Related papers: Asymptotics of eigenvalues of the Schroedinger ope…

We consider a generalized Schr\"odinger operator in $L^2(\R^2)$ with an attractive strongly singular interaction of $\delta'$ type characterized by the coupling parameter $\beta>0$ and supported by a $C^4$-smooth closed curve $\Gamma$ of…

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…

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 consider a singular Schr\"odinger operator in $L^2(\mathbb{R}^2)$ written formally as $-\Delta - \beta\delta(x-\gamma)$ where $\gamma$ is a $C^4$ smooth open arc in $\mathbb{R}^2$ of length $L$ with regular ends. It is shown that the…

We investigate the two-dimensional Aharonov-Bohm operator $H_{c_0,\beta} = {(-i\nabla -A)}^{2}-\beta\delta(.-\Gamma),$ where $\Gamma$ is a smooth loop and $A$ is the vector potential which corresponds to Aharonov-Bohm potential. The…

Let $\Gamma\subset\mathbb{R}^2$ be a piecewise smooth closed curve with corners. We discuss the asymptotic behavior of the individual eigenvalues of the two-dimensional Schr\"odinger operator $-\Delta-\alpha\delta_\Gamma$ for…

Let $\Gamma\subset \mathbb{R}^2$ be a simple closed curve which is smooth except at the origin, at which it has a power cusp and coincides with the curve $|x_2|=x_1^p$ for some $p>1$. We study the eigenvalues of the Schr\"odinger operator…

We investigate the two-dimensional magnetic operator $H_{c_0,B,\beta} = {(-i\nabla -A)}^{2}-\beta\delta(.-\Gamma),$ where $\Gamma$ is a smooth loop. The vector potential has the form $A=c_0\bigg(\frac{-y}{{x^2+y^2}}; \frac{x}{{x^2+y^2}}…

We investigate the two-dimensional magnetic Schr\"odinger operator $H_{B,\beta}=(-i\nabla-A)^2 -\beta\delta(\cdot-\Gamma)$, where $\Gamma$ is a smooth loop and the vector potential $A$ corresponds to a homogeneous magnetic field $B$…

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…

We investigate the operator $-\Delta -\alpha \delta (x-\Gamma)$ in $L^2(\mathbb{R}^3)$, where $\Gamma$ is a smooth surface which is either compact or periodic and satisfies suitable regularity requirements. We find an asymptotic expansion…

Let $S\subset\mathbb{R}^3$ be a $C^4$-smooth relatively compact orientable surface with a sufficiently regular boundary. For $\beta\in\mathbb{R}_+$, let $E_j(\beta)$ denote the $j$th negative eigenvalue of the operator associated with the…

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…

We consider self-adjoint Schr\"odinger operators in $L^2 (\mathbb{R}^d)$ with a $\delta$-interaction of strength $\alpha$ and a $\delta'$-interaction of strength $\beta$, respectively, supported on a hypersurface, where $\alpha$ and…

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…

We study the Schr\"odinger operator $-\Delta -\alpha \delta (x-\Gamma)$ in $L^2(\R^3)$ with a $\delta$ interaction supported by an infinite non-planar surface $\Gamma$ which is smooth, admits a global normal parameterization with a…

We consider compact Hankel operators realized in $ \ell^2(\mathbb Z_+)$ as infinite matrices $\Gamma$ with matrix elements $h(j+k)$. Roughly speaking, we show that if $h(j)\sim (b_{1}+ (-1)^j b_{-1}) j^{-1}(\log j)^{-\alpha}$ as $j\to…

We investigate a class of generalized Schr\"{o}dinger operators in $L^2(\mathbb{R}^3)$ with a singular interaction supported by a smooth curve $\Gamma$. We find a strong-coupling asymptotic expansion of the discrete spectrum in case when…

We consider the Schr\"odinger operator $H_{\eta W} = -\Delta + \eta W$, self-adjoint in $L^2({\mathbb R}^d)$, $d \geq 1$. Here $\eta$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. We study…

We show that a Schr\"odinger operator $A_{\delta, \alpha}$ with a $\delta$-interaction of strength $\alpha$ supported on a bounded or unbounded $C^2$-hypersurface $\Sigma \subset \mathbb{R}^d$, $d\ge 2$, can be approximated in the norm…