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Sylwia Kondej

Publications and source records attributed to Sylwia Kondej.

At least 19 recordsLinked to original sources
arXiv2026

Resonances in a Dirichlet quantum waveguide coupled to a cavity

We consider a Dirichlet waveguide in $\mathbb{R}^n$ ($n = 2,3$) with an attached cavity. We show that if the cavity admits a small gap, then the original embedded eigenvalues turn into resonances. The main question we address is how the size of the gap affects the resonant properties, in particular the imaginary part of the resonant pole. For example, in the case of a two dimensional waveguide with a gap of size $\varepsilon$, we show that the leading order term of the resonance behaves as $\mathcal O (\varepsilon^2)$. In the three-dimensional case, if the aperture is defined by a rectangular opening with volume proportional to $\varepsilon^2$, the resonant component behaves as $\mathcal{O}(\varepsilon^4)$. This shows that, in the analyzed class of models, the characteristic time scale associated with the resonances is generically of order $\mathcal{O}((\mathrm{vol}_\varepsilon)^{-2})$, where $\mathrm{vol}_\varepsilon$ denotes the volume of the aperture inducing the resonance.

math-ph↗
arXiv2024

Fermi's golden rule in tunneling models with quantum waveguides perturbed by Kato class measures

In this paper we consider two dimensional quantum system with an infinite waveguide of the width $d$ and a transversally invariant profile. Furthermore, we assume that at a distant $ρ$ there is a perturbation defined by the Kato measure. We show that, under certain conditions, the resolvent of the Hamiltonian has the second sheet pole which reproduces the resonance at $z(ρ)$ with the asymptotics $z(ρ)=\mathcal E_{β; n}+\mathcal O \Big(\frac{ \exp(-\sqrt{2 |\mathcal E_{β;n}| } ρ)}{ρ}\Big)$ for $ρ$ large and with the resonant energy $\mathcal E_{β;n}$. Moreover, we show that the imaginary component of $z(ρ)$ satisfies Fermi's golden rule which we explicitly derive.

math-ph↗
arXiv2022

Bound states of weakly deformed soft waveguides

In this paper we consider the two-dimensional Schrödinger operator with an attractive potential which is a multiple of the characteristic function of an unbounded strip-shaped region, whose thickness is varying and is determined by the function $\mathbb{R}\ni x \mapsto d+\varepsilon f(x)$, where $d > 0$ is a constant, $\varepsilon > 0$ is a small parameter, and $f$ is a compactly supported continuous function. We prove that if $\int_{\mathbb{R}} f \,\mathsf{d} x > 0$, then the respective Schrödinger operator has a unique simple eigenvalue below the threshold of the essential spectrum for all sufficiently small $\varepsilon >0$ and we obtain the asymptotic expansion of this eigenvalue in the regime $\varepsilon\rightarrow 0$. An asymptotic expansion of the respective eigenfunction as $\varepsilon\rightarrow 0$ is also obtained. In the case that $\int_{\mathbb{R}} f \,\mathsf{d} x < 0$ we prove that the discrete spectrum is empty for all sufficiently small $\varepsilon > 0$. In the critical case $\int_{\mathbb{R}} f \,\mathsf{d} x = 0$, we derive a sufficient condition for the existence of a unique bound state for all sufficiently small $\varepsilon > 0$.

math.SP↗
arXiv2020

Soft quantum waveguides with an explicit cut-locus

We consider two-dimensional Schroedinger operators with an attractive potential in the form of a channel of a fixed profile built along an unbounded curve composed of a circular arc and two straight semi-lines. Using a test-function argument with help of parallel coordinates outside the cut-locus of the curve, we establish the existence of discrete eigenvalues. This is a special variant of a recent result of Exner in a non-smooth case and via a different technique which does not require non-positive constraining potentials.

math-ph↗
arXiv2019

Spectral optimization for strongly singular Schrödinger operators with a star-shaped interaction

