Spectral properties of Landau Hamiltonians with non-local potentials

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 ${\rm Op}^{\rm w}({\mathcal V})$ with real-valued Weyl symbol ${\mathcal V}$, such that ${\rm Op}^{\rm w}({\mathcal V}) H_0^{-1}$ is compact. We study the spectral properties of the perturbed operator $H_{{\mathcal V}} = H_0 + {\rm Op}^{\rm w}({\mathcal V})$. 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 ${\mathcal T}_q({\mathcal V}) = p_q {\rm Op}^{\rm w}({\mathcal V}) p_q$, where $p_q$ is the orthogonal projection onto ${\rm Ker}(H_0 - \Lambda_q I)$, and investigate its spectral asymptotics.