发表机构
St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院列别杰夫数学研究所圣彼得堡分部）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对三维声学动力系统，证明若哈密顿量存在束缚态，在特定点重新聚焦的系统会失去可控性，进而产生无混响的波散射效应，其数学意义优于波前反转。
AI 中文摘要
三维空间中，动力声学散射系统由波动方程$u_{tt}-\Delta u+qu=0$描述，其中势函数$q$具有紧支集，无穷远处的源（控制项）$f$会产生入射球面波$u=u^f(x,t)$，且满足$u^f\big|_{|x|<-t,\\,\\,t<0}=0$。这些波在$x=0$处聚焦，并在$t=0$时刻充满整个空间。若所有有限能量控制$f$产生的波$u^f(\cdot,0)$构成的集合覆盖空间$L_2(\mathbb{R}^3)$，则该系统是可控的。研究表明，若哈密顿量$H=-\Delta+q$存在束缚态，则空间中会出现点$a$，使得在$x=a$处重新聚焦的系统失去可控性。这一特性导致一种物理效应：存在有限能量的波$u^f$，它们同时在过去锥$|x|<-t$和未来锥$|x|<t$中消失，且无混响地离开$q$的非均匀区域。该效应与波前反转（时间反转镜）有一定相似性，但从数学角度看更具意义。
英文摘要
The dynamic acoustic scattering system is governed by the wave equation $u_{tt}-Δu+qu=0$ in $\Bbb R^3,\,\,\,-\infty<t<\infty$, with a compactly supported potential $q$ and infinitely distant sources (controls) $f$, which initiate incoming spherical waves $u=u^f(x,t)$ provided $u^f\big|_{|x|<-t,\,\,\,t<0}=0$. These waves are focused at $x=0$ and fill up the whole space at the moment $t=0$. The system is {\it controllable} if the set of waves $u^f(\cdot,0)$ produced by all finite energy controls $f$, covers the space $L_2(\Bbb R^3)$. As we show, if the Hamiltonian $H=-Δ+q$ has the bound states, then in the space the points $a$ appear such that the system, being refocused at $x=a$, loses controllability. The latter leads to a physical effect: the waves $u^f$ of finite energy appear, which vanish simultaneously in the past and future cones $|x|<\pm\, t$ and leave the region of inhomogeneity of $q$ without reverberation. This effect has some similarities with the wavefront reversal (Time Reversing Mirror), but is more meaningful from a mathematical point of view.
