Quantum models with spectrum generated by the flows of polynomial zeros

A class R of purely bosonic models is characterized having the following properties in a Hilbert space of analytic functions: (i) wave function ψ(ϵ,z)=∑n=0∞ ϕn(ϵ)zn is the generating function for orthogonal polynomials ϕn(ϵ) of a discrete energy variable ϵ, (ii) any Hamiltonian H-hat b∈R has nondegenerate purely point spectrum that corresponds to infinite discrete support of measure dν(x) in the orthogonality relation of the polynomials ϕn, (iii) the support is determined exclusively by the points of discontinuity of ν(x), (iv) the spectrum of H-hat b∈R can be numerically determined as fixed points of monotonic flows of the zeros of orthogonal polynomials ϕn(ϵ), (v) one can compute practically an unlimited number of energy levels (e.g. 253 in double precision). If a model of R 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. (paper)