Abstract
A class of purely bosonic models is characterized having the following properties in a Hilbert space of analytic functions: (i) wave function is the generating function for orthogonal polynomials of a discrete energy variable ϵ, (ii) any Hamiltonian has nondegenerate purely point spectrum that corresponds to infinite discrete support of measure in the orthogonality relation of the polynomials , (iii) the support is determined exclusively by the points of discontinuity of , (iv) the spectrum of can be numerically determined as fixed points of monotonic flows of the zeros of orthogonal polynomials , (v) one can compute practically an unlimited number of energy levels (e.g. in double precision). If a model of is exactly solvable, its spectrum can only assume one of four qualitatively different types. The results are applied to spin-boson quantum models that are, at least partially, diagonalizable and have at least single one-dimensional irreducible component in the spin subspace. Examples include the Rabi model and its various generalizations.
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