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 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.

11 pages, RevTex - introduction and a part of Sec. II redrafted to take into account Haydock's work; 3 references added

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Cited by in corpus (5)

• The quantum Rabi model: solution and dynamics
• Generalized Rabi models: diagonalization in the spin subspace and differential operators of Dunkl type
• Constraint polynomial approach -- an alternative to the functional Bethe Ansatz method?
• A unified treatment of polynomial sectors and constraint polynomials of the Rabi models
• On uniqueness of Heine-Stieltjes polynomials for second order finite-difference equations