We introduce four types of SU(2M + 1) spin chains which can be regarded as the BC N 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.

We compute the partition function of the su͑m͒ Polychronakos-Frahm spin chain of BC N type by means of the freezing trick. We use this partition function to study several statistical properties of the spectrum, which turn out to be analogous to those of other spin chains of Haldane-Shastry type. In particular, we find that when the number of particles is sufficiently large the level density follows a Gaussian distribution with great accuracy. We also show that the distribution of ͑normalized͒ spacings between consecutive levels is of neither Poisson nor Wigner type but is qualitatively similar to that of the original Haldane-Shastry spin chain. This suggests that spin chains of Haldane-Shastry type are exceptional integrable models since they do not satisfy a well-known conjecture of Berry and Tabor, according to which the spacings distribution of a generic integrable system should be Poissonian. We derive a simple analytic expression for the cumulative spacings distribution of the BC N -type Polychronakos-Frahm chain using only a few essential properties of its spectrum such as the Gaussian character of the level density and the fact that the energy levels are equally spaced. This expression is shown to be in excellent agreement with the numerical data.

We introduce four types of SU(2M + 1) spin chains which can be regarded as the BC N 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.

Surfaces immersed in Lie algebras can be characterized by the so called fundamental forms. The coefficients of these forms satisfy a system of nonlinear partial differential equations (PDEs), the Gauss-Mainardi-Codazzi-Ricci equations. For particular surfaces, this system of PDEs belongs to a distinguished class of equations known as integrable equations. Such an example in R 3 is the class of surfaces of constant mean curvature which is associated with the sinh-Gordon equation. Here an explicit formula is presented which associates with a given system of integrable nonlinear PDEs infinitely many surfaces immersed in Lie algebras. This formula is based on the general construction of surfaces on Lie algebras introduced recently by the first two authors, and on the fact that integrable equations possess infinitely many symmetries. Several examples of surfaces immersed in the 3-dimensional Euclidean space are discussed, including the list of integrable surfaces recently presented by Bobenko and certain deformations thereof. Mathematics Subject Classification (2000). 35Q53, 35Q58, 53A05.

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. 0 1996 American Institute ofPhysics.