Sylwia Kondej - Academia.edu (original) (raw)

Papers by Sylwia Kondej

Analysis and Operator Theory, 2019

We consider the scattering problem for a class of strongly singular Schrödinger operators in L 2 ... more We consider the scattering problem for a class of strongly singular Schrödinger operators in L 2 (R 3) which can be formally written as H α,Γ = −∆ + δ α (x − Γ) where α ∈ R is the coupling parameter and Γ is an infinite curve which is a local smooth deformation of a straight line Σ ⊂ R 3. Using Kato-Birman method we prove that the wave operators Ω ± (H α,Γ , H α,Σ) exist and are complete.

Journal of Physics A: Mathematical and Theoretical, 2021

We consider two-dimensional Schrödinger operators with an attractive potential in the form of a c... more We consider two-dimensional Schrödinger 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 [3] in a non-smooth case and via a different technique which does not require non-positive constraining potentials.

Journal of Physical Studies, 2011

We study the Hamiltonian in two dimensional system with an interaction supported by a set of codi... more We study the Hamiltonian in two dimensional system with an interaction supported by a set of codimension one. The perturbation in our model is given by the appropriated operator. We derive the formula for the eigenvalues and the corresponding eigenfunctions. Moreover we analyze the functions counting the number of discrete and embedded points of the spectrum depending on coupling constants involved in the model. Finally, we study generalized eigenfunctions.

Journal of Mathematical Analysis and Applications, 2017

We consider the Schrödinger operator with a complex delta interaction supported by two parallel h... more We consider the Schrödinger 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.

Letters in Mathematical Physics, 2019

We discuss the spectral properties of singular Schrödinger operators in three dimensions with the... more 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 maximized when the arms are arranged in one of the known five sharp configurations known as solutions of the closely related Thomson problem.

Journal of Mathematical Physics, 2018

In this paper we consider the three-dimensional Schrödinger operator with a δ-interaction of stre... more 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 R 2 ∋ x → (x, βf (x)), where β ∈ [0, ∞) and f : R 2 → R, f ≡ 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 [− 1 4 α 2 , +∞). 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 → 0+. In particular, this eigenvalue tends to − 1 4 α 2 exponentially fast as β → 0+.

Journal of Physics A: Mathematical and Theoretical, 2015

The main question studied in this paper concerns the weak-coupling behavior of the geometrically ... more 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.

Reports on Mathematical Physics, 2016

We consider Schrödinger operators with a strongly attractive singular interaction supported by a ... more We consider Schrödinger operators with a strongly attractive singular interaction supported by a finite curve Γ of lenghth L in R 3. We show that if Γ is C 4-smooth and has regular endpoints, the j-th eigenvalue of such an operator has the asymptotic expansion λj (Hα,Γ) = ξα +λj(S)+O(e πα) as the coupling parameter α → ∞, where ξα = −4 e 2(−2πα+ψ(1)) and λj(S) is the j-th eigenvalue of the Schrödinger operator S = − d 2 ds 2 − 1 4 γ 2 (s) on L 2 (0, L) with Dirichlet condition at the interval endpoints in which γ is the curvature of Γ.

Mathematische Nachrichten

We investigate the eigengenvalues problem for self-adjoint operators with the singular perturbati... more We investigate the eigengenvalues problem for self-adjoint operators with the singular perturbations. The general results presented here include weakly as well as strongly singular cases. We illustrate these results on two models which correspond to so-called additive strongly singular perturbations.

Journal of Physical Studies

Journal of Mathematical Analysis and Applications, 2014

We consider a self-adjoint two-dimensional Schrödinger operator Hαµ, which corresponds to the for... more We consider a self-adjoint two-dimensional Schrödinger operator Hαµ, which corresponds to the formal differential expression −∆ − αµ, where µ is a finite compactly supported positive Radon measure on R 2 from the generalized Kato class and α > 0 is the coupling constant. It was proven earlier that σess(Hαµ) = [0, +∞). We show that for sufficiently small α the condition ♯σ d (Hαµ) = 1 holds and that the corresponding unique eigenvalue has the asymptotic expansion λ(α) = −(Cµ + o(1)) exp − 4π αµ(R 2) , α → 0+, with a certain constant Cµ > 0. We obtain also the formula for the computation of Cµ. The asymptotic expansion of the corresponding eigenfunction is provided. The statements of this paper extend Simon's results, see [Si76], to the case of potentials-measures. Also for regular potentials our results are partially new.

XIVth International Congress on Mathematical Physics, 2006

We discuss a model of a leaky quantum wire and a family of quantum dots described by Laplacian in... more We discuss a model of a leaky quantum wire and a family of quantum dots described by Laplacian in L 2 (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.

Operator Methods in Ordinary and Partial Differential Equations, 2002

We are interested in Schrodinger operator with the singular perturbations given by operators whic... more We are interested in Schrodinger operator with the singular perturbations given by operators which act in space with measure supported by null set N. We insist on methods which are additive in the case of weakly — as well as strongly singular perturbations. Both cases are illustrated on the examples corresponding to the Schrodinger operator in L2(ℝ2,dx) and L2(ℝ3, dx) with the singular perturbations living on circle. We also solve the problem of the boundary conditions for two above mentioned models.

Computational Methods in Science and Technology, 2010

We study the two dimensional quantum system governedby the Schrödinger operator with delta type p... more We study the two dimensional quantum system governedby the Schrödinger operator with delta type potential. The interaction is supported by line Γ which coincides with a straight at infinity and which admits two widely separated deformations. The aim of this paper is to express the number of bound states of our system by the number of bound states of the system with single deformation.

Journal of Physics A: Mathematical and Theoretical, 2014

We study a straight infinite planer waveguide with, so called, leaky wire attached to the walls o... more 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. The widths of resonances are calculated explicitly in the lowest order perturbation term.

Reviews in Mathematical Physics, 2004

We investigate a class of generalized Schrödinger operators in L2(ℝ3) with a singular interaction... more We investigate a class of generalized Schrödinger operators in L2(ℝ3) with a singular interaction supported by a smooth curve Γ. We find a strong-coupling asymptotic expansion of the discrete spectrum in the case when Γ is a loop or an infinite bent curve which is asymptotically straight. It is given in terms of an auxiliary one-dimensional Schrödinger operator with a potential determined by the curvature of Γ. In the same way, we obtain asymptotics of spectral bands for a periodic curve. In particular, the spectrum is shown to have open gaps in this case if Γ is not a straight line and the singular interaction is strong enough.

Reports on Mathematical Physics, 2011

ABSTRACT We study a two-dimensional quantum system governed by the Schrödinger operator with a de... more ABSTRACT We study a two-dimensional quantum system governed by the Schrödinger operator with a delta type potential. The interaction in our model is supported by a line Γ which coincides with a straight line at infinity. The aim of this paper is to derive a method which allows to find an upper bound for the number of bound states. The method presented here is based on the Birman–Schwinger technics. Finally, we express the mentioned upper bound in terms of geometrical properties of Γ.

Publications of the Research Institute for Mathematical Sciences, 2013

We consider a non-relativistic quantum particle interacting with a singular potential supported b... more 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 Hardytype inequalities. For a class of our models admitting mirror symmetry, we also establish the existence of embedded eigenvalues and show that they turn into resonances after introducing a small perturbation.

Letters in Mathematical Physics, 2006

Dedicated to Pavel Exner on the occasion of his 60 th birthday.