Toda Soliton Limits on General Backgrounds

Starting from an arbitrary background solution of the Toda lattice, we study limits ofN-soliton solutions on this given background asNtends to infinity. This yields a new class of solutions of the Toda lattice. Simultaneously, we solve an inverse spectral problem for one dimensional Jacobi operators–we explicitly construct Jacobi operators whose spectrum contains a given (countable, bounded) set of eigenvalues and whose absolutely continuous spectrum coincides with that of a given background operator.