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Mikhail Dubashinskiy

Publications and source records attributed to Mikhail Dubashinskiy.

5 recordsLinked to original sources
arXiv2025

Coexact 1-Laplacian spectral gap and exponential growth of a group

Let $Γ$ be a discrete finitely presented group. Pick any system $S$ of generators in $Γ$. In Cayley graph $\mathrm{Cay}(Γ)=\mathrm{Cay}(Γ, S)$ with edge set $E$, glue with oriented polygons all the group relations translated to all the points of $Γ$; denote the obtained simply connected complex by $\mathrm{Cay}^{(2)}(Γ)$. We study non-negative Hodge--Laplace operator $Δ_1$ on edge functions which is defined via complex $\mathrm{Cay}^{(2)}(Γ)$; $Δ_1$ acts on $$ \ell^2_{0,c}(E):= \mathrm{clos}_{\ell^2(E)} \left\{\mbox{finitely supported closed $1$-(co)chains in }\mathrm{Cay}^{}(Γ)\right\}. $$ We prove the following implication in the spirit of Kesten Theorem: if $Δ_1|_{\ell_{0,c}^2(E)}$ has a spectral gap then $Γ$ either has exponential growth or is virtually $\mathbb Z$.

math.SP↗
arXiv2025

Growth and nodal current of complexified horocycle eigenfunctions

We study horocycle eigenfunctions at Lobachevsky plane. They are functions $u\colon \mathbb H=\mathbb C^+=\{z\in\mathbb C\colon \Im z>0\}\to\mathbb C$ such that $\left(-y^2\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+ 2iτy\frac{\partial}{\partial x}\right)u(x+iy)=s^2 u(x+iy)$, $x+iy\in\mathbb C^+$, with $τ,s\in\mathbb R$, $τ$ large and $s/τ$ small. In other words, we study eigenfunctions of magnetic quantum Hamiltonian on hyperbolic plane. By Bohr semiclassical correspondence principle, the asymptotic behavior of such functions is related to horocycle flow on $T\mathbb H$. Let $u^{\mathbb C}$ be analytic continuation of function $u$ to Grauert tube; the latter is an open neighbourhood of $\mathbb H$ in the complexified Lobachevsky plane $\mathbb H^{\mathbb C}$. If a sequence of horocycle functions possesses microlocal quantum ergodicity at the admissible energy level (with $\hbar=1/τ$) then we may find asymptotic distribution of divisor of $u^{\mathbb C}$. This is done by establishing the asymptotic estimates on $|u^{\mathbb C}|$ in $\mathbb H^{\mathbb C}$. Under imaginary-time horocycle flow, microlocalization of $u$ in $T^*\mathbb H$ is taken to localization of $u^{\mathbb C}$ on $\mathbb H^{\mathbb C}$. The growth of functions $u^{\mathbb C}$ as $τ\to\infty$ turns to be governed by the growth of complexified gauge factor occurring in $τ$-automorphic kernels for functions on $\mathbb H$.

math.SP↗
arXiv2020

Infinite ascension limit: horocyclic chaos

What will be if, given a pure stationary state on a compact hyperbolic surface, we start applying raising operator every $\hbar$ "adiabatic" second? It turns that during adiabatic time comparable to 1 wavefunction will change as a wave traveling with a finite speed (with respect to the adiabatic time), whereas the semiclassical measure of the system will undergo a controllable transformation. If adiabatic time goes to infinity then, by quantized Furstenberg Theorem, the system will become quantum uniquely ergodic. Thus, infinite ascension of a closed system leads to quantum chaos.

math.DS↗
arXiv2015

Interpolation by periods in planar domain

Let $Ω\subset\mathbb R^2$ be a countably connected domain. To any closed differential form of degree $1$ in $Ω$ with components in $L^2(Ω)$ one associates the sequence of its periods around holes in $Ω$, that is around bounded connected components of $\mathbb R^2\setminus Ω$. For which $Ω$ the collection of such period sequences coincides with $\ell^2$? We give the answer in terms of metric properties of holes in $Ω$.

math.CA↗