Quantum models with spectrum generated by the flows of polynomial zeros
Abstract
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 spinboson quantum models that are, at least partially, diagonalizable and have at least single onedimensional irreducible component in the spin subspace. Examples include the Rabi model and its various generalizations.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 December 2014
 DOI:
 10.1088/17518113/47/49/495204
 arXiv:
 arXiv:1403.3773
 Bibcode:
 2014JPhA...47W5204M
 Keywords:

 Mathematical Physics;
 Quantum Physics
 EPrint:
 11 pages, RevTex  introduction and a part of Sec. II redrafted to take into account Haydock's work