Linear systems with Schrodinger operators and their transfer functions

We study linear, conservative, stationary dynamic systems (rigged operator colligations) and their transfer functions. The main operator of such systems is an extension with exit in a rigged Hilbert space of a Schrodinger operator on a half-line with non-self-adjoint boundary conditions. The description of all systems with accretive main operator in terms of transfer functions and their linear-fractional transformations and connections with Stieltjes and inverse Stieltjes functions are obtained. Using the bi-extension theory of symmetric operators in rigged Hilbert spaces and the system theory approach, we establish new properties of spectral functions of distributions (and corresponding Stieltjes integrals) of nonnegative self-adjoint extensions of a nonnegative Schrodinger operators on a half-line, new relations between spectral functions of the Friedrichs and Krein-von Neumann extremal, nonnegative self-adjoint extensions, as well as new sharp inequalities involving pointwise functionals and extremal nonnegative extensions.