Spectral properties of Landau Hamiltonians with non-local potentials

Abstract

We consider the Landau Hamiltonian ${\mathit{H}}_0$ , self-adjoint in ${\mathit{L}}^2\mathrm{(}{\mathbb{R}}^2\mathrm{)}$ , whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues ${\mathrm{\Lambda }}_{\mathit{q}}$ , $\mathit{q}\in {\mathbb{Z}}_{+}$ . We perturb ${\mathit{H}}_0$ by a non-local potential written as a bounded pseudo-differential operator ${\mathrm{Op}}^{\mathrm{w}}\mathrm{(}\mathcal{V}\mathrm{)}$ with real-valued Weyl symbol $\mathcal{V}$ , such that ${\mathrm{Op}}^{\mathrm{w}}\mathrm{(}\mathcal{V}\mathrm{)}{\mathit{H}}_0^{-1}$ is compact. We study the spectral properties of the perturbed operator ${\mathit{H}}_{\mathcal{V}}={\mathit{H}}_0+{\mathrm{Op}}^{\mathrm{w}}\mathrm{(}\mathcal{V}\mathrm{)}$ . First, we construct symbols $\mathcal{V}$ , possessing a suitable symmetry, such that the operator ${\mathit{H}}_{\mathcal{V}}$ admits an explicit eigenbasis in ${\mathit{L}}^2\mathrm{(}{\mathbb{R}}^2\mathrm{)}$ , 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 ${\mathit{H}}_{\mathcal{V}}$ adjoining any given ${\mathrm{\Lambda }}_{\mathit{q}}$ . We find that the effective Hamiltonian in this context is the Toeplitz operator ${\mathcal{T}}_{\mathit{q}}\mathrm{(}\mathcal{V}\mathrm{)}={\mathit{p}}_{\mathit{q}}{\mathrm{Op}}^{\mathrm{w}}\mathrm{(}\mathcal{V}\mathrm{)}{\mathit{p}}_{\mathit{q}}$ , where ${\mathit{p}}_{\mathit{q}}$ is the orthogonal projection onto $Ker\mathrm{(}{\mathit{H}}_0-{\mathrm{\Lambda }}_{\mathit{q}}\mathit{I}\mathrm{)}$ , and investigate its spectral asymptotics.