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arXiv · 1004.2502

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

Mikhail Belishev·Aleksei Vakulenko
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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 $u_{tt}-Δu+qu=0$ 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 $a \in Υ_q$) 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 $Υ_q$: we reveal its relation to the factorization of the $S$-matrix, connections with the discrete spectrum of the Schr$\ddot{\rm o}$dinger operator $-Δ+q$ and the jet degeneration of the polynomially growing solutions to the equation ${(-Δ+q)} p=0$.

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Mikhail Belishev, Aleksei Vakulenko. 2010-04-14. $s$-points in $3\rm d$ acoustical scattering. https://arxiv.org/abs/1004.2502

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