Related papers: One-dimensional Schroedinger operators with delta-…
A closed form solution for the one-dimensional Schr\"{o}dinger equation with a finite number of $\delta$-interactions \[ \mathbf{L}_{q,\mathfrak{I}_{N}}y:=-y^{\prime\prime}+\left( q(x)+\sum _{k=1}^{N}\alpha_{k}\delta(x-x_{k})\right)…
We consider singular self-adjoint extensions for the Schr\"{o}dinger operator of spin-$1/2$ particle in one dimension. The corresponding boundary conditions at a singular point are obtained. There are boundary conditions with the spin-flip…
This paper is concerned with the spectral analysis of a Hamiltonian with a $\delta$-interaction supported along a broken line with angle $\theta$. The bound states with energy slightly below the threshold of the essential spectrum are…
We consider a generalized Schr\"odinger operator in $L^2(\mathbb R^2)$ describing an attractive $\delta'$ interaction in a strong coupling limit. $\delta'$ interaction is characterized by a coupling parameter $\beta$ and it is supported by…
We study Hamiltonians with point interactions in spaces of vector-valued functions. Using some information from the theory of quantum graphs we describe a class of the operators which can be reduced to the direct sum of several…
Interactions between electrons in one-dimension are fully described at low energies by only a few parameters of the Tomonaga-Luttinger model which is based on linearisation of the spectrum. We consider a model of spinless fermions with a…
Given $n\geq 2$, we put $r=\min\{i\in\mathbb{N}; i>n/2 \}$. Let $\Sigma$ be acompact, $C^{r}$-smooth surface in $\mathbb{R}^{n}$ which contains the origin. Let further $\{S_{\epsilon}\}_{0\le\epsilon<\eta}$ be a family of measurable subsets…
The representations of the kernels of the transmutation operator and of its inverse relating the one-dimensional Schr\"odinger operator with the second derivative are obtained in terms of the eigenfunctions of a corresponding…
The spectral and scattering theory for 1-dimensional Dirac operators with mass $m$ and with zero-range interactions are fully investigated. Explicit expressions for the wave operators and for the scattering operator are provided. These new…
We study Schr\"odinger operators on $\R$ with measures as potentials. Choosing a suitable subset of measures we can work with a dynamical system consisting of measures. We then relate properties of this dynamical system with spectral…
We study two- and three-dimensional matrix Schr\"odinger operators with $m\in \mathbb N$ point interactions. Using the technique of boundary triplets and the corresponding Weyl functions, we complete and generalize the results obtained by…
We consider semi-infinite Jacobi matrices corresponding to a point interaction for the discrete Schr\"odinger operator. Our goal is to find explicit expressions for the spectral measure, the resolvent and other spectral characteristics of…
We consider the Schroedinger operator with a complex delta interaction supported by two parallel hypersurfaces in the Euclidean space of any dimension. We analyse spectral properties of the system in the limit when the distance between the…
This paper deals with the approximation of a magnetic Schr\"odinger operator with a singular $\delta$-potential that is formally given by $(i \nabla + A)^2 + Q + \alpha \delta_\Sigma$ by Schr\"odinger operators with regular potentials in…
In this study, we define discrete fractional Sturm-Liouville (DFSL) operators within Riemann-Liouville and Gr\"unwald-Letnikov fractional operators with both delta and nabla operators. We show selfadjointness of the DFSL operator for the…
This paper proves new results on spectral and scattering theory for matrix-valued Schr\"odinger operators on the discrete line with non-compactly supported perturbations whose first moments are assumed to exist. In particular, a Levinson…
In this paper we consider the three-dimensional Schr\"{o}dinger operator with a $\delta$-interaction of strength $\alpha > 0$ supported on an unbounded surface parametrized by the mapping $\mathbb{R}^2\ni x\mapsto (x,\beta f(x))$, where…
For the pair $\{-\Delta, -\Delta-\alpha\delta_\mathcal{C}\}$ of self-adjoint Schr\"{o}dinger operators in $L^2(\mathbb{R}^n)$ a spectral shift function is determined in an explicit form with the help of (energy parameter dependent)…
The paper deals with singular Schr\"odinger operators of the form \begin{gather*} -{\mathrm{d}^2\over \mathrm{d} x^2 } + \sum_{k\in\mathbb{Z} }\gamma_k \delta(\cdot-z_k),\quad \gamma_k\in\mathbb{R}, \end{gather*} in…
We investigate the spectrum of three-dimensional Schr\"{o}dinger operators with $\delta$-interactions of constant strength supported on circular cones. As shown in earlier works, such operators have infinitely many eigenvalues below the…