Classical and quantum mechanical systems of Toda-lattice type

Abstract

In a previous paper it was shown that certain Schrödinger operatorsH=Δ − V onR ^ℓ such as the Hamiltonians for the quantized one-dimensional lattice systems of Toda type (either non-periodic or periodic) are part of a family of mutually commuting differential operatorsH=L ₁, ...,L _ℓ onR ^ℓ. The potentialV in these cases is associated with a finite root system of rank ℓ, and the top-order symbols of the operatorsL _i are a set of functionally independent polynomials that generate the polynomial invariants for the Weyl groupW of the root system. In this paper it is proved that the spaces of joint eigenfunctions for the family of operatorsL _i have dimension |W|. In the case of the periodic Toda lattices it is shown that the Hamiltonian has only bound states. An integrable holomorphic connection with periodic coefficients is constructed on a trivial |W|-dimensional vector bundle over ℂ^ℓ, and it is shown that the joint eigenfunctions correspond exactly to the covariant constant sections of this bundle. Hence the eigenfunctions can be calculated (in principle) by integrating a system of ordinary differential equations. These eigenfunctions are holomorphic functions on ℂ^ℓ, and are multivariable generalizations of the classical Whittaker functions and Mathieu functions. A generalization of Hill's determinant method is used to analyze the monodromy of the connection.