THE SPECTRUM OF LIOUVILLE OPERATORS AND MULTIPARTICLE HAMILTONIANS ASSOCIATED TO ONE-PARTICLE HAMILTONIANS WITH SINGULAR CONTINUOUS-SPECTRUM

We study the structure of spectrum of the Liouville operator H- = H X I - I X H and the two-particle Hamiltonian H+ = H X I + I X H in some model situations when the corresponding one-particle Hamiltonian H has singular continuous spectrum, a Hamiltonian H with singular continuou s spectrum of Hausdorff dimension one is constructed such that the abs olutely continuous spectrum of the operators H- and H+ is empty, On th e other hand, we prove the existence of a Hamiltonian H with singular continuous spectrum of Hausdorff dimension zero such that the operator s H- and H+ have nonempty absolutely continuous spectrum. Thus the Hau sdorff dimension of the support cannot serve as characteristic of the singular measure of a one-body Hamiltonian that determines the spectra l type of the corresponding Liouvillians or two-body Hamiltonians. (C) 1997 American Institute of Physics.