arXiv subjects

Mikhail I. Belishev

Publications and source records attributed to Mikhail I. Belishev.

5 recordsLinked to original sources

arXiv2026

Controllability, returning waves and scattering without reverberation in 3D acoustic dynamic system

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.

math-ph↗

arXiv2026

On a stability of time-optimal version of the Boundary Control method

Let $Ω$ be a Riemannian manifold with boundary. The time-optimal version of the BC-method determines the parameters in the $T$-neigh\-bor\-hood $Ω^T$ of $\partialΩ$ from the boundary observations (response operator) $R^{2T}$ on the time segment $[0,2T]$. It visualizes the invisible waves supported in $Ω^T$, by reconstructing the operator $W^T$ that creates these waves. The visualization is based on the triangular factorization of the operator $C^T:=W^{T\,*}W^T$ in the form $C^T:=F^{T\,*}F^T$ with a factor $F^T=U^{T}W^T$, where $U^T$ is a unitary operator. The factorization $C^T\mapsto F^T$ has certain continuity properties, due to which the time-optimal reconstruction $R^{2T}\mapsto C^T\mapsto F^T\mapsto W^T$ turns out to be continuous (stable) in the sense of relevant operator topologies (convergences). As an example, determination of the potential $q$ in the wave equation $u_{tt}-Δu+qu=0$ from $R^{2T}$ is considered. We show that $R^{2T}_j\to R^{2T}$ implies $q_j\to q$ in $H^{-2}(Ω^T)$. However, the question of quantitative estimates of stability (the rate of convergence) remains open.

math-ph↗

arXiv2019

On algebraic and uniqueness properties of 3d harmonic quaternion fields

Let $Ω$ be a smooth compact oriented 3-dimensional Riemannian manifold with boundary. A quaternion field is a pair $q=\{α,u\}$ of a function $α$ and a vector field $u$ on $Ω$. A field $q$ is {\it harmonic} if $α, u$ are continuous in $Ω$ and $\nablaα={\rm rot\,}u,\,{\rm div\,}u=0$ holds into $Ω$. The space ${\mathscr Q}(Ω)$ of harmonic fields is a subspace of the Banach algebra $\mathscr C\left(Ω\right)$ of continuous quaternion fields with the point-wise multiplication $qq'=\{αα'-u\cdot u',\,αu'+α'u+u\wedge u'\}$. We prove a Stone-Weierstrass type theorem: the subalgebra $\vee{\mathscr Q}(Ω)$ generated by harmonic fields is dense in $\mathscr C\left(Ω\right)$. Some results on 2-jets of harmonic functions and the uniqueness sets of harmonic fields are provided.

math.FA↗

arXiv2017

On algebras of harmonic quaternion fields in ${\mathbb R}^3$

Let ${\mathscr A}(D)$ be an algebra of functions continuous in the disk $D=\{z\in{\mathbb C}\,|\,\,\,|z|\leqslant 1\}$ and {\it holomorphic} into $D$. The well-known fact is that the set ${\mathscr M}$ of its characters (homomorphisms ${\mathscr A}(D)\to\mathbb C$) is exhausted by the Dirac measures $\{δ_{z_0}\,|\,\,z_0\in D\}$ and a homeomorphism ${\mathscr M}\cong D$ holds. We present a 3d analog of this classical result as follows. Let $B=\{x\in{\mathbb R}^3\,|\,\,|x|\leqslant 1\}$. A quaternion field is a pair $p=\{α,u\}$ of a function $α$ and vector field $u$ in the ball $B$. A field $p$ is {\it harmonic} if $α, u$ are continuous in $B$ and $\nablaα={\rm rot\,}u,\,{\rm div\,}u=0$ holds into $B$. The space ${\mathscr Q}(B)$ of such fields is not an algebra w.r.t. the relevant (point-wise quaternion) multiplication. However, it contains the commutative algebras ${\mathscr A}_ω(B)=\{p\in{\mathscr Q}(B)\,|\,\,\nabla_ωα=0,\,\nabla_ωu=0\}\,\,(ω\in S^2)$, each ${\mathscr A}_ω(B)$ being isometrically isomorphic to ${\mathscr A}(D)$. This enables one to introduce a set ${\mathscr M}^{\mathbb H}$ of the $\mathbb H$-valued linear functionals on ${\mathscr Q}(B)$ ({\it $\mathbb H$-characters}), which are multiplicative on each ${\mathscr A}_ω(B)$, and prove that ${\mathscr M}^{\mathbb H}=\{δ^{\mathbb H}_{x_0}\,|\,\,x_0\in B\}\cong B$, where $δ^{\mathbb H}_{x_0}(p)=p(x_0)$.

math.FA↗

arXiv2016

On a calculus of variations problem

The paper is of scientific-methodical character. The classical soap film shape (minimal surface) problem is considered, the film being stretched between two parallel coaxial rings. An analytical approach based on relations to the Sturm-Liouville problem is proposed. An energy terms interpretation of the classical Goldschmidt condition is discussed. Appearance of the soliton potential in course of the second variation analysis is noticed.

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