Related papers: Weyl type bound on positive Interior Transmission …
Let $A$ be a densely defined symmetric operator with equal deficiency indices in a Hilbert space. We introduce the notion of a Weyl function $M(z)$ of $A$ corresponding to an ordinary boundary triplet of the operator $A^*$ and then…
We introduce an analog of the Dirichlet-to-Neumann map for the Maxwell equation in a bounded domain. We show that it can be approximated by a pseudodifferential operator on the boundary with a matrix-valued symbol and we compute the…
Two-term Weyl-type asymptotic law for the eigenvalues of one-dimensional fractional Laplace operator (-d^2/dx^2)^(alpha/2) (0 < alpha < 2) in the interval (-1,1) is given: the n-th eigenvalue is equal to (n pi/2 - (2 - alpha) pi/8)^alpha +…
We propose a reformulation of the boundary integral equations for the Helmholtz equation in a domain in terms of incoming and outgoing boundary waves. We obtain transfer operator descriptions which are exact and thus incorporate features…
We review recent results on the semiclassical behaviour of Schr\"{o}dinger operators with Neumann boundary conditions. In this setting, the validity of Weyl's law requires additional conditions on the potential. We will explain the…
The purpose of this article is to establish new lower bounds for the sums of powers of eigenvalues of the Dirichlet fractional Laplacian operator $(-\Delta)^{\alpha/2}|_{\Omega}$ restricted to a bounded domain $\Omega\subset{\mathbb R}^d$…
This paper deals with the inverse spectral problem for a non-self-adjoint Sturm-Liouville operator with discontinuous conditions inside the interval. We obtain that if the potential $q$ is known a priori on a subinterval $ \left[ b,\pi…
We prove the existence, uniqueness, and sharp bilateral pointwise estimates for positive bounded solutions to the Lane--Emden type problem \[ \begin{cases} L u = \sum\limits_{i=1}^{m}\sigma_{i} u^{q_{i}}+\sigma_0, \quad u\geq0 & \text{in }…
The (interior) transmission eigenvalue problems are a type of non-elliptic, non-selfadjoint and nonlinear spectral problems that arise in the theory of wave scattering. They connect to the direct and inverse scattering problems in many…
In this paper, we present a Spectral-Galerkin Method to approximate the zero-index transmission eigenvalues with a conductive boundary condition. This is a new eigenvalue problem derived from the scalar inverse scattering problem for an…
We present upper and lower bounds for Steklov eigenvalues for domains in $\mathbb{R}^{N+1}$ with $C^2$ boundary compatible with the Weyl asymptotics. In particular, we obtain sharp upper bounds on Riesz-means and the trace of corresponding…
We present a finite element algorithm that computes eigenvalues and eigenfunctions of the Laplace operator for two-dimensional problems with homogeneous Neumann or Dirichlet boundary conditions or combinations of either for different parts…
When an eigenvector of a semi-bounded operator is positive, we show that a remarkably simple argument allows to obtain upper and lower bounds for its associated eigenvalue. This theorem is a substantial generalization of Barta-like…
Two boundary value problems for the Helmholtz equation in a semi-infinite strip are considered. The main feature of these problems is that, in addition to the function and its normal derivative on the boundary, the functionals of the…
Cakoni and Nguyen recently proposed very general conditions on the coefficients of Maxwell equations for which they established the discreteness of the set of eigenvalues of the transmission problem and studied their locations. In this…
This paper investigates an initial-Neumann boundary value problem for a Keller--Segel system with parabolic-parabolic-ODE coupling. The model incorporates a signal-dependent, non-increasing motility function that, through indirect signal…
The eigenvalue problem for the Sen--Witten operator on closed spacelike hypersurfaces is investigated. The (square of its) eigenvalues are shown to be given exactly by the 3-surface integral appearing in the expression of the total…
By taking the Weyl equation with external electro-magnetic potentials as the simplest representative for a system of PDOs, we give a new method of treating non-commutativity of coefficients matrices. More precisely, we construct a Fourier…
The aim of this work is to provide an upper bound on the eigenvalues counting function $N(\mathbb{R}^n,-\Delta+V,e)$ of a Sch\"odinger operator $-\Delta +V$ on $\mathbb{R}^n$ corresponding to a potential $V\in…
This paper investigates a multilayered Helmholtz model in $\mathbb{R}^d$ ($d \ge 2$) characterized by concentric layers of materials with alternating positive and negative refractive indices. To overcome the loss of coercivity induced by…