Spectral properties of Landau Hamiltonians with non-local potentials

Abstract

We consider the Landau Hamiltonian $H_0$, self-adjoint in $L^2({\mathbb{R}}^2)$, whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues ${\Lambda }_q$, $q\in {\mathbb{Z}}_{+}$. We perturb $H_0$ by a non-local potential written as a bounded pseudo-differential operator ${\mathrm{Op}}^{\mathrm{w}}(\mathcal{V})$ with real-valued Weyl symbol $\mathcal{V}$, such that ${\mathrm{Op}}^{\mathrm{w}}(\mathcal{V})H_0^{-1}$ is compact. We study the spectral properties of the perturbed operator $H_{\mathcal{V}}=H_0+{\mathrm{Op}}^{\mathrm{w}}(\mathcal{V})$. First, we construct symbols $\mathcal{V}$, possessing a suitable symmetry, such that the operator $H_{\mathcal{V}}$ admits an explicit eigenbasis in $L^2({\mathbb{R}}^2)$, and calculate the corresponding eigenvalues. Moreover, for $\mathcal{V}$ which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of $H_{\mathcal{V}}$ adjoining any given ${\Lambda }_q$. We find that the effective Hamiltonian in this context is the Toeplitz operator ${\mathcal{T}}_q(\mathcal{V})=p_q{\mathrm{Op}}^{\mathrm{w}}(\mathcal{V})p_q$, where $p_q$ is the orthogonal projection onto $\mathrm{Ker}(H_0-{\Lambda }_{q}I)$, and investigate its spectral asymptotics.

31 pages, a co-author added, introduction expanded, Proposition 5.6 and Corollary 5.7 added, Theorem 5.2 proved under more general hypotheses