Some results on semiclassical spectral analysis of magnetic Schrödinger operators
Yuri A. Kordyukov
TL;DR
This work develops a comprehensive semiclassical spectral analysis for magnetic Schrödinger operators on $\mathbb{R}^d$, formulating the problem in terms of the operator $H_{\hbar}$ with smooth bounded magnetic and electric potentials. It introduces model operators $\mathcal{H}^{(x_0)}$ obtained by freezing coefficients at each $x_0$ and derives the associated Landau-type spectra $\Lambda_{\mathbf{k}}(x_0)$; under a full-rank magnetic field, the spectrum of $H_{\hbar}$ in $[0,K\hbar]$ is shown to lie near $\hbar\Sigma$, with $\Sigma$ built from the $\Sigma_{x_0}$, and a refined $\hbar^{5/4}$-type proximity bound is established. The paper then develops a Bergman-type off-diagonal expansion for smoothing functions $\varphi(H_{\hbar}/\hbar)$, giving a detailed asymptotic expansion of the Schwartz kernel near the diagonal with leading term given by the model operator kernels and providing explicit formulas for the leading coefficients. Finally, it proves localization results for spectral projections and eigenfunctions in spectral gaps, including discreteness of the local spectrum and exponential decay of eigenfunctions away from the classical allowed set, using a gauge-covariant perturbation framework and Agmon-type estimates.
Abstract
In our recent papers, we studied semiclassical spectral problems for the Bochner-Schrödinger operator on a manifold of bounded geometry. We survey some results of these papers in the setting of the magnetic Schrödinger operator in the Euclidean space and describe some ideas of the proofs.