On the Spectral Properties of the Landau Hamiltonian Perturbed by a moderately Decaying Magnetic Field

Discrete spectrum distribution of the Landau Operator Perturbed by an Expanding Electric Potential

Under a perturbation by a decaying electric potential, the Landau Hamiltonian acquires some discrete eigenvalues between the Landau levels. We study the perturbation by an "expanding" electric potential $V(t^{-1}x)$, $t>0$, and derive a quasi-classical formula for the counting function of the discrete spectrum as $t\to \infty$.

Eigenvalue asymptotics of perturbed periodic Dirac systems in the slow-decay limit

A perturbation decaying to 0 at infinity and not too irregular at 0 introduces at most a discrete set of eigenvalues into the spectral gaps of a one-dimensional Dirac operator on the half-line. We show that the number of these eigenvalues in a compact subset of a gap in the essential spectrum is given by a quasi-semiclassical asymptotic formula in the slow-decay limit, which for power-decaying perturbations is equivalent to the large-coupling limit. This asymptotic behaviour elucidates the origin of the dense point spectrum observed in spherically symmetric, radially periodic three-dimensional…

受扰周期薛定谔算子本征函数的精确衰减率

Sharp decay rate for eigenfunctions of perturbed periodic Schrödinger operators

本文研究$\u2124^d$上受扰周期薛定谔算子的唯一性，建立了方程仅存平凡解的定量条件，得出杂质超指数衰减时算子无嵌入本征值等结论，拓展了相关研究。

Lifshitz tails for alloy type models in a constant magnetic field

In this note, we study Lifshitz tails for a 2D Landau Hamiltonian perturbed by a random alloy-type potential constructed with single site potentials decaying at least at a Gaussian speed. We prove that, if the Landau level stays preserved as a band edge for the perturbed Hamiltonian, at the Landau levels, the integrated density of states has a Lifshitz behavior of the type $e^{-\log^2|E-2bq|}$.

Non-random perturbations of the Anderson Hamiltonian

The Anderson Hamiltonian $H_0=-Δ+V(x,ω)$ is considered, where $V$ is a random potential of Bernoulli type. The operator $H_0$ is perturbed by a non-random, continuous potential $-w(x) \leq 0$, decaying at infinity. It will be shown that the borderline between finitely, and infinitely many negative eigenvalues of the perturbed operator, is achieved with a decay of the potential $-w(x)$ as $O(\ln^{-2/d} |x|)$.

Eigenvalues behaviours for self-adjoint Pauli operators with unsigned perturbations and admissible magnetic fields

We investigate the behaviour of the eigenvalues of two-dimensional Pauli operators with nonconstant magnetic fields perturbed by a sign-indefinite decaying electric potential V. We prove new eigenvalues asymptotics.