Positons for the Toda lattice and related spectral problems

The concept of positons is introduced for the Toda lattice equation. It is shown that these multiparametric oscillating and slowly decaying solutions, when inserted as potentials in the finite-difference Schrodinger equation of the corresponding Lax pair, lead to a trivial S-matrix. The resulting eigenvalues are embedded in the continuous spectrum of this infinite Jacobi matrix. The singularities connected with the one-positon solution are discussed and compared with those of the positons of continuous integrable models. The special features of the soliton-positon interaction are analysed.