Solution of the Finite Toda Lattice with Self-Consistent Source

In the present paper it is showed that the finite Toda lattice with a self-consistent source is also an important theoretical model as it is a completely integrable system. Namely, it will be determined the time evolution of the spectral data for the Jacobi matrix associated with the solution of a finite Toda chain with a self-consistent source. Then, using the solution of the inverse spectral problem with respect to the time-dependent spectral data, we recover the time-dependent Jacobi matrix and the desired solution of the finite Toda chain with a self-consistent source. For the case , explicit formulas for the solution of the problem under consideration are obtained.