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Vesselin Petkov

Publications and source records attributed to Vesselin Petkov.

At least 19 recordsLinked to original sources
arXiv2026

Absence of eigenvalues of dissipative operator for strictly convex obstacles

We study the wave equation in the exterior of a strictly convex bounded domain $K \subset {\mathbb R}^d, d \geq 3,$ odd, with dissipative boundary condition $\partial_{\nu} u - \gamma(x) \partial_t u = 0$ on the boundary $\Gamma$ and $0 < \gamma(x) <1, \:\forall x \in \Gamma.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, \: t \geq 0.$ In [10] we established that for $\gamma \equiv const$ and $K = \{x \in {\mathbb R}^3: \:|x| \leq 1\}$ the operator $G$ has no eigenvalues and we conjectured that the same result holds for every strictly convex obstacle. In this paper we prove this conjecture.

math.AP↗
arXiv2026

Length spectrum of periodic rays for billard flow

We study for several compact strictly convex disjoint obstacles the length spectrum $\mathcal L$ formed by the lengths of all primitive periodic reflecting rays. We prove the existence of sequences $\{\ell_j\},\: \{m_j\}$ with $\ell_j \in \mathcal L,\: m_j \in \mathbb N$ such that the condition (LB) related to the dynamical zeta function $η_D(s)$ is satisfied. This condition implies the existence of lower bounds for the number of the scattering resonances for Dirichlet Laplacian. We construct such sequences under some separation condition for a small subset of $\mathcal L$ corresponding to lengths of the periodic rays with even reflexions. Our separation condition is weaker than the assumption of exponentially separated length spectrum $\mathcal L.$ Moreover, we show that the periodic orbits in the phase space are exponentially separated.

math.DS↗
arXiv2025

Dirichlet dynamical zeta function for billiard flow

We study the Dirichlet dynamical zeta function $η_D(s)$ for billiard flow corresponding to several strictly convex disjoint obstacles. For large ${\rm Re}\: s$ we have $η_D(s) =\sum_{n= 1}^{\infty} a_n e^{-λ_n s}, \: a_n \in \mathbb R$ and $η_D$ admits a meromorphic continuation to $\mathbb C$. We obtain some conditions of the frequencies $λ_n$ and some sums of coefficients $a_n$ which imply that $η_D$ cannot be prolonged as entire function.

math.DS↗
arXiv2025

On the number of poles of the dynamical zeta functions for billiard flow

We study the number of the poles of the meromorphic continuation of the dynamical zeta functions $η_N$ and $η_D$ for several strictly convex disjoint obstacles satisfying non-eclipse condition. We obtain a strip $\{z \in \mathbb C:\: {\rm Re}\: s > β\}$ with infinite number of poles. For $η_D$ we prove the same result assuming the boundary real analytic. Moreover, for $η_N$ we obtain a characterisation of $β$ by the pressure $P(2G)$ of some function $G$ on the space $Σ_A^f$ related to the dynamical characteristics of the obstacle.

math.DS↗
arXiv2025

Eigenvalues and resonances of dissipative acoustic operator for strictly convex obstacles

We examine the wave equation in the exterior of a strictly convex bounded domain $K$ with dissipative boundary condition $\partial_ν u - γ(x) \partial_t u = 0$ on the boundary $Γ$ and $0 < γ(x) <1, \:\forall x \in Γ.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, \: t \geq 0.$ The poles $λ$ of the meromorphic incoming resolvent $(G - λ)^{-1}: \:{ \mathcal H}_{comp} \rightarrow {\mathcal D}_{loc}$ are eigenvalues of G if ${\rm Re}\: λ< 0$ and incoming resonances if ${\rm Re}\: λ> 0$. We obtain sharper results for the location of the eigenvalues of $G$ and incoming resonances in $Λ= \{λ\in \mathbb C:\: |{\rm Re}\: λ| \leq C_2(1 + |{\rm Im}\: λ|)^{-2},\: |{\rm Im}\: λ| \geq A_2 > 1\}$ and we prove a Weyl formula for their asymptotic. For $K = \{x \in {\mathbb R}^3:\:|x| \leq 1\}$ and $γ$ constant we show that $G$ has no eigenvalues so the Weyl formula concerns only the incoming resonances.