We discuss the spectral properties of singular Schrödinger operators in three dimensions with the interaction supported by an equilateral star, finite or infinite. In the finite case the discrete spectrum is nonempty if the star arms are long enough. Our main result concerns spectral optimization: we show that the principal eigenvalue is uniquely maxi\-mized when the arms are arranged in one of the known five sharp configurations known as solutions of the closely related Thomson problem.

math-ph↗
arXiv2018

Scattering on leaky wires in dimension three

We consider the scattering problem for a class of strongly singular Schrödinger operators in $L^2(\mathbb{R}R^3)$ which can be formally written as $H_{α,Γ}= -Δ+ δ_α(x-Γ)$ where $α\in\mathbb{R}$ is the coupling parameter and $Γ$ is an infinite curve which is a local smooth deformation of a straight line $Σ\subset\mathbb{R}^3$. Using Kato-Birman method we prove that the wave operators $Ω_\pm(H_{α,Γ}, H_{α,Σ})$ exist and are complete.

math-ph↗
arXiv2017

Asymptotics of the bound state induced by $δ$-interaction supported on a weakly deformed plane

In this paper we consider the three-dimensional Schrödinger operator with a $δ$-interaction of strength $α> 0$ supported on an unbounded surface parametrized by the mapping $\mathbb{R}^2\ni x\mapsto (x,βf(x))$, where $β\in [0,\infty)$ and $f\colon \mathbb{R}^2\rightarrow\mathbb{R}$, $f\not\equiv 0$, is a $C^2$-smooth, compactly supported function. The surface supporting the interaction can be viewed as a local deformation of the plane. It is known that the essential spectrum of this Schrödinger operator coincides with $[-\frac14α^2,+\infty)$. We prove that for all sufficiently small $β> 0$ its discrete spectrum is non-empty and consists of a unique simple eigenvalue. Moreover, we obtain an asymptotic expansion of this eigenvalue in the limit $β\rightarrow 0+$. In particular, this eigenvalue tends to $-\frac14α^2$ exponentially fast as $β\rightarrow 0+$.

math.SP↗
arXiv2017

Aharonov and Bohm vs. Welsh eigenvalues

We consider a class of two-dimensional Schrödinger operator with a singular interaction of the $δ$ type and a fixed strength $β$ supported by an infinite family of concentric, equidistantly spaced circles, and discuss what happens below the essential spectrum when the system is amended by an Aharonov-Bohm flux $α\in [0,\frac12]$ in the center. It is shown that if $β\ne 0$, there is a critical value $α_\mathrm{crit} \in(0,\frac12)$ such that the discrete spectrum has an accumulation point when $α<α_\mathrm{crit} $, while for $α\geα_\mathrm{crit} $ the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed $α\in (0,\frac12)$ and $|β|$ small enough.

math.SP↗
arXiv2017

Straight quantum layer with impurities inducing resonances

We consider a straight three dimensional quantum layer with singular potential supported on a straight wire which is localized perpendicularly to the walls and connects them. We prove that the infinite number of embedded eigenvalues appears in this system. Furthermore, we show that after introducing a small surface impurity to the layer, the embedded eigenvalues turn to the second sheet resolvent poles which state resonances. We discuss the asymptotics of the imaginary component of the resolvent pole with respect to the surface area.

math-ph↗
arXiv2016

Asymptotic spectral analysis in colliding leaky quantum layers

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 hypersurfaces tends to zero. We establish the norm-resolvent convergence to a limiting operator and derive first-order corrections for the corresponding eigenvalues.

math-ph↗
arXiv2015

Gap asymptotics in a weakly bent leaky quantum wire

The main question studied in this paper concerns the weak-coupling behavior of the geometrically induced bound states of singular Schrödinger operators with an attractive $δ$ interaction supported by a planar, asymptotically straight curve $Γ$. We demonstrate that if $Γ$ is only slightly bent or weakly deformed, then there is a single eigenvalue and the gap between it and the continuum threshold is in the leading order proportional to the fourth power of the bending angle, or the deformation parameter. For comparison, we analyze the behavior of a general geometrical induced eigenvalue in the situation when one of the curve asymptotes is wiggled.

