Quantum models with spectrum generated by the flows of polynomial zeros

A class {\cal R}_p of purely bosonic models is characterized having the following properties in the Bargmann Hilbert space of analytic functions: (i) wave function ψ(ε,z)=\sum_{n=0}^\infty ϕ_n(ε) z^n is the {\em generating function} for orthogonal polynomials ϕ_n(ε) of a discrete energy variable ε, (ii) any Hamiltonian \hat{H}_b\in {\cal R}_p has nondegenerate purely point spectrum that corresponds to infinite discrete support of measure dν(x) in the orthogonality relation of the polynomials ϕ_n, (iii) the support is determined exclusively by the points of discontinuity of ν(x), (iv) the spectrum of \hat{H}_b\in {\cal R}_p can be numerically determined as fixed points of monotonic flows of the zeros of orthogonal polynomials ϕ_n(\upepsilon), (v) one can compute practically an unlimited number of energy levels (e.g. 2^{53} in double precision). If a model of {\cal R}_p 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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