Additions for Jacobi Operators and the Toda Hierarchy of Lattice Equations

Abstract

We develop a class of Darboux transformations called additions for Jacobi operators. We show that by conjugating by a reflection, an addition may be inverted by another addition with the same spectral parameter. This leads to the development of an “infinitesimal addition”, which allows the transformation to be interpreted as a vector field on a space of Jacobi operators rather than a discrete transformation. We show that in an appropriate limit, this vector field generates the flows of the Toda hierarchy of lattice equations, in analogy to the known fact that an infinitesimal addition on Schrödinger operators can generate the Korteweg-de Vries hierarchy of PDEs. Furthermore, in the case of scattering-type operators, the same vector field appears as a gradient of the transmission coefficient, indicating that the values of the transmission coefficent form a commuting family of functionals with respect to the Poisson bracket corresponding to the Toda hierarchy.