Affiliations: [a] Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago de Chile, Chile. E-mails: [email protected], [email protected], [email protected] | [b] Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., bl. 8, 1113 Sofia, Bulgaria
Abstract: We consider the Landau Hamiltonian H0, self-adjoint in L2(R2), whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues Λq, q∈Z+. We perturb H0 by a non-local potential written as a bounded pseudo-differential operator Opw(V) with real-valued Weyl symbol V, such that Opw(V)H0−1 is compact. We study the spectral properties of the perturbed operator HV=H0+Opw(V). First, we construct symbols V, possessing a suitable symmetry, such that the operator HV admits an explicit eigenbasis in L2(R2), and calculate the corresponding eigenvalues. Moreover, for V which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of HV adjoining any given Λq. We find that the effective Hamiltonian in this context is the Toeplitz operator Tq(V)=pqOpw(V)pq, where pq is the orthogonal projection onto Ker(H0−ΛqI), and investigate its spectral asymptotics.
