Recent developments in quantum mechanics with magnetic fields

Abstract

We present a review on the recent developments concerning rigorous mathematical results on Schrödinger operators with magnetic fields. This paper is dedicated to the sixtieth birthday of Barry Simon. AMS 2000 Subject Classification 81Q10, 81Q70Running title: Recent developments on magnetic fields * Partially supported by EU-IHP Network "Analysis and Quantum" HPRN-CT-2002-0027 and by Harvard University relativistic theory. Although the framework for QED has been clear since the 30's, the mathematical difficulties even to formulate the theory rigorously have not yet been resolved. In the low energy regime, however, massive quantum particles can be described non-relativistically. Electric and magnetic fields, with a good approximation, can be considered decoupled. Since typical magnetic fields in laboratory are relatively weak, as a first approximation one can completely neglect magnetic fields and concentrate only on quantum point particles interacting via electric potentials.The rigorous mathematical theory of Schrödinger operators has therefore started with studying the operator H = 1 2m p 2 + V (x) on L 2 (R d ) and its multi-particle analogues. Here x ∈ R d is the location of the particle in the d-dimensional configuration space, p = −i∇ x is the momentum operator and m is the mass, that can be set m = 1 2 with convenient units. The Laplace operator describes the kinetic energy of the particle and the real-valued function V (x) is the electric potential. Although both the kinetic and potential energy operators are very simple to understand separately, their sum exhibits a rich variety of complex phenomena which differ from their classical counterparts in many aspects. The mathematical theory of this operator is the most developed and most extensive in mathematical physics: the best recent review is by Simon [152].As a next approximation, classical magnetic fields are included in the theory, but spins are neglected. The kinetic energy operator is modified from p 2 to (p + A) 2 by the minimal substitution rule: p → p − eA and we set the charge to be e = −1. Here A : R d → R d is the magnetic vector potential that generates the magnetic field B according to classical electrodynamics. In d = 2 or d = 3 dimensions B = ∇ × A is a scalar or a vector field, respectively. In d = 1 dimension the vector potential can be removed by a unitary gauge transformation, e iϕ (p + A) 2 e −iϕ = p 2 , ϕ = A, therefore magnetic phenomena in R 1 are absent (they are present in the case of S 1 ).We will call the operator (p + A) 2 + V the magnetic Schrödinger operator. In general, even the kinetic energy part contains noncommuting operators, [(p + A) k , (p + A) ℓ ] = 0, and the theory of (p + A) 2 itself is more complicated than that of p 2 + V . The simplest case of constant magnetic field, B = const, is explicitly solvable. The resulting Landau-spectrum consists, in two dimensions, of infinitely degenerate eigenvalues at energies (2n+1)|B|, n = 0, 1, . . .. Notice that the magnetic spectrum is characteristically different from that ...

Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions

“…In [20] and [8] Strichartz and smoothing estimates were obtained for small A and V . For more background and many references on magnetic operators see Erdös's survey [7].…”

Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in $\mathbb{R}^3$

“…More recent results on scattering by magnetic potentials are [24] and [30]. The review [7] contains a long list of references. There has been much activity surrounding dispersive estimates for the case A = 0 under suitable decay (and also regularity when d ≥ 4) assumptions on V .…”

Compactness of the solution operator to ∂¯ in weighted L2-spaces

“…Finally, we also point out some interesting connections to the theory of Dirac and Pauli operators, when discussing the case of non-compact resolvents (see [10,11,26,39,49]). …”

“…In this case an interesting connection to Dirac and Pauli operators is of importance (see [10,11,26,39,49]). Let us first consider the real two-dimensional case.…”