On the Spectral Properties of the Perturbed Landau Hamiltonian

The Landau Hamiltonian governing the behavior of a quantum particle in dimension 2 in a constant magnetic field is perturbed by a compactly supported magnetic field and a similar electric field. We describe how the spectral subspaces change and how the Landau levels split under this perturbation.     

Stability of the Magnetic Schrödinger Operator in a Waveguide

Abstract

The spectrum of the Schrödinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any local enlargement of the waveguide produces eigenvalues beneath the continuous spectrum. Also, if the waveguide is bent, eigenvalues will arise below the continuous spectrum. In this paper a magnetic field is added into the system. The spectrum of the magnetic Schrödinger operator is proved to be stable under small local deformations and also under small bending of the waveguide. The proof includes a magnetic Hardy-type inequality in the waveguide, which is interesting in its own right.     

A Time-Independent Approach for the Study of the Spectral Shift Function and an Application to Stark Hamiltonians

In this paper we develop a time-independent approach for the study of the spectral shift function (SSF for short). We apply this method for the perturbed Stark Hamiltonian. We obtain a weak and a Weyl-type asymptotics with optimal remainder estimate of the SSF of the operator pair (P = P₀ + V(x), P₀ = ? h²Δ +x₁), x = (x₁,…, x_n) where V(x) ∈ 𝒞^∞(?ⁿ, ?) decays sufficiently fast at infinity, and h is a small positive parameter. Near a non-trapping energy λ, we give a pointwise asymptotic expansions in powers of h of the derivative of the SSF, and we compute explicitly the two leading terms.     

Asymptotics of resonances for a Schrödinger operator with vector values

We show that the resonance counting function for a Schrödinger operator in dimension one has an asymptotic expansion and calculate an explicit expression for the leading term in some situations.     

On eigenfunction expansions of piecewise smooth functions,associated with the power of the Schrödinger operator

The eigenfunction expansions of an integer power of the Schrödinger operator in an arbitrary two-dimensional domain are considered. The convergence of the corresponding expansions of piecewise smooth functions is proved. When the dimension of the domain is greater than two, then it is well known that this result is not valid any more.     

Gauge Optimization and Spectral Properties of Magnetic Schrödinger Operators

We establish new necessary and sufficient conditions for the discreteness of spectrum and strict positivity of magnetic Schrödinger operators with positive scalar potentials. We also derive two-sided estimates for the bottoms of the spectrum and essential spectrum. The main idea is to optimize the gauges of the magnetic field on cubes, thus reducing the quadratic form on the cubes to ones without magnetic field (but with appropriately adjusted scalar potentials).     

Stiff convergence of force-gradient operator splitting methods

We consider force-gradient, also called modified potential, operator splitting methods for problems with unbounded operators. We prove that force-gradient operator splitting schemes retain their classical orders of accuracy for linear time-dependent partial differential equations of parabolic and Schrödinger types, provided that the solution is sufficiently regular.     

Non-random perturbations of the Anderson Hamiltonian in the 1-D case

Recently (S. Molchanov and B. Vainberg, Non-random perturbations of the Anderson Hamiltonian, J. Spectral Theory 50 (2) (2011), pp. 179–195), two of the authors applied the Lieb method to the study of the negative spectrum for particular operators of the form H?=?H ₀???W. Here, H ₀ is the generator of the positive stochastic (or sub-stochastic) semigroup, W(x)?≥?0 and W(x)?→?0 as x?→?∞ on some phase space X. They used the general results in several ‘exotic’ situations, among them the Anderson Hamiltonian H ₀. In the 1-D case, the subject of this article, we will prove similar, but more precise results.     

Semiclassical Spectral Asymptotics for a Two-Dimensional Magnetic Schrödinger Operator II: The Case of Degenerate Wells

We continue our study of a magnetic Schrödinger operator on a two-dimensional compact Riemannian manifold in the case when the minimal value of the module of the magnetic field is strictly positive. We analyze the case when the magnetic field has degenerate magnetic wells. The main result of the paper is an asymptotics of the groundstate energy of the operator in the semiclassical limit. The upper bounds are improved in the case when we have a localization by a miniwell effect of lowest order. These results are applied to prove the existence of an arbitrary large number of spectral gaps in the semiclassical limit in the corresponding periodic setting.     

Strichartz and Smoothing Estimates for Dispersive Equations with Magnetic Potentials

We prove global smoothing and Strichartz estimates for the Schrödinger, wave, Klein–Gordon equations and for the massless and massive Dirac systems, perturbed with singular electromagnetic potentials. We impose a smallness condition on the magnetic part, while the electric part can be large. The decay and regularity assumptions on the coefficients are close to critical.