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Pavel Exner

Publications and source records attributed to Pavel Exner.

At least 163 records · Page 9Linked to original sources
arXiv2005

Absolute continuity of the spectrum for periodically modulated leaky wires in $\mathbb{R}^3$

We consider a model of leaky quantum wire in three dimensions. The Hamiltonian is a singular perturbation of the Laplacian supported by a line with the coupling which is bounded and periodically modulated along the line. We demonstrate that such a system has a purely absolutely continuous spectrum and its negative part has band structure with an at most finite number of gaps. This result is extended also to the situation when there is an infinite number of the lines supporting the perturbations arranged periodically in one direction.

math.SP↗
arXiv2005

Approximations of permutation-symmetric vertex couplings in quantum graphs

We consider boundary conditions at the vertex of a star graph which make Schroedinger operators on the graph self-adjoint, in particular, the two-parameter family of such conditions invariant with respect to permutations of graph edges. It is proved that the corresponding operators can be approximated in the norm-resolvent sense by elements of another Schroedinger operator family on the same graph in which the delta coupling is imposed at the vertex and an additional point interaction is placed at each edge provided the coupling parameters are properly chosen.

math-ph↗
arXiv2005

Sufficient conditions for the anti-Zeno effect

The ideal anti-Zeno effect means that a perpetual observation leads to an immediate disappearance of the unstable system. We present a straightforward way to derive sufficient conditions under which such a situation occurs expressed in terms of the decaying states and spectral properties of the Hamiltonian. They show, in particular, that the gap between Zeno and anti-Zeno effects is in fact very narrow.

quant-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↗
arXiv2005

An isoperimetric problem for leaky loops and related mean-chord inequalities

We consider a class of Hamiltonians in $L^2(\R^2)$ with attractive interaction supported by piecewise $C^2$ smooth loops $Γ$ of a fixed length $L$, formally given by $-Δ-αδ(x-Γ)$ with $α>0$. It is shown that the ground state of this operator is locally maximized by a circular $Γ$. We also conjecture that this property holds globally and show that the problem is related to an interesting family of geometric inequalities concerning mean values of chords of $Γ$.

math-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↗
arXiv2004

Convergence of spectra of graph-like thin manifolds

We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace-Beltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand, if the the shrinking at the vertex parts of the manifold is sufficiently slower comparing to that of the edge parts, the limiting spectrum corresponds to decoupled edges with Dirichlet boundary conditions at the endpoints. At the borderline between the two regimes we have a third possibility when the limiting spectrum can be described by a nontrivial coupling at the vertices.

math-ph↗
arXiv2004

An isoperimetric problem for point interactions

We consider Hamiltonian with $N$ point interactions in $\R^d, d=2,3,$ all with the same coupling constant, placed at vertices of an equilateral polygon $\PP_N$. It is shown that the ground state energy is locally maximized by a regular polygon. The question whether the maximum is global is reduced to an interesting geometric problem.

math-ph↗
arXiv2004

An approximation to $δ'$ couplings on graphs

We discuss a general parametrization for vertices of quantum graphs and show, in particular, how the $δ'_s$ and $δ'$ coupling at an $n$ edge vertex can be approximated by means of $n+1$ couplings of the $δ$ type provided the latter are properly scaled.

quant-ph↗
arXiv2004

Lieb-Thirring inequalities for geometrically induced bound states

We prove new inequalities of the Lieb-Thirring type on the eigenvalues of Schrödinger operators in wave guides with local perturbations. The estimates are optimal in the weak-coupling case. To illustrate their applications, we consider, in particular, a straight strip and a straight circular tube with either mixed boundary conditions or boundary deformations.

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↗
arXiv2003

Absolute continuity in periodic thin tubes and strongly coupled leaky wires

Using a perturbative argument, we show that in any finite region containing the lowest transverse eigenmode, the spectrum of a periodically curved smooth Dirichlet tube in two or three dimensions is absolutely continuous provided the tube is sufficiently thin. In a similar way we demonstrate absolute continuity at the bottom of the spectrum for generalized Schrödinger operators with a sufficiently strongly attractive $δ$ interaction supported by a periodic curve in $\mathbb{R}^d, d=2,3$.

math.SP↗
arXiv2003

Spectral properties of Schroedinger operators with a strongly attractive delta interaction supported by a surface

We investigate the operator $-Δ-αδ(x-Γ)$ in $L^2(\mathbb{R}^3)$, where $Γ$ is a smooth surface which is either compact or periodic and satisfies suitable regularity requirements. We find an asymptotic expansion for the lower part of the spectrum as $α\to\infty$ which involves a ``two-dimensional'' comparison operator determined by the geometry of the surface $Γ$. In the compact case the asymptotics concerns negative eigenvalues, in the periodic case Floquet eigenvalues. We also give a bandwidth estimate in the case when a periodic $Γ$ decomposes into compact connected components. Finally, we comment on analogous systems of lower dimension and other aspects of the problem.

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↗