2019/01/14 by Cárdenas, Esteban, Raikov, Georgi, Tejeda, Ignacio
#35P20 #81Q10 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1901.04370
We consider the Landau Hamiltonian H0, self-adjoint in L2(\mathbb R2), whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues Λq, q ∈ \mathbb Z+. We perturb H0 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) H0-1 is compact. We study the spectral properties of the perturbed operator H\mathcal V = H0 + \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 L2(\mathbb R2), 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 Λq. We find that the effective Hamiltonian in this context is the Toeplitz operator \mathcal Tq(\mathcal V) = pq \rm Op\rm w(\mathcal V) pq, where pq is the orthogonal projection onto \rm Ker(H0 - Λq I), and investigate its spectral asymptotics.