On the Boundary Control Approach to Invese Spectral and Scattering Theory for Schrodinger Operators

We link boundary control theory and inverse spectral theory for the Schrödinger operatorH = @2 x+q (x) on L2 (0;1) with Dirichlet boundary condition at x = 0: This provides a shortcut to some results on inverse spectral theory due to Simon, Gesztesy-Simon and Remling. The approach also has a clear physical interpritation in terms of boundary control theory for the wave equation. 1. Introduction This methodological paper links together a several developments of the late 1990s and earlier 2000s related to classical but still important issues of the inverse problems for the one dimensional wave and Schrodinger equations. In 1986 Belishev put forward a very powerful approach to boundary value inverse problems (see his 2007 review [7] and the extensive literature therein). His approach is based upon deep connections between inverse problems and boundary control theory and is now referred to as the BC method. It is however much less known in the Schrodinger operator community (including inverse problems). Likewise, the boundary control community does not appear to have tested the BC method in the direct/inverse spectral/scattering theory. In [3] we showed that the BC method can be applied to the study of the Titchmarsh-Weyl m-function. In this short note we demonstrate yet another application of the BC method to inverse problems for the one-dimensional Schrodinger equation. In terms relevant to our situation, the main idea of the BC method is to study the (dynamic) Dirichlet-to-Neumann map u (0; t)! @xu (0; t) for the wave equation @ t u @ xu+ q (x)u = 0; x > 0; t > 0 with zero initial conditions. The map u (0; t) ! @xu (0; t) turns out to be the so-called response operator a natural object available in physical experiments. The kernel of this operator, the response function r (t), reconstructs the potential q (x) on (0; l) by r (t) on (0; l) through an elegant procedure; every step of which being physically motivated. Beside its transparency, this method is also essentially local, i.e. instead of studying the problem on (0;1) at a time (as Gelfand-Levitan type methods require) one can solve it on (0; l). The …nal integral equation (2:16), Date : February 16, 2008. 2000 Mathematics Subject Classi…cation. Primary 34B20, 34E05, 34L25, 34E40, 47B20.