34 research outputs found
Exact solutions of an elliptic Calogero--Sutherland model
A model describing N particles on a line interacting pairwise via an elliptic
function potential in the presence of an external field is partially solved in
the quantum case in a totally algebraic way. As an example, the ground state
and the lowest excitations are calculated explicitly for N=2.Comment: 4 pages, 3 figures, typeset with RevTeX 4b3 and AMS-LaTe
Families of superintegrable Hamiltonians constructed from exceptional polynomials
We introduce a family of exactly-solvable two-dimensional Hamiltonians whose
wave functions are given in terms of Laguerre and exceptional Jacobi
polynomials. The Hamiltonians contain purely quantum terms which vanish in the
classical limit leaving only a previously known family of superintegrable
systems. Additional, higher-order integrals of motion are constructed from
ladder operators for the considered orthogonal polynomials proving the quantum
system to be superintegrable
Multiple algebraisations of an elliptic Calogero-Sutherland model
Recently, Gomez-Ullate et al. (1) have studied a particular N-particle
quantum problem with an elliptic function potential supplemented by an external
field. They have shown that the Hamiltonian operator preserves a finite
dimensional space of functions and as such is quasi exactly solvable (QES). In
this paper we show that other types of invariant function spaces exist, which
are in close relation to the algebraic properties of the elliptic functions.
Accordingly, series of new algebraic eigenfunctions can be constructed.Comment: 9 Revtex pages, 3 PS-figures; Summary, abstract and conclusions
extende
Connection between the Green functions of the supersymmetric pair of Dirac Hamiltonians
The Sukumar theorem about the connection between the Green functions of the
supersymmetric pair of the Schr\"odinger Hamiltonians is generalized to the
case of the supersymmetric pair of the Dirac Hamiltonians.Comment: 12 pages,Latex, no figure
Understanding complex dynamics by means of an associated Riemann surface
We provide an example of how the complex dynamics of a recently introduced
model can be understood via a detailed analysis of its associated Riemann
surface. Thanks to this geometric description an explicit formula for the
period of the orbits can be derived, which is shown to depend on the initial
data and the continued fraction expansion of a simple ratio of the coupling
constants of the problem. For rational values of this ratio and generic values
of the initial data, all orbits are periodic and the system is isochronous. For
irrational values of the ratio, there exist periodic and quasi-periodic orbits
for different initial data. Moreover, the dependence of the period on the
initial data shows a rich behavior and initial data can always be found such
the period is arbitrarily high.Comment: 25 pages, 14 figures, typed in AMS-LaTe
PT-Symmetric, Quasi-Exactly Solvable matrix Hamiltonians
Matrix quasi exactly solvable operators are considered and new conditions are
determined to test whether a matrix differential operator possesses one or
several finite dimensional invariant vector spaces. New examples of -matrix quasi exactly solvable operators are constructed with the emphasis
set on PT-symmetric Hamiltonians.Comment: 14 pages, 1 figure, one equation corrected, results adde