In this research, we discuss some important properties of half line Titcchmarsh-Weyl m functions associated to the vector-valued discrete Schrodinger operators induced by the second order difference e..

In this research, we discuss some important properties of half line Titcchmarsh-Weyl m functions associated to the vector-valued discrete Schrodinger operators induced by the second order difference expression. The Titchmarsh-Weyl m functions provide explicit description of absolutely continuous, singular continuous and pure point spectrum of corresponding Schrodinger operators. The Remling's theorem utilizes these m functions to describe the absolutely continuous spectrum. We have established that these m functions are matrix-valued Herglotz functions maping complex upper half plane to Siegel Space, a generalization of complex upper half plane. We then define a metric on the Siegel space as a generalization of the hyperbolic metric on complex upper half plane. Then we establish the distance decreasing property of these m functions with respect to the metric we defined, for vector-valued discrete Schrodinger operators. This property of functions is essential to prove the Remling's theorem.