Related papers: Resonances for Euler-Bernoulli operator on the hal…
We investigate a two-dimensional Schr\"odinger operator, $-h^2 \Delta +iV(x)$, with a purely complex potential $iV(x)$. A rigorous definition of this non-selfadjoint operator is provided for bounded and unbounded domains with common…
We consider a Schroedinger operator on the axis with a bipartite potential consisting of two compactly supported complex-valued functions, whose supports are separated by a large distance. We show that this operator possesses a sequence of…
We consider the massless Dirac operator $H=\alpha \cdot D+Q(x)$ on the Hilbert space $L^{2}(\mathbb{R}^{3},\mathbb{C}^{4})$, where $Q(x)$ is a $4\times4$ Hermitian matrix valued function which suitably decays at infinity. We show that the…
We study spectral properties of the Neumann-Poincar\'e operator on planar domains with corners with particular emphasis on existence of continuous spectrum and pure point spectrum. We show that the rate of resonance at continuous spectrum…
In this article, we conduct a study of integral operators defined in terms of non-convolution type kernels with singularities of various degrees. The operators that fall within our scope of research include fractional integrals, fractional…
We consider the meromorphic operator-valued function 1-K(z) = 1-A(z)/z where A(z) is holomorphic on the domain D, and has values in the class of compact operators acting in a given Hilbert space. Under the assumption that A(0) is a…
We consider resonances in the semi-classical limit, generated by a single closed hyperbolic orbit, for an operator on ${\bf R}^2$. We determine all such resonancess in a domain independent of the semi-classical parameter As an application…
We investigate the behavior of eigenvalues for a magnetic Aharonov-Bohm operator with half-integer circulation and Dirichlet boundary conditions in a planar domain. We provide sharp asymptotics for eigenvalues as the pole is moving in the…
We consider the operator $H={d^4dt^4}+{ddt}p{ddt}+q$ with 1-periodic coefficients on the real line. The spectrum of $H$ is absolutely continuous and consists of intervals separated by gaps. We describe the spectrum of this operator in terms…
This paper is on magnetic Schrodinger operators in two dimensional domains with corners. Semiclassical formulas are obtained for the sum and number of eigenvalues. The obtained results extend former formulas for smooth domains in \cite{Fr,…
We consider an elliptic self-adjoint first order pseudodifferential operator acting on columns of m complex-valued half-densities over a connected compact n-dimensional manifold without boundary. The eigenvalues of the principal symbol are…
We consider a class of Hamiltonian PDEs that can be split into a linear unbounded operator and a regular non linear part, and we analyze their numerical discretizations by symplectic methods when the initial value is small in Sobolev norms.…
We study the dynamics of resonances of analytic perturbations of 0th order pseudodifferential operators $P(s)$. In particular, we prove a Fermi golden rule for resonances of $P(s)$ at embedded eigenvalues of $P=P(0)$. We also study the…
We obtain new results about the high-energy distribution of resonances for the one-dimensional Schr\"odinger operator. Our primary result is an upper bound on the density of resonances above any logarithmic curve in terms of the singular…
In this note we review some results regarding higher order elliptic differential operators on manifolds without boundary.
We show that the Dirichlet problem at infinity is unsolvable for the p-Laplace equation for any nonconstant continuous boundary data, for certain range of p>n, on an n-dimensional Cartan-Hadamard manifold constructed from a complete…
We consider the operator ${d^4dt^4}+V$ on the real line with a real periodic potential $V$. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define a Lyapunov function which is analytic…
Recent decades have provided a host of examples and applications motivating the study of nonlocal differential operators. We discuss a class of such operators acting on bounded domains, focusing on those with integrable kernels having…
We consider the fourth order Schr\"odinger operator $H=\Delta^2+V$ and show that if there are no eigenvalues or resonances in the absolutely continuous spectrum of $H$ that the solution operator $e^{-itH}$ satisfies a large time integrable…
Wave-like partial differential equations occur in many engineering applications. Here the engineering setup is embedded into the Hilbert space framework of functional analysis of modern mathematical physics. The notion wave-like is a…