62 research outputs found
Quasi-exact solvability beyond the SL(2) algebraization
We present evidence to suggest that the study of one dimensional
quasi-exactly solvable (QES) models in quantum mechanics should be extended
beyond the usual \sla(2) approach. The motivation is twofold: We first show
that certain quasi-exactly solvable potentials constructed with the \sla(2)
Lie algebraic method allow for a new larger portion of the spectrum to be
obtained algebraically. This is done via another algebraization in which the
algebraic hamiltonian cannot be expressed as a polynomial in the generators of
\sla(2). We then show an example of a new quasi-exactly solvable potential
which cannot be obtained within the Lie-algebraic approach.Comment: Submitted to the proceedings of the 2005 Dubna workshop on
superintegrabilit
Exceptional orthogonal polynomials and the Darboux transformation
We adapt the notion of the Darboux transformation to the context of
polynomial Sturm-Liouville problems. As an application, we characterize the
recently described Laguerre polynomials in terms of an isospectral
Darboux transformation. We also show that the shape-invariance of these new
polynomial families is a direct consequence of the permutability property of
the Darboux-Crum transformation.Comment: corrected abstract, added references, minor correction
Low-lying spectra in anharmonic three-body oscillators with a strong short-range repulsion
Three-body Schroedinger equation is studied in one dimension. Its two-body
interactions are assumed composed of the long-range attraction (dominated by
the L-th-power potential) in superposition with a short-range repulsion
(dominated by the (-K)-th-power core) plus further subdominant power-law
components if necessary. This unsolvable and non-separable generalization of
Calogero model (which is a separable and solvable exception at L = K = 2) is
presented in polar Jacobi coordinates. We derive a set of trigonometric
identities for the potentials which generalizes the well known K=2 identity of
Calogero to all integers. This enables us to write down the related partial
differential Schroedinger equation in an amazingly compact form. As a
consequence, we are able to show that all these models become separable and
solvable in the limit of strong repulsion.Comment: 18 pages plus 6 pages of appendices with new auxiliary identitie
Exceptional orthogonal polynomials and new exactly solvable potentials in quantum mechanics
In recent years, one of the most interesting developments in quantum
mechanics has been the construction of new exactly solvable potentials
connected with the appearance of families of exceptional orthogonal polynomials
(EOP) in mathematical physics. In contrast with families of (Jacobi, Laguerre
and Hermite) classical orthogonal polynomials, which start with a constant, the
EOP families begin with some polynomial of degree greater than or equal to one,
but still form complete, orthogonal sets with respect to some positive-definite
measure. We show how they may appear in the bound-state wavefunctions of some
rational extensions of well-known exactly solvable quantum potentials. Such
rational extensions are most easily constructed in the framework of
supersymmetric quantum mechanics (SUSYQM), where they give rise to a new class
of translationally shape invariant potentials. We review the most recent
results in this field, which use higher-order SUSYQM. We also comment on some
recent re-examinations of the shape invariance condition, which are independent
of the EOP construction problem.Comment: 21 pages, no figure; communication at the Symposium Symmetries in
Science XV, July 31-August 5, 2011, Bregenz, Austri
Quasi-Exact Solvability and the direct approach to invariant subspaces
We propose a more direct approach to constructing differential operators that
preserve polynomial subspaces than the one based on considering elements of the
enveloping algebra of sl(2). This approach is used here to construct new
exactly solvable and quasi-exactly solvable quantum Hamiltonians on the line
which are not Lie-algebraic. It is also applied to generate potentials with
multiple algebraic sectors. We discuss two illustrative examples of these two
applications: an interesting generalization of the Lam\'e potential which
posses four algebraic sectors, and a quasi-exactly solvable deformation of the
Morse potential which is not Lie-algebraic.Comment: 17 pages, 3 figure
Exchange operator formalism for N-body spin models with near-neighbors interactions
We present a detailed analysis of the spin models with near-neighbors
interactions constructed in our previous paper [Phys. Lett. B 605 (2005) 214]
by a suitable generalization of the exchange operator formalism. We provide a
complete description of a certain flag of finite-dimensional spaces of spin
functions preserved by the Hamiltonian of each model. By explicitly
diagonalizing the Hamiltonian in the latter spaces, we compute several infinite
families of eigenfunctions of the above models in closed form in terms of
generalized Laguerre and Jacobi polynomials.Comment: RevTeX, 31 pages, no figures; important additional conten
Exact solutions of a new elliptic CalogeroSutherland model, Phys
Abstract A quantum Hamiltonian describing N particles on a line interacting pairwise via an elliptic function potential in the presence of an external field is introduced. For a discrete set of values of the strength of the external potential, it is shown that a finite number of eigenfunctions and eigenvalues of the model can be exactly computed in an algebraic way. 2001 Elsevier Science B.V. All rights reserved. PACS: 03.65.Fd; 71.10.Pm; 11.10.Lm It is well known that the class of exactly solvable problems does not include most physical problems. The development of computer science in the last decades has made possible the use of numerical methods to approximate exact solutions in a wide variety of situations. Yet, the study of exactly solvable models still deserves attention, not only because the knowledge of exact solutions can be used to test approximate methods, but also in its own right, due to the simplicity and mathematical beauty of the models, and the wide range of connections with other fields of physical and mathematical research. This is illustrated by the renewed interest in the Calogero-Sutherland (CS) models of interacting particles in one dimension, which have been recently applied to many different fields such us quantum spin chains with long range interaction The first example of a non-trivial integrable quantum many-body problem was found by Caloger
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
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