Solution of the finite complex Toda lattice by the method of inverse spectral problem

We show that the finite Toda lattice with complex-valued initial data can be integrated by the method of inverse spectral problem. For this goal spectral data for complex Jacobi matrices are introduced and an inverse spectral problem with respect to the spectral data is solved. The time evolution of the spectral data for the Jacobi matrix associated with the solution of the Toda lattice is computed. Using the solution of the inverse spectral problem with respect to the time-dependent spectral data we reconstruct the time-dependent Jacobi matrix and hence the desired solution of the finite complex Toda lattice.

► Inverse problem for finite complex Jacobi matrices is solved.
► Time evolution of the spectral data associated with the finite Toda lattice is computed.
► The inverse spectral method is applied to solve the finite Toda lattice.
► All complex solutions to the finite Toda lattice are constructed.