Semiboundedness and Asimptotic Behavior of Spectrum of three Quantum Hamiltonians

For three Hamiltonians of quantum optics conservation laws depending on particle number operators, boundedness from below and asimptotic behavior of the spectrum are investigated. For generalized Hamiltonian of Raman scattering sufficient conditions of boundedness from below and absence the same in dependence on type of quantum algebra, which describes quantum excitations of medium, are proved. For two versions of Quantum Nonlinear Schrodinger Equation estimates of eigenvalues are proved.