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 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.
- Publication:
-
Journal of Physics A Mathematical General
- Pub Date:
- December 2014
- DOI:
- 10.1088/1751-8113/47/49/495204
- arXiv:
- arXiv:1403.3773
- Bibcode:
- 2014JPhA...47W5204M
- Keywords:
-
- Mathematical Physics;
- Quantum Physics
- E-Print:
- 11 pages, RevTex - introduction and a part of Sec. II redrafted to take into account Haydock's work