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A class {\cal R}_p of purely bosonic models is characterized having thefollowing properties in the Bargmann Hilbert space of analytic functions: (i)wave function \psi(\epsilon,z)=\sum_{n=0}^\infty \phi_n(\epsilon) z^n is the{\em generating function} for orthogonal polynomials \phi_n(\epsilon) of adiscrete energy variable \epsilon, (ii) any Hamiltonian \hat{H}_b\in {\cal R}_phas nondegenerate purely point spectrum that corresponds to infinite discretesupport of measure d\nu(x) in the orthogonality relation of the polynomials\phi_n, (iii) the support is determined exclusively by the points ofdiscontinuity of \nu(x), (iv) the spectrum of \hat{H}_b\in {\cal R}_p can benumerically determined as fixed points of monotonic flows of the zeros oforthogonal polynomials \phi_n(\upepsilon), (v) one can compute practically anunlimited number of energy levels (e.g. 2^{53} in double precision). If a modelof {\cal R}_p is exactly solvable, its spectrum can only assume one of fourqualitatively different types. The results are applied to spin-boson quantummodels that are, at least partially, diagonalizable and have at least singleone-dimensional irreducible component in the spin subspace. Examples includethe Rabi model and its various generalizations.

Quantum models with spectrum generated by the flows of polynomial zeros