math-ph↗
arXiv2014

Strong coupling asymptotics for a singular Schroedinger operator with an interaction supported by an open arc

We consider a singular Schrödinger operator in $L^2(\mathbb{R}^2)$ written formally as $-Δ- βδ(x-γ)$ where $γ$ is a $C^4$ smooth open arc in $\mathbb{R}^2$ of length $L$ with regular ends. It is shown that the $j$th negative eigenvalue of this operator behaves in the strong-coupling limit, $β\to +\infty$, asymptotically as \[ E_j(β)=-\frac{β^2}{4} +μ_j +\mathcal{O}\Big(\dfrac{\logβ}β\Big), \] where $μ_j$ is the $j$th Dirichlet eigenvalue of the operator \[ -\frac{d^2}{ds^2} -\frac{κ(s)^2}{4}\, \] on $L^2(0,L)$ with $κ(s)$ being the signed curvature of $γ$ at the point $s\in(0,L)$.

math-ph↗
arXiv2014

A straight waveguide with a wire inducing resonances

We study a straight infinite planer waveguide with, so called, leaky wire attached to the walls of the waveguide. The wire is modelled by an attractive delta interaction supported by a finite segment. If the wire is placed perpendicularly then the system preserves mirror symmetry which leads the embedded eigenvalues phenomena. We show that if we break the symmetry the corresponding resolvent poles turn to resonances.

math-ph↗
arXiv2014

Weakly coupled bound state of 2D Schrödinger operator with potential-measure

We consider a self-adjoint two-dimensional Schrödinger operator $H_{αμ}$, which corresponds to the formal differential expression \[ -Δ- αμ, \] where $μ$ is a finite compactly supported positive Radon measure on ${\mathbb R}^2$ from the generalized Kato class and $α>0$ is the coupling constant. It was proven earlier that $σ_{\rm ess}(H_{αμ}) = [0,+\infty)$. We show that for sufficiently small $α$ the condition $\sharpσ_{\rm d}(H_{αμ}) = 1$ holds and that the corresponding unique eigenvalue has the asymptotic expansion $$ λ(α) = -(C_μ+ o(1))\exp\Big(-\tfrac{4π}{αμ({\mathbb R}^2)}\Big), \qquad α\rightarrow 0+, $$ with a certain constant $C_μ> 0$. We obtain also the formula for the computation of $C_μ$. The asymptotic expansion of the corresponding eigenfunction is provided. The statements of this paper extend Simon's results, see \cite{Si76}, to the case of potentials-measures. Also for regular potentials our results are partially new.

math.SP↗
arXiv2013

Spectral analysis of a quantum system with a double line singular interaction

We consider a non-relativistic quantum particle interacting with a singular potential supported by two parallel straight lines in the plane. We locate the essential spectrum under the hypothesis that the interaction asymptotically approaches a constant value and find conditions which guarantee either the existence of discrete eigenvalues or Hardy-type inequalities. For a class of our models admitting a mirror symmetry, we also establish the existence of embedded eigenvalues and show that they turn into resonances after introducing a small perturbation.

math.SP↗
arXiv2007

Hiatus perturbation for a singular Schrödinger operator with an interaction supported by a curve in \mathbb{R}^3

We consider Schrödinger operators in $L^2(\mathbb{R}^3)$ with a singular interaction supported by a finite curve $Γ$. We present a proper definition of the operators and study their properties, in particular, we show that the discrete spectrum can be empty if $Γ$ is short enough. If it is not the case, we investigate properties of the eigenvalues in the situation when the curve has a hiatus of length $2ε$. We derive an asymptotic expansion with the leading term which a multiple of $ε\lnε$.

math-ph↗
arXiv2006

Spectral gap of segments of periodic waveguides

We consider a periodic strip in the plane and the associated quantum waveguide with Dirichlet boundary conditions. We analyse finite segments of the waveguide consisting of $L$ periodicity cells, equipped with periodic boundary conditions at the ``new'' boundaries. Our main result is that the distance between the first and second eigenvalue of such a finite segment behaves like $L^{-2}$.

math-ph↗