s -Points in Three-Dimensional Acoustical Scattering

Source abstract

The notion of s-points was introduced by the authors in [SIAM J. Math. Anal., 39 (2008), pp. 1821–1850] in connection with the control problem for the dynamical system governed by the three-dimensional acoustical equation $u_{tt}-\Delta u+qu=0$ with a real potential $q\in C^\infty_0(\mathbb{R}^3)$ and controlled by incoming spherical waves. In the generic case, this system is controllable in the relevant sense, whereas $a\in\mathbb{R}^3$ is called an s-point (we write $a\in\Upsilon_q$) if the system with the shifted potential $q_a=q(\,\cdot-a)$ is not controllable. Such a lack of controllability is related to the subtle physical effect: in the system with the potential $q_a$, there exist the finite energy waves vanishing in the past and future cones simultaneously. The subject of this paper is the set $\Upsilon_q$: we reveal its relation to the factorization of the S-matrix, connections with the discrete spectrum of the Schrödinger operator $-\Delta+q$, and the jet degeneration of the polynomially growing solutions to the equation $(-\Delta+q)p=0$.