10 research outputs found
On the families of orthogonal polynomials associated to the Razavy potential
We show that there are two different families of (weakly) orthogonal
polynomials associated to the quasi-exactly solvable Razavy potential V(x)=(\z
\cosh 2x-M)^2 (\z>0, ). One of these families encompasses the
four sets of orthogonal polynomials recently found by Khare and Mandal, while
the other one is new. These results are extended to the related periodic
potential U(x)=-(\z \cos 2x -M)^2, for which we also construct two different
families of weakly orthogonal polynomials. We prove that either of these two
families yields the ground state (when is odd) and the lowest lying gaps in
the energy spectrum of the latter periodic potential up to and including the
gap and having the same parity as . Moreover, we show
that the algebraic eigenfunctions obtained in this way are the well-known
finite solutions of the Whittaker--Hill (or Hill's three-term) periodic
differential equation. Thus, the foregoing results provide a Lie-algebraic
justification of the fact that the Whittaker--Hill equation (unlike, for
instance, Mathieu's equation) admits finite solutions.Comment: Typeset in LaTeX2e using amsmath, amssymb, epic, epsfig, float (24
pages, 1 figure
Short-range oscillators in power-series picture
A class of short-range potentials on the line is considered as an
asymptotically vanishing phenomenological alternative to the popular confining
polynomials. We propose a method which parallels the analytic Hill-Taylor
description of anharmonic oscillators and represents all our Jost solutions
non-numerically, in terms of certain infinite hypergeometric-like series. In
this way the well known solvable Rosen-Morse and scarf models are generalized.Comment: 23 pages, latex, submitted to J. Phys. A: Math. Ge
Nonlinear Oscillations in New Anharmonic Potential
The classical equation of motion for a particle moving in the new double-well potential V(x) = ½ V(A cosh ax - 1) is solved exactly for different values of the parameter A and the energy constant E. The solutions in various special cases are discussed
On quantum phase crossovers in finite systems
In this work we define a formal notion of a quantum phase crossover for certain Bethe ansatz solvable models. The approach we adopt exploits an exact mapping of the spectrum of a many-body integrable system, which admits an exact Bethe ansatz solution, into the quasi-exactly solvable spectrum of a one-body Schrodinger operator. Bifurcations of the minima for the potential of the Schrodinger operator determine the crossover couplings. By considering the behaviour of particular ground state correlation functions, these may be identified as quantum phase crossovers in the many-body integrable system with finite particle number. In this approach the existence of the quantum phase crossover is not dependent on the existence of a thermodynamic limit, rendering applications to finite systems feasible. We study two examples of bosonic Hamiltonians which admit second-order crossovers