THE PERIODIC TODA CHAIN AND A MATRIX GENERALIZATION OF THE BESSEL-FUNCTION RECURSION-RELATIONS

We obtain the quantization conditions of the periodic Toda lattice in the Baxter form: LAMBDA(u)Q(u) = i(N) Q(u + ihBAR) + i(-N) Q(u - ihBAR) LAMBDA is the 'transfer matrix' containing the information about the spectrum and Q is an integral operator commuting with LAMBDA. The logarithms of the matrix elements of Q are the generating functions of the canonical Backlund transformation. The requirement that Q is analytic and vanishes when u goes to infinity completely determines the spectrum of LAMBDA.