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 \psi (ɛ ,z)=\sum _n=0^∞ {{φ }_n}(ɛ ){{z}ⁿ} is the generating function for orthogonal polynomials {{φ }_n}(ɛ ) of a discrete energy variable ɛ, (ii) any Hamiltonian {{\hat{H}}_b}\in R 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 R can be numerically determined as fixed points of monotonic flows of the zeros of orthogonal polynomials {{φ }_n}(ɛ ), (v) one can compute practically an unlimited number of energy levels (e.g. {{2}⁵³} 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.