$s$-points in $3\rm d$ acoustical scattering

Abstract

The notion of $s$-points has been introduced by the authors (SIAM JMA, 39 (2008), 1821--1850) in connection with the control problem for the dynamical system governed by the $3\rm d$ 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 a {\it $s$-point} (we write $a \in \Upsilon_q$) if the system with the shifted potential $q_a=q(\,\cdot-a)$ {\it 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 the paper is the set $\Upsilon_q$: we reveal its relation to the factorization of the $S$-matrix, connections with the discrete spectrum of the Schr$\ddot{\rm o}$dinger operator $-\Delta+q$ and the jet degeneration of the polynomially growing solutions to the equation ${(-\Delta+q)} p=0$.