On the discrete spectrum of Schrödinger operators with strong magnetic fields of compact support.

Rainer Hempel TU Braunschweig

We discuss the discrete eigenvalues of magnetic Schrödinger operators in R^{2}

H(λa⃗) = (−i∇ − λa⃗)^{2} + V (x), λ ≥ 0,

where V is a bounded potential and a⃗ a magnetic vector potential of class C^{1} in R^{2} which is such that the associated magnetic field B = curl a⃗ has compact support consisting in a finite number of components. The interesting case is when the flux ∫ B dx is non-zero. As for the unperturbed operator H = −Δ + V, we consider the periodic case as well as the case where H has compact resolvent, like the harmonic oscillator.

More precisely, we study the (signed) flow of spectral multiplicity across a fixed energy level E ∈ R\σ(H), where σ(H) is the spectrum of H. It is shown that there exists a sequence of couplings λ_{k} → ∞ such that the total signed spectral flow for H(λ_{k}a⃗) corresponds to the spectral flow for the family of operators H + λχ_{Ω}, as λ → ∞, where Ω = {B(x) ≠ 0}.