Quantum models with spectrum generated by the flows of polynomial zeros

Abstract

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 \psi(\epsilon,z)=\sum_{n=0}^\infty \phi_n(\epsilon) z^n is the {\em generating function} for orthogonal polynomials \phi_n(\epsilon) of a discrete energy variable \epsilon, (ii) any Hamiltonian \hat{H}_b\in {\cal R}_p has nondegenerate purely point spectrum that corresponds to infinite discrete support of measure d\nu(x) in the orthogonality relation of the polynomials \phi_n, (iii) the support is determined exclusively by the points of discontinuity of \nu(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 \phi_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.