Local spectral asymptotics for metric perturbations of the Landau Hamiltonian

Abstract

We consider metric perturbations of the Landau Hamiltonian. We investigate the asymptotic behaviour of the discrete spectrum of the perturbed operator near the Landau levels, for perturbations of compact support, and of exponential or powerlike decay at infinity.

The fate of Landau levels under $δ$-interactions

“…Singular Toeplitz operators as in (2.11) play an important role in modern operator theory and are also of independent interest. They were already considered in [2], and in connection with magnetic Laplacians with different types of boundary conditions these types of operators appear in [20,21,37]; we also refer the reader to [3,10,11,13,16,32,36,41,42] for some other recent related works in this context. Note that the operator T q (υδ Γ ) corresponds to the quadratic form (2.12)…”

Eigenvalue Counting Function for Robin Laplacians on Conical Domains

“…These Schrödinger operators are also used as effective Hamiltonians for more complex problems, see [1,14] for δ-interaction supported on conical domains, [8,9] for Dirichlet Laplacian in a conical layer, [2,26] for twisted waves guides, [3] for magnetic Hamiltonian on a strip, [4] for magnetic Hamiltonian on the half plan with Neumann condition, [16] for a model of quantum Hall effect, or [11] for Schrödinger operators with degenerate kinetic terms. Another family of results concerns perturbations of magnetic hamiltonians where the accumulation is governed by some Toeplitz operators (see [10,[21][22][23][24]) or by Anti-Wick pseudo-differential operators as in [4,5,15]. The recent paper [13] studied the Weyl asymptotics for the negative discrete spectrum of Q Ω α for bounded smooth Ω and large α, but the nature of the asymptotics is completely different.…”

Threshold Singularities of the Spectral Shift Function for Geometric Perturbations of Magnetic Hamiltonians

“…Unitary equivalence of M 6,q and M 7,q . In our proof of the unitary equivalence of the operators M 6,q and M 7,q (see Subsection 5.4) we will follow closely the argument of the proof of [25,Proposition 4.1]. Set (A.1)…”