WIAS Preprint No. 2033, (2014)

On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case



Authors

  • Muminov, Mukhiddin
  • Neidhardt, Hagen
  • Rasulov, Tulkin

2010 Mathematics Subject Classification

  • 81Q10 35P20 47N50

Keywords

  • spin-boson Hamiltonian, block operator matrix, bosonic Fock space, annihilation and creation operators, Birman-Schwinger principle, essential spectrum, point and discrete spectrum

Abstract

A lattice model of radiative decay (so-called spin-boson model) of a two level atom and at most two photons is considered. The location of the essential spectrum is described. For any coupling constant the finiteness of the number of eigenvalues below the bottom of its essential spectrum is proved. The results are obtained by considering a more general model H for which the lower bound of its essential spectrum is estimated. Conditions which guarantee the finiteness of the number of eigenvalues of H below the bottom of its essential spectrum are found. It is shown that the discrete spectrum might be infinite if the parameter functions are chosen in a special form.

Appeared in

  • J. Math. Phys., 56 (2015) pp. 053507/1--053507/24.

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