On Realization of the Original Weyl-Titchmarsh Functions by Shrodinger L-systems

摘要

We study realizations generated by the original Weyl-Titchmarsh functions m(infinity)(z) and m(alpha)(z). It is shown that the Herglotz-Nevanlinna functions (-m(infinity)(z)) and (1/m(infinity)(z)) can be realized as the impedance functions of the corresponding Shrodinger L-systems sharing the same main dissipative operator. These L-systems are presented explicitly and related to Dirichlet and Neumann boundary problems. Similar results but related to the mixed boundary problems are derived for the Herglotz-Nevanlinna functions (-m(alpha)(z)) and (1/m(alpha)(z)). We also obtain some additional properties of these realizations in the case when the minimal symmetric Shrodinger operator is non-negative. In addition to that we state and prove the uniqueness realization criteria for Shrodinger L-systems with equal boundary parameters. A condition for two Shrodinger L-systems to share the same main operator is established as well. Examples that illustrate the obtained results are presented in the end of the paper.