Artemio Gonzalez-Lopez

Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators

We completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi-exactly solvable Schrödinger operator on the line. Methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasi-exactly solvable Hamiltonians and explicit normalizability conditions in general coordinate systems.

Global properties of the spectrum of the Haldane-Shastry spin chain

Departamento de F´isica Te´orica II, Universidad Complutense, 28040 Madrid, Spain(Dated: May 23, 2005; revised September 1, 2005)We derive an exact expression for the partition function of the su(m) Haldane–Shastry spin chain,which we use to study the density of levels and the distribution of the spacing between consecutivelevels. Our computations show that when the number of sites N is large enough the level density isGaussian to a very high degree of approximation. More surprisingly, we also find that the nearest-neighbor spacing distribution is not Poissonian, so that this model departs from the typical behaviorfor an integrable system. We show that the cumulative spacing distribution of the model can bewell approximated by a simple functional law involving only three parameters.

Symmetry and integrability by quadratures of ordinary differential equations

Abstract In this paper, the connection between point symmetries and the integrability by quadratures of second-order ordinary differential equations is discussed. An example is given of a family of second-order ordinary differential equations integrable by quadratures whose point symmetry group is, nevertheless, trivial. This refutes the widespread belief that the existence of nontrivial point symmetries is a necessary condition for the integrability by quadratures of ordinary differential equations. The significance of dynamical (versus point) symmetries in this field is illustrated with a few recent results.

AN-Type Dunkl Operators and New Spin Calogero–Sutherland Models

Abstract: A new family of AN-type Dunkl operators preserving a polynomial subspace of finite dimension is constructed. Using a general quadratic combination of these operators and the usual Dunkl operators, several new families of exactly and quasi-exactly solvable quantum spin Calogero–Sutherland models are obtained. These include, in particular, three families of quasi-exactly solvable elliptic spin Hamiltonians.

Quasi‐exactly solvable potentials on the line and orthogonal polynomials

In this paper we show that a quasi‐exactly solvable (normalizable or periodic) one‐dimensional Hamiltonian satisfying very mild conditions defines a family of weakly orthogonal polynomials which obey a three‐term recursion relation. In particular, we prove that (normalizable) exactly solvable one‐dimensional systems are characterized by the fact that their associated polynomials satisfy a two‐term recursion relation. We study the properties of the family of weakly orthogonal polynomials defined by an arbitrary one‐dimensional quasi‐exactly solvable Hamiltonian, showing in particular that its associated Stieltjes measure is supported on a finite set. From this we deduce that the corresponding moment problem is determined, and that the kth moment grows like the kth power of a constant as k tends to infinity. We also show that the moments satisfy a constant coefficient linear difference equation, and that this property actually characterizes weakly orthogonal polynomial systems.

New quasi-exactly solvable Hamiltonians in two dimensions

Quasi-exactly solvable Schrödinger operators have the remarkable property that a part of their spectrum can be computed by algebraic methods. Such operators lie in the enveloping algebra of a finite-dimensional Lie algebra of first order differential operators—the “hidden symmetry algebra”. In this paper we develop some general techniques for constructing quasi-exactly solvable operators. Our methods are applied to provide a wide variety of new explicit two-dimensional examples (on both flat and curved spaces) of quasi-exactly solvable Hamiltonians, corresponding to both semisimple and more general classes of Lie algebras.

Newtonian systems of differential equations, integrable via quadratures, with trivial group of point symmetries

Abstract Examples are given of systems of second order ordinary differential equations integrable via quadratures with trivial symmetry group of local point transformations.

New spin Calogero–Sutherland models related to BN-type Dunkl operators

Abstract We construct several new families of exactly and quasi-exactly solvable BCN-type Calogero–Sutherland models with internal degrees of freedom. Our approach is based on the introduction of a new family of Dunkl operators of BN type which, together with the original BN-type Dunkl operators, are shown to preserve certain polynomial subspaces of finite dimension. We prove that a wide class of quadratic combinations involving these three sets of Dunkl operators always yields a spin Calogero–Sutherland model, which is (quasi-)exactly solvable by construction. We show that all the spin Calogero–Sutherland models obtainable within this framework can be expressed in a unified way in terms of a Weierstrass ℘ function with suitable half-periods. This provides a natural spin counterpart of the well-known general formula for a scalar completely integrable potential of BCN type due to Olshanetsky and Perelomov. As an illustration of our method, we exactly compute several energy levels and their corresponding wavefunctions of an elliptic quasi-exactly solvable potential for two and three particles of spin 1/2.

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.

Haldane-Shastry spin chains of BCN type

We introduce four types of SU(2M + 1) spin chains which can be regarded as the BCN versions of the celebrated Haldane–Shastry chain. These chains depend on two free parameters and, unlike the original Haldane–Shastry chain, their sites need not be equally spaced. We prove that all four chains are solvable by deriving an exact expression for their partition function using Polychronakos’s “freezing trick”. From this expression we deduce several properties of the spectrum, and advance a number of conjectures that hold for a wide range of values of the spin M and the number of particles. In particular, we conjecture that the level density is Gaussian, and provide a heuristic derivation of general formulas for the mean and the standard deviation of the energy.