math.AP↗
arXiv2024

Dynamical zeta functions for billiards

Let $D \subset {\mathbb R}^d,\: d \geqslant 2,$ be the union of a finite collection of pairwise disjoint strictly convex compact obstacles. Let $μ_j \in {\mathbb C},\: {\rm Im}\: μ_j > 0,$ be the resonances of the Laplacian in the exterior of $D$ with Neumann or Dirichlet boundary condition on $\partial D$. For $d$ odd, $u(t) = \sum_j e^{i |t| μ_j}$ is a distribution in $ \mathcal{D}'({\mathbb R} \setminus \{0\})$ and the Laplace transforms of the leading singularities of $u(t)$ yield the dynamical zeta functions $η_{\mathrm N},\: η_{\mathrm D}$ for Neumann and Dirichlet boundary conditions, respectively. These zeta functions play a crucial role in the analysis of the distribution of the resonances. Under the non-eclipse condition (1.1), for $d \geqslant 2$ we show that $η_{\mathrm N}$ and $η_\mathrm D$ admit a meromorphic continuation to the whole complex plane. In the particular case when the boundary $\partial D$ is real analytic, by using a result of Fried (1995), we prove that the function $η_\mathrm{D}$ cannot be entire. Following the result of Ikawa (1988), this implies the existence of a strip $\{z \in {\mathbb C}: \: 0 < {\rm Im}\: z \leqα\}$ containing an infinite number of resonances $μ_j$ for the Dirichlet problem. Moreover, for $α\gg 1$ we obtain a lower bound for the resonances lying in this strip.

math.DS↗
arXiv2023

Asymptotic of the dissipative eigenvalues of Maxwell's equations

Let $Ω= \mathbb R^3 \setminus \bar{K}$, where $K$ is an open bounded domain with smooth boundary $Γ$. Let $V(t) = e^{tG_b},\: t \geq 0,$ be the semigroup related to Maxwell's equations in $Ω$ with dissipative boundary condition $ν\wedge (ν\wedge E)+ γ(x) (ν\wedge H) = 0, γ(x) > 0, \forall x \in Γ.$ We study the case when $γ(x) \neq 1, \: \forall x \in Γ,$ and we establish a Weyl formula for the counting function of the eigenvalues of $G_b$ in a polynomial neighbourhood of the negative real axis.

math.AP↗
arXiv2021

Weyl formula for the eigenvalues of the dissipative acoustic operator

We study the wave equation in the exterior of a bounded domain $K$ with dissipative boundary condition $\partial_ν u - γ(x) \partial_t u = 0$ on the boundary $Γ$ and $γ(x) > 0.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, \: t \geq 0.$ The eigenvalues $λ_k$ of $G$ with ${\rm Re}\: λ_k < 0$ yield asymptotically disappearing solutions $u(t, x) = e^{λ_k t} f(x)$ having exponentially decreasing global energy. We establish a Weyl formula for these eigenvalues in the case $\min_{x\in Γ} γ(x) > 1.$ For strictly convex obstacles $K$ this formula concerns all eigenvalues of $G.$

math.AP↗
arXiv2020

Absence of embedded eigenvalues for Hamiltonian with crossed magnetic and electric fields

In the presence of the homogeneous electric field ${\bf E}$ and the homogeneous perpendicular magnetic field ${\bf B}$, the classical trajectory of a quantum particle on ${\mathbb R}^2$ moves with drift velocity $α$ which is perpendicular to the electric and magnetic fields. For such Hamiltonians the absence of the embedded eigenvalues of perturbed Hamiltonian has been conjectured. In this paper one proves this conjecture for the perturbations $V(x, y)$ which have sufficiently small support in direction of drift velocity.

math.SP↗
arXiv2020

Sharp large deviations for hyperbolic flows

For hyperbolic flows $φ_t$ we examine the Gibbs measure of points $w$ for which $$\int_0^T G(φ_t w) dt - a T \in (- e^{-εn}, e^{- εn})$$ as $n \to \infty$ and $T \geq n$, provided $ε> 0$ is sufficiently small. This is similar to local central limit theorems. The fact that the interval $(- e^{-εn}, e^{- εn})$ is exponentially shrinking as $n \to \infty$ leads to several difficulties. Under some geometric assumptions we establish a sharp large deviation result with leading term $C(a) ε_n e^{γ(a) T}$ and rate function $γ(a) \leq 0.$ The proof is based on the spectral estimates for the iterations of the Ruelle operators with two complex parameters and on a new Tauberian theorem for sequence of functions $g_n(t)$ having an asymptotic as $ n \to \infty$ and $t \geq n.$

math.DS↗
arXiv2019

Polynomial bounds on the Sobolev norms of the solutions of the nonlinear wave equation with time dependent potential

