Semiclassical Spectral Invariants for Schrödinger Operators

Spectral Gaps for Periodic Schrödinger Operators with Strong Magnetic Fields

We consider Schrödinger operators H = (ih d + A)∗(ih d + A) with the periodic magnetic field B = dA on covering spaces of compact manifolds. Under some assumptions on B, we prove that there are arbitrarily large number of gaps in the spectrum of these operators in the semiclassical limit of strong magnetic field h → 0.

Semiclassical Asymptotics and Gaps in the Spectra of Periodic Schrödinger Operators with Magnetic Wells

We show that, under some very weak assumption of effective variation for the magnetic field, a periodic Schrödinger operator with magnetic wells on a noncompact Riemannian manifold M such that H(M, R) = 0 equipped with a properly disconnected, cocompact action of a finitely generated, discrete group of isometries has an arbitrarily large number of spectral gaps in the semi-classical limit.

Band Invariants for Perturbations of the Harmonic Oscillator

We study the direct and inverse spectral problems for semiclassical operators of the form S = S0 + ~V , where S0 = 12 ( −~∆Rn + |x| ) is the harmonic oscillator and V : R → R is a tempered smooth function. We show that the spectrum of S forms eigenvalue clusters as ~ tends to zero, and compute the first two associated “band invariants”. We derive several inverse spectral results for V , under v...

Semiclassical Spectral Asymptotics for a Two–Dimensional Magnetic Schrödinger Operator: The Case of Discrete Wells

Tunneling Estimates for Magnetic Schrödinger Operators

We study the behavior of eigenfunctions in the semiclassical limit for Schrödinger operators with a simple well potential and a (non-zero) constant magnetic field. We prove an exponential decay estimates on the low-lying eigenfunctions, where the exponent depends explicitly on the magnetic field strength.