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

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.

Analytic description of the essential spectrum of a family of 3×3 operator matrices

“…The family of operator matrices of this form play a key role for the study of the energy operator of the spin-boson model with two bosons on the torus. In fact, the latter is a 6 × 6 operator matrix which is unitary equivalent to a 2×2 block diagonal operator with two copies of a particular case of H(K) on the diagonal, see [14]. Consequently, the essential spectrum and finiteness of discrete eigenvalues of the spin-boson Hamiltonian are determined by the corresponding spectral information on the operator matrix H(K) in (2.1).…”

“…In statistical physics [11], solid-state physics [12] and the theory of quantum fields [5], one considers systems, where the number of quasi-particles is not fixed. Their number can be unbounded as in the case of full spin-boson models (infinite operator matrix) [7] or bounded as in the case of "truncated" spin-boson models (finite operator matrix) [6,11,14,16,17,24]. Often, the number of particles can be arbitrary large as in cases involving photons, in other cases, such as scattering of spin waves on defects, scattering massive particles and chemical reactions, there are only participants at any given time, though their number can be change.…”

Threshold analysis for a family of 2×2 operator matrices

“…One special class of operator matrices are Hamiltonians associated with the systems of non-conserved number of quasi-particles on a lattice. In such systems the number of particles can be unbounded as in the case of spin-boson models [5,18] or bounded as in the case of "truncated" spin-boson models [7,9,11,12]. They arise, for example, in the theory of solid-state physics [8], quantum field theory [4] and statistical physics [6,7].…”

Spectral analysis of the spin-boson Hamiltonian with two bosons for arbitrary coupling and bounded dispersion relation

“…For the photon case, spectral and scattering properties of the full spin-boson Hamiltonian as well as of its finite photon approximations have been investigated extensively and as a by-product sophisticated 1 techniques have been developed. The corresponding literature is enormous and we limit ourselves to citing [47,50,23,36,20,7,12,44,10,6,16,22,19,5,1,21,15,31,9,13] (and the related work [38,37,39,40]). In these studies, the dispersion of the free field is taken to be either the relativistic dispersion ω(k) = √ k 2 + m 2 or its limiting cases ω(k) = |k|, ω(k) = k 2 2m , and sometimes as a general unbounded and almost everywhere continuous function preserving all the features of these physical photon dispersion relations.…”