相关论文: On the spectrum of the Schr\"odinger operator on $…

We are concerned with the non-normal Schr\"odinger operator $$ H=-\Delta+V $$ on $ L^2(\mathbb R^n)$, where $V\in W^{1,\infty}_{\text{loc}}(\mathbb{R}^n)$ and $\operatorname{Re} (V(x))\ge c|x|^2-d$ for some $c,d>0$. The spectrum of this…

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…

We consider Schr\"odinger operators $H=-\Delta+V({\mathbf x})$ in ${\mathbb R}^d$, $d\geq2$, with quasi-periodic potentials $V({\mathbf x})$. We prove that the absolutely continuous spectrum of a generic $H$ contains a semi-axis…

We consider a family $$ \widehat H_{a,b}(\mu)=\widehat H_0 +\mu \widehat V_{a,b}\quad \mu>0, $$ of Schr\"odinger-type operators on the two dimensional lattice $\mathbb{Z}^2,$ where $\widehat H_0$ is a Laurent-Toeplitz-type convolution…

Let $\mathcal{H}=-\Delta_{\mathbb{H}}+V$ be the Schr\"odinger operator on the Heisenberg group $\mathbb{H}^n$, where $\Delta_{\mathbb{H}}$ is the full laplacian on $\mathbb{H}^n$ and $V$ is a positive smooth potential, bounded below and…

We consider the Dirichlet realization of the operator $-h^2\Delta+iV$ in the semi-classical limit $h\to0$, where $V$ is a smooth real potential with no critical points. For a one dimensional setting, we obtain the complete asymptotic…

The construction of "sparse potentials", suggested in \cite{RS09} for the lattice $\Z^d,\ d>2$, is extended to a wide class of combinatorial and metric graphs whose global dimension is a number $D>2$. For the Schr\"odinger operator $-\D-\a…

We study the spectral properties of discrete Schr\"odinger operator $$ \widehat H_\mu=\widehat H_0 + \mu \widehat{V},\qquad \mu\ge0, $$ associated to a one-particle system in $d$-dimensional lattice $\mathbb{Z}^d, $ $d=1,2,$ where the…

Let $(\lambda_-,\lambda_+)$ be a spectral gap of a periodic Schr\"odinger operator $A$ on the lattice ${\mathbb Z}^d$. Consider the operator $A(\alpha)=A-\alpha V$ where $V$ is a decaying positive potential on ${\mathbb Z}^d$. We study 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 $H_{\beta}$…

We study the Schr\"odinger operator with a potential given by the sum of the potentials for harmonic oscillator and imaginary cubic oscillator and we focus on its pseudospectral properties. A summary of known results about the operator and…

Commutator relations are used to investigate the spectra of Schr\"odinger Hamiltonians, $H = -\Delta + V({x}),$ acting on functions of a smooth, compact $d$-dimensional manifold $M$ immersed in $\bbr^{\nu}, \nu \geq d+1$. Here $\Delta$…

A model operator $H$ corresponding to a three-particle discrete Schr\"odinger operator on a lattice $\Z^3$ is studied. The essential spectrum is described via the spectrum of two Friedrichs models with parameters $h_\alpha(p),$…

We prove an asymptotic expansion for the eigenvalues and eigenfunctions of Schr\"{o}dinger-type operator with a confining potential and with principle part a periodic elliptic operator in divergence form. We compare the spectrum to the…

We analyze spectral properties of the operator $H=\frac{\partial^2}{\partial x^2} -\frac{\partial^2}{\partial y^2} +\omega^2y^2-\lambda y^2V(x y)$ in $L^2(\mathbb{R}^2)$, where $\omega\ne 0$ and $V\ge 0$ is a compactly supported and…

We consider one dimensional Schr\"{o}dinger operators $H_\lambda=-\frac{d^2}{dx^2}+U+ \lambda V_\lambda$ with nonlinear dependence on the parameter $\lambda$ and study the small $\lambda$ behaviour of eigenvalues. The potentials $U$ and…

We consider Schr\"odinger operators with a strongly attractive singular interaction supported by a finite curve $\Gamma$ of lenghth $L$ in $\R^3$. We show that if $\Gamma$ is $C^4$-smooth and has regular endpoints, the $j$-th eigenvalue of…

In this article we consider asymptotics for the spectral function of Schr\"odinger operators on the real line. Let $P:L^2(\mathbb{R})\to L^2(\mathbb{R})$ have the form $$ P:=-\tfrac{d^2}{dx^2}+W, $$ where $W$ is a self-adjoint first order…

Consider a two-dimensional domain shaped like a wire, not necessarily of uniform cross section. Let $V$ denote an electric potential driven by a voltage drop between the conducting surfaces of the wire. We consider the operator ${\mathcal…

In this paper, we develop a systematic framework to study the dispersion surfaces of Schr{\"o}dinger operators $ -\Delta + V$, where the potential $V \in C^\infty(\mathbb{R}^n,\mathbb{R})$ is periodic with respect to a lattice $\Lambda…