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Journal of Global Research in Mathematical Archives(JGRMA)

BOUND AND GROUND STATES OF A SPIN-BOSON MODEL WITH AT MOST ONE PHOTON: NON-INTEGER LATTICE CASE

Nigora Nasirova

Published Sep 29, 2019• 0 citations
OpenAlex

Abstract

A non-integer lattice model of radiative decay (so-called spin-boson model) of a two level atom and at most one photon is considered. The number and location of the eigenvalues (bound and ground states) of this model are studied.

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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.
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A class of bound-state problems which represent the interaction between a system of N two-level atoms and an electromagnetic radiation via an m-photon process is studied, with m = 1; 2; 3;.... We also evaluate some nonlinear effects usually related with the polarisability of the medium and with the dependence on the intensity of the radiation field on the matter-radiation interaction. We obtain exact values for the eigenstates and eigenvalues for all values of total angular momentum of the system and all possible number m of photons involved in the interaction. We give explicit analytic expressions for small numbers of atoms and discuss some aspects of the composition of the eigenstates and of the spectra obtained in these cases.
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