arXiv subjects

Vesselin Petkov

Publications and source records attributed to Vesselin Petkov.

At least 37 records · Page 2Linked to original sources

arXiv2016

Eigenvalues for Maxwell's equations with dissipative boundary conditions

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 prove that if $γ(x)$ is nowhere equal to 1, then for every $0 < ε\ll 1$ and every $N \in {\mathbb N}$ the eigenvalues of $G_b$ lie in the region $Λ_ε \cup {\mathcal R}_N,$ where $Λ_ε = \{ z \in {\mathbb C}:\: |\Re z | \leq C_ε (|\Im z|^{\frac{1}{2} + ε} + 1), \: \Re z < 0\},$ ${\mathcal R}_N = \{z \in {\mathbb C}:\: |\Im z| \leq C_N (|\Re z| + 1)^{-N},\: \Re z < 0\}.$

math.AP↗

arXiv2016

Location of eigenvalues for the wave equation with dissipative boundary conditions

We examine the location of the eigenvalues of the generator $G$ of a semi-group $V(t) = e^{tG},\: t \geq 0,$ related to the wave equation in an unbounded domain $Ω\subset {\mathbb R}^d$ with dissipative boundary condition $\partial_νu - γ(x) \partial_t u = 0$ on $Γ= \partial Ω.$ We study two cases: $(A): \: 0 < γ(x) < 1,\: \forall x \in Γ$ and $(B):\: 1 < γ(x), \: \forall x \in Γ.$ We prove that for every $0 < ε\ll 1,$ the eigenvalues of $G$ in the case $(A)$ lie in the region $Λ_ε = \{z \in {\mathbb C}:\: |\Re z | \leq C_ε (|\Im z|^{\frac{1}{2} + ε} + 1), \: \Re z < 0\},$ while in the case $(B)$ for every $0 < ε\ll 1$ and every $N \in {\mathbb N}$ the eigenvalues lie in $Λ_ε \cup {\mathcal R}_N,$ where ${\mathcal R}_N = \{z \in {\mathbb C}:\: |\Im z| \leq C_N (|\Re z| + 1)^{-N},\: \Re z < 0\}.$

math.AP↗

arXiv2015

Cauchy problem for effectively hyperbolic operators with triple characteristics of variable multiplicity

We study a class of third order hyperbolic operators $P$ in $G = \{(t, x):0 \leq t \leq T, x \in U \Subset {\mathbb R}^{n}\}$ with triple characteristics at $ρ= (0, x_0, ξ), ξ\in {\mathbb R}^n \setminus \{0\}$. We consider the case when the fundamental matrix of the principal symbol of $P$ at $ρ$ has a couple of non-vanishing real eigenvalues. Such operators are called {\it effectively hyperbolic}. V. Ivrii introduced the conjecture that every effectively hyperbolic operator is {\it strongly hyperbolic}, that is the Cauchy problem for $P + Q$ is locally well posed for any lower order terms $Q$. This conjecture has been solved for operators having at most double characteristics and for operators with triple characteristics in the case when the principal symbol admits a factorization. A strongly hyperbolic operator in $G$ could have triple characteristics in $G$ only for $t = 0$ or for $t = T$. We prove that the operators in our class are strongly hyperbolic if $T$ is small enough. Our proof is based on energy estimates with a loss of regularity.

math.AP↗

arXiv2014

Asymptotics of the number of the interior transmission eigenvalues

We prove a Weyl asymptotics $N(r) = c r^d + {\mathcal O}_ε(r^{d - κ+ ε})$, $\forall\, 0< ε\ll 1$, for the counting function $N(r) = \sharp\{λ_j \in {\mathbb C} \setminus \{0\}:\: |λ_j| \leq r^2\}$, $r>1$, of the interior transmission eigenvalues (ITE), $λ_j$. Here $0<κ\leq 1$ is such that there are no (ITE) in the region $\{λ\in {\mathbb C}:\: |{\rm Im}\:λ|\geq C(| {\rm Re}\:λ|+1)^{1-\fracκ{2}}\}$ for some $C>0$.

math.SP↗

arXiv2014

Spectral estimates for Ruelle transfer operators with two parameters and applications

For $C^2$ weak mixing Axiom A flow $ϕ_t: M \longrightarrow M$ on a Riemannian manifold $M$ and a basic set $Λ$ for $ϕ_t$ we consider the Ruelle transfer operator $L_{f - s τ+ z g}$, where $f$ and $g$ are real-valued Hölder functions on $Λ$, $τ$ is the roof function and $s, z$ are complex parameters. Under some assumptions about $ϕ_t$ we establish estimates for the iterations of this Ruelle operator in the spirit of the estimates for operators with one complex parameter (see \cite{D}, \cite{St2}, \cite{St3}). Two cases are covered: (i) for arbitrary Hölder $f,g$ when $|\Im z| \leq B |\Im s|^μ$ for some constants $B > 0$, $0 < μ< 1$ ($μ= 1$ for Lipschitz $f,g$), (ii) for Lipschitz $f,g$ when $|\Im s| \leq B_1 |\Im z|$ for some constant $B > 0$ . Applying these estimates, we obtain a non zero analytic extension of the zeta function $ζ(s, z)$ for $P_f - ε< \Re (s) < P_f$ and $|z|$ small enough with simple pole at $s = s(z)$. Two other applications are considered as well: the first concerns the Hannay-Ozorio de Almeida sum formula, while the second deals with the asymptotic of the counting function $π_F(T)$ for weighted primitive periods of the flow $ϕ_t.$

math.DS↗

arXiv2014

Spectral problems for non elliptic symmetric systems with dissipative boundary conditions