We consider the Cauchy problem for the nonlinear wave equation $u_{tt} - Δ_x u +q(t, x) u + u^3 = 0$ with smooth potential $q(t, x) \geq 0$ having compact support with respect to $x$. The linear equation without the nonlinear term $u^3$ and potential periodic in $t$ may have solutions with exponentially increasing as $ t \to \infty$ norm $H^1({\mathbb R}^3_x)$. In [2] it was established that adding the nonlinear term $u^3$ the $H^1({\mathbb R}^3_x)$ norm of the solution is polynomially bounded for every choice of $q$. In this paper we show that $H^k({\mathbb R}^3_x)$ norm of this global solution is also polynomially bounded. To prove this we apply a different argument based on the analysis of a sequence $\{Y_k(nτ_k)\}_{n = 0}^{\infty}$ with suitably defined energy norm $Y_k(t)$ and $0 < τ_k <1.$

math.AP↗
arXiv2018

Spectral estimates for Ruelle operators with two parameters and sharp large deviations

We obtain spectral estimates for the iterations of Ruelle operator $L_{f + (a + ıb)τ+ (c + ıd) g}$ with two complex parameters and Hölder functions $f,\: g$ generalizing the case $\Pr(f) =0$ studied in [PeS2]. As an application we prove a sharp large deviation theorem concerning exponentially shrinking intervals which improves the result in [PeS1].

math.DS↗
arXiv2018

On the nonlinear wave equation with time periodic potential

It is known that for some time periodic potentials $q(t, x) \geq 0$ having compact support with respect to $x$ some solutions of the Cauchy problem for the wave equation $\partial_t^2 u - Δ_x u + q(t,x)u = 0$ have exponentially increasing energy as $t \to \infty$. We show that if one adds a nonlinear defocusing interaction $|u|^ru, 2\leq r < 4,$ then the solution of the nonlinear wave equation exists for all $t \in {\mathbb R}$ and its energy is polynomially bounded as $t \to \infty$ for every choice of $q$. Moreover, we prove that the zero solution of the nonlinear wave equation is instable if the corresponding linear equation has the property mentioned above.

math.AP↗
arXiv2017

Cauchy problem for effectively hyperbolic operators with triple characteristics

We study the Cauchy problem for effectively hyperbolic operators $P$ with principal symbol $p(t, x,τ,ξ)$ having triple characteristics on $t = 0$. Under a condition (E) we show that such operators are strongly hyperbolic, that is the Cauchy problem is well posed for $p(t, x,D_t, D_x) + Q(t, x, D_t, D_x)$ with arbitrary lower order term $Q$. The proof is based on energy estimates with weight $t^{-N}$ for a first order pseudo-differential system, where $N$ depends on lower order terms. For our analysis we construct a non-negative definite symmetrizer $S(t)$ and we prove a version of Fefferman-Phong type inequality for ${\rm Re}\, (S(t)U, U)_{L^2({\mathbb R}^n)}$ with a lower bound $-C t^{-1}\|\langle D \rangle^{-1}U\|_{L^2(\mathbb R^n)}$.

math.AP↗
arXiv2017

Weyl formula for the negative dissipative eigenvalues of Maxwell's equations

Let $V(t) = e^{tG_b},\: t \geq 0,$ be the semigroup generated by Maxwell's equations in an exterior domain $Ω\subset {\mathbb R}^3$ with dissipative boundary condition $E_{tan}- γ(x) (ν\wedge B_{tan}) = 0, γ(x) > 0, \forall x \in Γ= \partial Ω.$ We study the case when $Ω= \{x \in {\mathbb R^3}:\: |x| > 1\}$ and $γ\neq 1$ is a constant. We establish a Weyl formula for the counting function of the negative real eigenvalues of $G_b.$

math.AP↗
arXiv2017

Localization of the interior transmission eigenvalues for a ball

We study the localization of the interior transmission eigenvalues (ITEs) in the case when the domain is the unit ball $\{x \in {\mathbb R}^d:\: |x| \leq 1\}, \: d\geq 2,$ and the coefficients $c_j(x), \: j =1,2,$ and the indices of refraction $n_j(x), \: j =1,2,$ are constants near the boundary $|x| = 1$. We prove that in this case the eigenvalue-free region obtained in [16] for strictly concave domains can be significantly improved. In particular, if $c_j(x), n_j(x), j = 1,2$ are constants for $|x| \leq 1$, we show that all (ITEs) lie in a strip $\{ λ\in {\mathbb C}:\:|{\rm Im}\: λ| \leq C\}$.

math.AP↗
arXiv2016

Location and Weyl formula for the eigenvalues of some non self-adjoint operators

We present a survey of some recent results concerning the location and the Weyl formula for the complex eigenvalues of two non self-adjoint operators. We study the eigenvalues of the generator $G$ of the contraction semigroup $e^{tG}, \: t \geq 0,$ related to the wave equation in an unbounded domain $Ω$ with dissipative boundary conditions on $\partial Ω$. Also one examines the interior transmission eigenvalues (ITE) in a bounded domain $K$ obtaining a Weyl formula with remainder for the counting function $N(r)$ of complex (ITE). The analysis is based on a semi-classical approach.

math.SP↗