Related papers: Note on 2D Schr\"odinger operators with delta-inte…
Boundedness of wave operators for Schr\"odinger operators in one space dimension for a class of singular potentials, admitting finitely many Dirac delta distributions, is proved. Applications are presented to, for example, dispersive…
We prove a Lieb--Thirring inequality for Schr\"odinger operators $-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+V$ on the semi-axis with Robin boundary condition at the origin. The result improves on a bound obtained by P.~Exner, A.~Laptev and…
We show that the spectrum of a discrete two-dimensional periodic Schr\"odinger operator on a square lattice with a sufficiently small potential is an interval, provided the period is odd in at least one dimension. In general, we show that…
We consider Sch\"odinger operators on the half-line, both discrete and continuous, and show that the absence of bound states implies the absence of embedded singular spectrum. More precisely, in the discrete case we prove that if $\Delta +…
A system of three quantum particles on the three-dimensional lattice $\Z^3$ with arbitrary "dispersion functions" having non-compact support and interacting via short-range pair potentials is considered. The energy operators of the systems…
We study the discrete spectrum of the two-particle Schr\"odinger operator $\hat H_{\mu\lambda}(K),$ $K\in\mathbb{T}^2,$ associated to the Bose-Hubbard Hamiltonian $\hat {\mathbb H}_{\mu\lambda}$ of a system of two identical bosons…
We introduce and study a family of quantum fields, associated to delta-interactions in one dimension. These fields are analogous to holonomic quantum fields of M. Sato, T. Miwa and M. Jimbo. Corresponding field operators belong to an…
Quantum systems composed of $N$ distinct particles in $\R^2$ with two-body contact interactions of TMS type are shown to arise as limits - in the norm resolvent sense - of Schr\"odinger operators with suitably rescaled pair potentials.
We prove sharp upper bounds for eigenvalues of Schr\"odinger operators on quantum graphs with $\delta$-coupling (also known as Robin) conditions at all vertices. The bounds depend on the geometry of the graph, on the potential, and the…
We construct a four-parameter point-interaction for a non-relativistic particle moving on a line as the limit of a short range interaction with range tending toward zero. For particular choices of the parameters, we can obtain a…
The norm resolvent convergence of discrete Schr\"odinger operators to a continuum Schr\"odinger operator in the continuum limit is proved under relatively weak assumptions. This result implies, in particular, the convergence of the spectrum…
We study the threshold behaviour of two dimensional Schr{\" o}dinger operators with finitely many local point interactions. We show that the resolvent can either be continuously extended up to the threshold, in which case we say that the…
In this paper we study the operator $H_{\beta}=-\Delta-\beta\delta(\cdot-\Gamma)$ in $L^{2}(\mathbb{R}^{2})$, where $\Gamma$ is a smooth periodic curve in $\mathbb{R}^{2}$. We obtain the asymptotic form of the band spectrum of $H_{\beta}$…
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 +…
In the present paper our aim is to explore some spectral properties of the family two-particle discrete Schr\"odinger operators $h^{\mathrm{d}}(k)=h^{\mathrm{d}}_0(k)+ \mathbf{v},$ $k\in \T^\mathrm{d},$ on the $\mathrm{d}$ dimensional…
A superspace version of the Schr\"odinger equation with a delta potential is studied using Fourier analysis. An explicit expression for the energy of the single bound state is found as a function of the super-dimension M in case M is…
We consider the three-dimensional Dirac operator coupled with a combination of electrostatic and Lorentz scalar $\delta$-shell interactions. We approximate this operator with general local interactions $V$. Without any hypotheses of…
We give an abstract definition of a one-dimensional Schr\"odinger operator with $\delta'$-interaction on an arbitrary set~$\Gamma$ of Lebesgue measure zero. The number of negative eigenvalues of such an operator is at least as large as the…
A $p$-adic Schr\"{o}dinger-type operator $D^{\alpha}+V_Y$ is studied. $D^{\alpha}$ ($\alpha>0$) is the operator of fractional differentiation and $V_Y=\sum_{i,j=1}^nb_{ij}<\delta_{x_j}, \cdot>\delta_{x_i}$ $(b_{ij}\in\mathbb{C})$ is a…
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…