This paper considers and extends spectral and scattering theory to dissipative symmetric systems that may have zero speeds and in particular to strictly dissipative boundary conditions for Maxwell's equations. Consider symmetric systems $\partial_t - \sum_{j=1}^n A_j \partial_{x_j}$ in ${\mathbb R}^n,\: n \geq 3$, $n$ odd, in a smooth connected exterior domain $Ω:= {\mathbb R}^n \setminus \bar{K}$. Assume that the rank of $A(ξ) = \sum_{j= 1}^n A_j ξ_j$ is constant for $ξ\not= 0.$ For maximally dissipative boundary conditions on $Ω:={\mathbb R}^n \setminus \bar{K}$ with bounded open domain $K$ the solution of the boundary problem in ${\mathbb R}^{+} \times Ω$ is described by a contraction semigroup $V(t) = e^{t G_b},\:t \geq 0.$ Assuming coercive conditions for $G_b$ and its adjoint $G_b^*$ on the complement of their kernels, we prove that the spectrum of $G_b$ in the open half plane $\Re z < 0$ is formed only by isolated eigenvalues with finite multiplicities.

math.FA↗

arXiv2014

Upper bound for the counting function of interior transmission eigenvalues

For the complex interior transmission eigenvalues (ITE) we study for small $θ> 0$ the counting function $$N(θ, r) = #\{λ\in \C:\: λ\: {\rm is} \: {\rm (ITE)},\: |λ| \leq r, \: 0 \leq \arg λ\leq θ\}.$$ We obtain for fixed $θ> 0$ an upper bound $N(θ, r) \leq C r^{n/2}, \: r \geq r(θ).$

math.SP↗

arXiv2013

Numerical Approximation of Asymptotically Disappearing Solutions of Maxwell's Equations

This work is on the numerical approximation of incoming solutions to Maxwell's equations with dissipative boundary conditions whose energy decays exponentially with time. Such solutions are called asymptotically disappearing (ADS) and they play an importarnt role in inverse back-scatering problems. The existence of ADS is a difficult mathematical problem. For the exterior of a sphere, such solutions have been constructed analytically by Colombini, Petkov and Rauch [7] by specifying appropriate initial conditions. However, for general domains of practical interest (such as Lipschitz polyhedra), the existence of such solutions is not evident. This paper considers a finite-element approximation of Maxwell's equations in the exterior of a polyhedron, whose boundary approximates the sphere. Standard Nedelec-Raviart-Thomas elements are used with a Crank-Nicholson scheme to approximate the electric and magnetic fields. Discrete initial conditions interpolating the ones chosen in [7] are modified so that they are (weakly) divergence-free. We prove that with such initial conditions, the approximation to the electric field is weakly divergence-free for all time. Finally, we show numerically that the finite-element approximations of the ADS also decay exponentially with time when the mesh size and the time step become small.

math.NA↗

arXiv2013

Scattering problems for symmetric systems with dissipative boundary conditions

We study symmetric systems with dissipative boundary conditions. The solutions of the mixed problems for such systems are given by a contraction semigroup $V(t)f = e^{tG_b}f, t \geq 0$. The solutions $u(t, x) = V(t)f$, where $f$ is an eigenfunction of the generator $G_b$ with eigenvalue $λ,\Re λ< 0,$ are called asymptotically disappearing (ADS). We prove that the wave operators are not complete if there exist (ADS). This is the case for Maxwell system with special boundary conditions in the exterior of the sphere. We obtain a representation of the scattering kernel and we examine the inverse back-scattering problem related to the leading term of the scattering kernel.

math-ph↗

arXiv2012

Semiclassical estimates of the cut-off resolvent for trapping perturbations

This paper is devoted to the study of a semiclassical "black box" operator $P$. We estimate the norm of its resolvent truncated near the trapped set by the norm of its resolvent truncated on rings far away from the origin. For $z$ in the unphysical sheet with $- h |ln h| < Im z < 0$, we prove that this estimate holds with a constant $h |Im z|^{-1} e^{C|Im z|/h}$. We also obtain analogous bounds for the resonances states of $P$. These results hold without any assumption on the trapped set neither any assumption on the multiplicity of the resonances.

math-ph↗

arXiv2012

Sharp large deviations for some hyperbolic systems

We prove a sharp large deviation principle concerning intervals shrinking with sub-exponential speed for certain models involving the Poincaré map related to a Markov family for an Axiom A flow restricted to a basic set $Λ$ satisfying some additional regularity assumptions.

math.DS↗

arXiv2012

Distribution of periods of closed trajectories in exponentially shrinking intervals

