The spectrum of Liouville operators and multiparticle Hamiltonians associated to one-particle Hamiltonians with singular continuous spectrum

Abstract

We study the structure of spectrum of the Liouville operator H−=H⊗I−I⊗H and the two-particle Hamiltonian H+=H⊗I+I⊗H in some model situations when the corresponding one-particle Hamiltonian H has singular continuous spectrum. A Hamiltonian H with singular continuous spectrum of Hausdorff dimension one is constructed such that the absolutely continuous spectrum of the operators H− and H+ is empty. On the other hand, we prove the existence of a Hamiltonian H with singular continuous spectrum of Hausdorff dimension zero such that the operators H− and H+ have nonempty absolutely continuous spectrum. Thus the Hausdorff dimension of the support cannot serve as characteristic of the singular measure of a one-body Hamiltonian that determines the spectral type of the corresponding Liouvillians or two-body Hamiltonians.