Related papers: Resolvent, spectrum and resonances for the acousti…

The paper studies properties of acoustic operators in bounded Lipschitz domains $\Omega$ with m-dissipative generalized impedance boundary conditions. We prove that such acoustic operators have a compact resolvent if and only if the…

We consider a bounded open subset $\Omega$ of ${\mathbb{R}}^n$ of class $C^{1,\alpha}$ for some $\alpha\in]0,1[$ and the space $V^{-1,\alpha}(\partial\Omega)$ of (distributional) normal derivatives on the boundary of $\alpha$-H\"{o}lder…

We give an alternative proof and improve upon a result of S.M. Kozlov \cite{ko}. It deals with the asymptotic of the integrated density of states of the acoustic operator $\displaystyle H_\omega=-\nabla\rho_\omega\nabla$, at the bottom of…

We obtain order sharp spectral estimates for the difference of resolvents of singularly perturbed elliptic operators $\mathbf{A}+\mathbf{V}_1$ and $\mathbf{A}+\mathbf{V}_2$ in a domain $\Omega\subseteq \mathbb{R}^\mathbf{N}$ with…

We study the boundary value problem $-{\rm div}((|\nabla u|^{p_1(x)-2}+|\nabla u|^{p_2(x)-2})\nabla u)=\lambda|u|^{q(x)-2}u$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\RR^N$ with smooth boundary,…

We study high energy resonances for the operator $-\Delta_{V,\partial\Omega}:=-\Delta+\delta_{\partial\Omega}\otimes V $ when $V$ has strong frequency dependence. The operator $-\Delta_{V,\partial\Omega}$ is a Hamiltonian used to model both…

We consider the discrete one-dimensional Schr\"{o}dinger operator $H=H_0+V$, where $(H_0x)[n]=-(x[n+1]+x[n-1]-2x[n])$ and $V$ is a self-adjoint operator on $\ell^2(\mathbb{Z})$ with a decay property given by $V$ extending to a compact…

The aim of the paper is to study the problem $u_{tt}-c^2\Delta u=0$ in $\mathbb{R}\times\Omega$, $\mu v_{tt}- \text{div}_\Gamma (\sigma \nabla_\Gamma v)+\delta v_t+\kappa v+\rho u_t =0$ on $\mathbb{R}\times \Gamma_1$, $v_t =\partial_\nu u$…

We provide a detailed study of the spectral properties of the linear operator $H(\varepsilon)=-(\varepsilon^{2}\chi_{\Omega_{\varepsilon}}+\chi_{\Omega^{c}_{\varepsilon}})\Delta$ modeling, through the wave equation…

In this article, we consider the following problem: $$ \quad \left\{ \begin{array}{lr} \quad (-\Delta)^s u = \alpha u^+ -\beta u^{-} + f(u) + h \; \text{in}\;\Omega \quad \quad \quad \quad u =0 \; \text{on}\; \mathbb{R}^n\setminus \Omega,…

We consider operator matrices {\bf H}= (A_0 B_{01} \\ B_{10} A_{1}) with self-adjoint entries A_i, i=0,1, and bounded B_{01}=B_{10}^*, acting in the orthogonal sum {\cal H}={\cal H}_0\oplus{\cal H}_1 of Hilbert spaces {\cal H}_0 and {\cal…

For $a,\alpha>0$ let $E(a,\alpha)$ be the set of all compact operators $A$ on a separable Hilbert space such that $s_n(A)=O(\exp(-an^\alpha))$, where $s_n(A)$ denotes the $n$-th singular number of $A$. We provide upper bounds for the norm…

The present paper addresses questions on resonances for a $1$D Schr\"{o}dinger operator with truncated periodic potential. Precisely, we consider the half-line operator $H^{\mathbb N}=-\Delta +V$ and $H^{\mathbb N}_L = -\Delta + V…

The spectral bound, s(a A + b V), of a combination of a resolvent positive linear operator A and an operator of multiplication V, was shown by Kato to be convex in b \in R. This is shown here, through an elementary lemma, to imply that s(a…

We consider the problem of essential self-adjointness of the drift-diffusion operator $H=-\frac{1}{\rho}\nabla\cdot \rho \mathbb D\nabla +V$ on domains $\Omega \subset \mathbb{R}^d$ with $\mathcal{C}^2$-boundary $\partial \Omega$ and for…

Let $\Omega$ be a Lipschitz domain in $\mathbb R^d$, and let $\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon)\nabla$ be a strongly elliptic operator on $\Omega$. We suppose that $\varepsilon$ is small and the function $A$ is…

We consider rotating wave solutions of the nonlinear wave equation \[ \left\{ \begin{aligned} \partial_{t}^2 v - \Delta v + m v & = |v|^{p-2} v \quad && \text{in $\mathbb{R} \times \textbf{B}$} \\ v & = 0 && \text{on $\mathbb{R} \times…

Let $H_0$ and $V$ be self-adjoint operators, such that $V$ admits a factorisation $V = F^*JF$ with bounded self-adjoint $J$ and $|H_0|^{1/2}$-compact $F.$ Coupling resonance functions, $r_j(z),$ of the pair $H_0$ and $V$ can be defined as…

Commutator methods are applied to get limiting absorption principles for the discrete standard and Molchanov-Vainberg Schr\"odinger operators $H_{\mathrm{std}}= \Delta+V$ and $H_{\mathrm{MV}} = D+V$ on $\ell^2(\mathbb{Z}^d)$, with emphasis…

The present paper is devoted to the study of resonances for a $1$D Schr\"{o}dinger operator with truncated periodic potential. Precisely, we consider the half-line operator $H^{\mathbb N}=-\Delta +V$ and $H^{\mathbb N}_{L}= -\Delta +…