On the Spectral Resolution of the Quantum Toda Lattice

Abstract

In this paper we study the solutions of the equation det(λ−L)ψ=0, where L is the Lax operator of the quantum Toda lattice. The solutions of the equation are determined by the eigenvectors of L, [formula]. In the classical case, there exists the canonical embedding of n-dimensional Toda lattice ↪n+1-dimensional Toda lattice. We show that the quantum analogue of this embedding exists. In the classical case, the Lax operator of the Toda lattice lies in sl(n). In the quantum case, this fact corresponds to the restriction of det(λ−L)ψ=0 to the hyperplane x1+…+xn=constant. We make clear the gap between the solution space of the restricted case and that of the non-restricted case. In the example of the 2-dimensional case, we show that the Bessel functions appear as the basis of the solution space of the above equation.