Discrete spectrum asymptotics for the Schrödinger operator in a moderate magnetic field

Abstract

Let a ∈ (L ^∞ (ℝ^d))^d, d ≥ 2 be a real vector-valued function. The Schrödinger operator H ₀ = H ₀ (h, µ) with the magnetic potential µa is defined as an operator associated with the quadratic form

$$ {{\left\| {( - ih\nabla - \mu a)u} \right\|}^{2}},u \in C_{0}^{\infty }({{\mathbb{R}}^{d}}), $$

, h ∈ (0, h ₀] and µ≥ 0 being the Planck constant and the intensity of the magnetic field respectively. We study spectral properties of the perturbed

$$ {{H}_{a}} = {{H}_{0}} + V,a = \left\{ {a,V} \right\} $$

, with a real-valued function V (electric potential). Precisely, we analyse the asymptotics as² h → 0, µh ≥ 0, µ h ≤ C of traces of the form

$$ {{M}_{s}}\left( {h,\mu } \right) = {{M}_{s}}\left( {h,\mu ;\psi ,a} \right) = tr\left\{ {\psi {{g}_{s}}\left( {{{H}_{a}}} \right)} \right\} $$

((1.1))

.