Factorization of random Jacobi operators and Bäcklund transformations (Q1210097)

The author shows that a positive definite random Jacobi operator \(L\) over an abstract dynamical system can be factorized as \(L=D^ 2\), where \(D\) is a random Jacobi operator defined over a dynamical system which is an integral extension of the initial system. An isospectral random Toda deformation of \(L\) corresponds to an isospectral random Volterra deformation of \(D\). The factorization leads to commuting Bäcklund transformations which can be written explicitly in terms of Titchmarsh- Weyl functions. In the periodic case, the Bäcklund transformations are time \(i\) maps of a Toda flow with a time dependent Hamiltonian. The factorization is applied to a description of a simple version of super symmetries.