger Operators with Strong Local Magnetic Perturbations: Existence of Eigenvalues in Gaps of the Essential Spectrum

Abstract

In the present note, we show that perturbations by strong magnetic fields of compact support may produce eigenvalues inside a spectral gap of a (periodic) Schrödinger operator. Here we will discuss the following situation:

In the Hilbertspace $\mathcal{H}\,{\text{ = }}\,L_2 (R^\nu )$, we consider the Schrödinger operator H = —Δ + V, with a fixed potential V : Rν → R, V bounded and V ≥ 1, where H is defined as the unique self-adjoint extension of $( - \Delta \, + \,V)|C_c^\infty (R^\nu )$. We shall make the basic assumption that H has a (non-trivial) gap in its essential spectrum; more precisely, we assume that there exist b > a > mi σ_ess}(H) with [a, b] ∩ σ(H) = 0. In view of the applications in solid state physics, one may think of H as a periodic Schrödinger operator.