The periodic Toda chain and a matrix generalization of the Bessel function recursion relations

The authors obtain the quantization conditions of the periodic Toda lattice in the Baxter form: Lambda (u)Q(u)=iNQ(u+i)+i-NQ(u-i) 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 .