We examine the asymptotics of the number of the closed trajectories $γ$ of hyperbolic flows $ϕ_t$ whose primitive periods $T_γ$ lie in exponentially shrinking intervals $(x - e^{-δx}, x + e^{-δx}),\:δ> 0,\: x \to + \infty.$ Our results holds for hyperbolic dynamical systems having a symbolic model with a non-lattice roof function $f$ under the assumption that the corresponding Ruelle operator related to $f$ satisfies strong spectral estimates. In particular, our analysis works for open billiard systems and for the geodesics flow on manifolds with constant negative curvature.

math.DS↗

arXiv2010

Incoming and disappearing solutions for Maxwell's equations

We prove that in contrast to the free wave equation in $\R^3$ there are no incoming solutions of Maxwell's equations in the form of spherical or modulated spherical waves. We construct solutions which are corrected by lower order incoming waves. With their aid, we construct dissipative boundary conditions and solutions to Maxwell's equations in the exterior of a sphere which decay exponentially as $t \to +\infty$. They are asymptotically disappearing. Disappearing solutions which are identically zero for $t \geq T > 0$ are constructed which satisfy maximal dissipative boundary conditions which depend on time $t$. Both types are invisible in scattering theory.

math-ph↗

arXiv2010

Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function

Let $s_0 < 0$ be the abscissa of absolute convergence of the dynamical zeta function $Z(s)$ for several disjoint strictly convex compact obstacles $K_i \subset \R^N, i = 1,..., κ_0,\: \ka_0 \geq 3,$ and let $R_χ(z) = χ(-Δ_D - z^2)^{-1}χ,\: χ\in C_0^{\infty}(\R^N),$ be the cut-off resolvent of the Dirichlet Laplacian $-Δ_D$ in $Ω= \bar{\R^N \setminus \cup_{i = 1}^{k_0} K_i}$. We prove that there exists $σ_1 < s_0$ such that $Z(s)$ is analytic for $\Re (s) \geq σ_1$ and the cut-off resolvent $R_χ(z)$ has an analytic continuation for $\Im (z) < - σ_1,\: |\Re (z)| \geq C > 0.$

math.AP↗

arXiv2010

Cauchy problem for hyperbolic operators with triple characteristics of variable multiplicity

We study a class of third order hyperbolic operators $P$ in $G = Ω\cap \{0 \leq t \leq T\},\: Ω\subset \R^{n+1}$ with triple characteristics on $t = 0$. We consider the case when the fundamental matrix of the principal symbol for $t = 0$ has a couple of non vanishing real eigenvalues and $P$ is strictly hyperbolic for $t > 0.$ We prove that $P$ is strongly hyperbolic, that is the Cauchy problem for $P + Q$ is well posed in $G$ for any lower order terms $Q$.

math.AP↗

arXiv2010

Spectral problems for operators with crossed magnetic and electric fields

We obtain a representation formula for the derivative of the spectral shift function $ξ(λ; B, ε)$ related to the operators $H_0(B,ε) = (D_x - By)^2 + D_y^2 + εx$ and $H(B, ε) = H_0(B, ε) + V(x,y), \: B > 0, ε> 0$. We prove that the operator $H(B, ε)$ has at most a finite number of embedded eigenvalues on $\R$ which is a step to the proof of the conjecture of absence of embedded eigenvalues of $H$ in $\R.$ Applying the formula for $ξ'(λ, B, ε)$, we obtain a semiclassical asymptotics of the spectral shift function related to the operators $H_0(h) = (hD_x - By)^2 + h^2D_y^2 + εx$ and $H(h) = H_0(h) + V(x,y).$

math-ph↗

arXiv2010

Spectral shift function for operators with crossed magnetic and electric fields

We obtain a representation formula for the derivative of the spectral shift function $ξ(λ; B, ε)$ related to the operators $H_0(B,ε) = (D_x - By)^2 + D_y^2 + εx$ and $H(B, ε) = H_0(B, ε) + V(x,y), \: B > 0, ε> 0$. We establish a limiting absorption principle for $H(B, ε)$ and an estimate ${\mathcal O}(ε^{n-2})$ for $ξ'(λ; B, ε)$, provided $λ\notin σ(Q)$, where $Q = (D_x - By)^2 + D_y^2 + V(x,y).$

math-ph↗

arXiv2010

Lessons from failures to achieve what was possible in the twentieth century physics

For several decades there has been no breakthrough in fundamental physics as revolutionary as relativity and quantum physics despite the amazing advancement of applied physics and technology. By discussing several examples of what physics could have achieved by now, but failed, I will argue that the present state of fundamental physics is not caused by the lack of talented physicists, but rather by problematic general views on how one should do physics. Although it appears to be widely believed that such general views cannot affect the advancement of physics I would like to draw the attention of the younger generation of physicists to three reasons that might have been responsible for failures in the past and might cause problems in the future: (i) misconceptions on the nature of physical theories, (ii) underestimation of the role of conceptual analyses so successfully employed by Galileo and Einstein, and (iii) overestimation of the predictive power of mathematics in physics.

physics.hist-ph↗