Related papers: 1-D Schr\"odinger operator on a star graph with no…
The spectrum of a selfadjoint second order elliptic differential operator in $L^2(\mathbb{R}^n)$ is described in terms of the limiting behavior of Dirichlet-to-Neumann maps, which arise in a multi-dimensional Glazman decomposition and…
The question of self-adjoint realizations of sign-indefinite second order differential operators is discussed in terms of a model problem. Operators of the type $-\frac{d}{dx} \sgn (x) \frac{d}{dx}$ are generalized to finite, not…
We consider the Schr\"odinger operator on a combinatorial graph consisting of a finite graph and a finite number of discrete half-lines, all jointed together, and compute an asymptotic expansion of its resolvent around the threshold $0$.…
The study of the scattering data for a star-shape network of LC-transmission lines is transformed into the scattering analysis of a Schr\"odinger operator on the same graph. The boundary conditions coming from the Kirchhoff rules ensure the…
In this article we investigate the distribution of eigenvalues of the Dirichlet pseudo-differential operator $\sum_{i=1}^{d}(-\partial_i^2)^{s}, \, s\in (1/2,1]$ on an open and bounded subdomain $\Omega \subset \mathbb{R}^d$ and predict…
The paper discusses the Nonlinear Dirac Equation with Kerr-type nonlinearity (i.e., $\psi^{p-2}\psi$) on noncompact metric graphs with a finite number of edges, in the case of Kirchhoff-type vertex conditions. Precisely, we prove local…
We consider the damped Schr\"odinger semigroup $e^{-it \frac{d^2}{dx^2}}$ on the tadpole graph ${\mathcal R}$. We first give a careful spectral analysis and an appropriate decomposition of the kernel of the resolvent. As a consequence and…
We study the discreteness of the spectrum of Schrodinger operators which are defined on N-dimensional rooted trees of a finite or infinite volume, and are subject to a certain mixed boundary condition. We present a method to estimate their…
We investigate the spectrum of a typical non-self-adjoint differential operator $AD=-d^2/dx^2\otimes A$ acting on $\Lp(0,1)\otimes \mathbb{C}^2$, where $A$ is a $2\times 2$ constant matrix. We impose Dirichlet and Neumann boundary…
The spectrum of discrete Schr\"odinger operator $L+V$ on the $d$-dimensional lattice is considered, where $L$ denotes the discrete Laplacian and $V$ a delta function with mass at a single point. Eigenvalues of $L+V$ are specified and the…
In this paper we study the spectrum of self-adjoint Schr\"odinger operators in $L^2(\mathbb{R}^2)$ with a new type of transmission conditions along a smooth closed curve $\Sigma\subseteq \mathbb{R}^2$. Although these $\textit{oblique}$…
This paper is dedicated to $L^p$ bounds on eigenfunctions of a Sch\"odinger-type operator $(-\Delta_g)^{\alpha/2} +V$ on closed Riemannian manifolds for critically singular potentials $V$. The operator $(-\Delta_g)^{\alpha/2}$ is defined…
The completeness of the system of eigenfunctions of the complex Schr\"odinger operator $\mathscr{L}_c=-d^2/dx^2+cx^\alpha$ on the semi-axis with Dirichlet boundary conditions is proved for $\alpha\in(0,2)$ and $|\arg…
We consider the two-dimensional Dirac operator with Lorentz-scalar $\delta$-shell interactions on each edge of a star-graph. An orthogonal decomposition is performed which shows such an operator is unitarily equivalent to an orthogonal sum…
We investigate dispersive and Strichartz estimates for the Schr\"{o}dinger time evolution propagator $\mathrm{e}^{-\mathrm{i}tH}$ on a star-shaped metric graph. The linear operator, $H$, taken into consideration is the self-adjoint…
We obtain asymptotic formulas for the spectral data of perturbed Stark operators associated with the differential expression \[ -\frac{d^2}{dx^2} + x + q(x), \quad x\in [0,\infty), \quad q\in L^1(0,\infty), \] and having either Dirichlet or…
We study the operator $L=-\Delta+q$ on a bounded domain $\Omega\subset\mathbb R^n$, where $q(x)$ is a distributional potential. We find sufficient conditions for $q(x)$ which guarantee that $L$ is well--defined with Dirichlet and…
We consider normalized Laplacians and their perturbations by periodic potentials (Schr\"odinger operators) on periodic discrete graphs. The spectrum of the operators consists of an absolutely continuous part (a union of a finite number of…
Spectral properties of 1-D Schr\"odinger operators $\mathrm{H}_{X,\alpha}:=-\frac{\mathrm{d}^2}{\mathrm{d} x^2} + \sum_{x_{n}\in X}\alpha_n\delta(x-x_n)$ with local point interactions on a discrete set $X=\{x_n\}_{n=1}^\infty$ are well…
We consider the magnetic Schrodinger operator in a two-dimensional strip. On the boundary of the strip the Dirichlet boundary condition is imposed except for a fixed segment (window), where it switches to magnetic Neumann boundary condition…