Searcharxiv⌕ Search

arXiv subjects

Sylwia Kondej

Publications and source records attributed to Sylwia Kondej.

26 records · Page 2Linked to original sources
arXiv2006

Lower bounds on the lowest spectral gap of singular potential Hamiltonians

We analyze Schrödinger operators whose potential is given by a singular interaction supported on a sub-manifold of the ambient space. Under the assumption that the operator has at least two eigenvalues below its essential spectrum we derive estimates on the lowest spectral gap. In the case where the sub-manifold is a finite curve in two dimensional Euclidean space the size of the gap depends only on the following parameters: the length, diameter and maximal curvature of the curve, a certain parameter measuring the injectivity of the curve embedding, and a compact sub-interval of the open, negative energy half-axis which contains the two lowest eigenvalues.

math-ph↗
arXiv2005

On relations between stable and Zeno dynamics in a leaky graph decay model

We use a caricature model of a system consisting of a quantum wire and a finite number of quantum dots, to discuss relation between the Zeno dynamics and the stable one which governs time evolution of the dot states in the absence of the wire. We analyze the weak coupling case and argue that the two time evolutions can differ significantly only at times comparable with the lifetime of the unstable system undisturbed by perpetual measurement.

quant-ph↗
arXiv2004

Scattering by local deformations of a straight leaky wire

We consider a model of a leaky quantum wire with the Hamiltonian $-Δ-αδ(x-Γ)$ in $L^2(\R^2)$, where $Γ$ is a compact deformation of a straight line. The existence of wave operators is proven and the S-matrix is found for the negative part of the spectrum. Moreover, we conjecture that the scattering at negative energies becomes asymptotically purely one-dimensional, being determined by the local geometry in the leading order, if $Γ$ is a smooth curve and $α\to\infty$.

math-ph↗
arXiv2003

Schroedinger operators with singular interactions: a model of tunneling resonances

We discuss a generalized Schrödinger operator in $L^2(\mathbb{R}^d), d=2,3$, with an attractive singular interaction supported by a $(d-1)$-dimensional hyperplane and a finite family of points. It can be regarded as a model of a leaky quantum wire and a family of quantum dots if $d=2$, or surface waves in presence of a finite number of impurities if $d=3$. We analyze the discrete spectrum, and furthermore, we show that the resonance problem in this setting can be explicitly solved; by Birman-Schwinger method it is cast into a form similar to the Friedrichs model.

math-ph↗
arXiv2003

Leaky quantum wire and dots: a resonance model

We discuss a model of a leaky quantum wire and a family of quantum dots described by Laplacian in $L^2(\mathbb{R}^2)$ with an attractive singular perturbation supported by a line and a finite number of points. The discrete spectrum is shown to be nonempty, and furthermore, the resonance problem can be explicitly solved in this setting; by Birman-Schwinger method it is reformulated into a Friedrichs-type model.

math-ph↗
arXiv2002

Bound states due to a strong $δ$ interaction supported by a curved surface

We study the Schrödinger operator $-Δ-αδ(x-Γ)$ in $L^2(\R^3)$ with a $δ$ interaction supported by an infinite non-planar surface $Γ$ which is smooth, admits a global normal parameterization with a uniformly elliptic metric. We show that if $Γ$ is asymptotically planar in a suitable sense and $α>0$ is sufficiently large this operator has a non-empty discrete spectrum and derive an asymptotic expansion of the eigenvalues in terms of a ``two-dimensional'' comparison operator determined by the geometry of the surface $Γ$. [A revised version, to appear in J. Phys. A]

math-ph↗
arXiv2002

Curvature-induced bound states for a $δ$ interaction supported by a curve in $\mathbb{R}^3$

We study the Laplacian in $L^2(\mathbb{R}^3)$ perturbed on an infinite curve $Γ$ by a $δ$ interaction defined through boundary conditions which relate the corresponding generalized boundary values. We show that if $Γ$ is smooth and not a straight line but it is asymptotically straight in a suitable sense, and if the interaction does not vary along the curve, the perturbed operator has at least one isolated eigenvalue below the threshold of the essential spectrum.

math-ph↗