Exact solutions of a new elliptic Calogero–Sutherland model

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 Calogero [8], and consists of a system of identical nonrelativistic particles interacting pairwise through an inverse-square potential v(r) = r −2 , so that the Hamiltonian is

Quasi-exact solvability beyond the sl(2) algebraization

“…Such a transformation always exists in one spatial dimension, but in more dimensions the equivalence problem remains open [10,11]. This has been a long standing obstacle to classify multidimensional quasi-exactly solvable Hamiltonians, where only a few families are known [12,13]. One important fact lies at the core of the Lie-algebraic method: it is ensured by Burnside's classical theorem that every differential operator which leaves the space P n (z) invariant belongs to the enveloping algebra U(sl(2)), since P n is an irreducible module for the sl(2) action.…”

Solvability of the Hamiltonians Related to Exceptional Root Spaces: Rational Case

“…Even for elliptic A 1 and BC n models the elliptic Weyl invariants allow to get the polynomials coefficients in front of derivatives (see [10] and [11] for details) rational models related to the exceptional root spaces. We introduce a general formalism which allows to study all these models on equal footing (Section 2).…”

“…(3.13), (4.9), (5.12), (6.11)). Hence P (E 8 ) = P (1,3,5,5,7,7,9,11) . The characteristic vector does not coincide with the highest root f highest root = (2, 2, 3, 3, 4, 4, 5, 6) suggested by Kac.…”

“…The most general polynomial transformation which preserves each linear space P (1,3,5 E 8 and moreover preserving the flag P (1,3,5,5,7,7,9,11) . Hence, the parameters can be chosen by following our convenience.…”

“…Hence, the parameters can be chosen by following our convenience. In a simply-minded way we set all of them equal to zero, It can be found one-parametric algebra of differential operators (in eight variables) for which P (1,3,5,5,7,7,9,11) n (see (7.8)) is a finite-dimensional irreducible representation space. Furthermore, the finite-dimensional representation spaces appear for different integer values of the algebra parameter.…”