Controllability, returning waves and scattering without reverberation in 3D acoustic dynamic system
Abstract
The dynamic acoustic scattering system is governed by the wave equation utt−Δu+qu=0 in Bbb R3, −∞<t<∞, with a compactly supported potential q and infinitely distant sources (controls) f, which initiate incoming spherical waves u=uf(x,t) provided uf||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 uf(·,0) produced by all finite energy controls f, covers the space L2(Bbb R3). 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 uf of finite energy appear, which vanish simultaneously in the past and future cones |x|<± 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.
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BibTeX
@misc{belishev2026controllability,
title = {{Controllability, returning waves and scattering without reverberation in 3D acoustic dynamic system}},
author = {Mikhail I. Belishev and Aleksei F. Vakulenko},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03536},
eprint = {2609.03536},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03536},
}
RIS
TY - UNPB
TI - Controllability, returning waves and scattering without reverberation in 3D acoustic dynamic system
AU - Mikhail I. Belishev
AU - Aleksei F. Vakulenko
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03536
UR - https://arxiv.org/abs/2609.03536
AB - The dynamic acoustic scattering system is governed by the wave equation u_tt−Δu+qu=0 in Bbb R^3, −∞<t<∞, 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|_|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(·,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|<± 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.
ER -