The complex high order Toda lattice and polynomials generated by three term recurrence relations

Se estudia la relación entre redes de Toda y sucesiones de polinomios dados por recuurencias a tres término

The Darboux transformation and the complex Toda lattice

It is well known that each solution of the Toda lattice can be represented by a tridiagonal matrix J(t). Under certain restrictions, it is possible to obtain some new solution by using the Darboux transformation of J(t) ¡ CI. Our goal is the extension of this fact, which is known for the real lattice, to high order complex Toda lattices as well as to the bi-infinite Toda lattice. In this latter case, we use the factorization LU for block-tridiagonal matrices

Spectrum and generation of the complex Toda lattice

Sufficient conditions for constructing a set of solutions of the Toda lattice are analyzed. First, under certain conditions the invariance of the spectrum of J t is established in the complex case. Second, given the tri-diagonal matrix J t defining a Toda lattice solution, the dynamic behavior of zeros of polynomials associated to J t is analyzed. Finally, it is shown by means of an example how to apply our results to generate complex solutions of the Toda lattice starting with a given solution