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Pavel Exner

Publications and source records attributed to Pavel Exner.

At least 109 records · Page 6Linked to original sources
arXiv2013

Approximations of quantum-graph vertex couplings by singularly scaled rank-one operators

We investigate approximations of the vertex coupling on a star-shaped graph by families of operators with singularly scaled rank-one interactions. We find a family of vertex couplings, generalizing the $δ'$-interaction on the line, and show that with a suitable choice of the parameters they can be approximated in this way in the norm-resolvent sense. We also analyze spectral properties of the involved operators and demonstrate convergence of the corresponding on-shell scattering matrices.

math.SP↗
arXiv2013

A regular version of Smilansky model

We discuss a modification of Smilansky model in which a singular potential `channel' is replaced by a regular, below unbounded potential which shrinks as it becomes deeper. We demonstrate that, similarly to the original model, such a system exhibits a spectral transition with respect to the coupling constant, and determine the critical value above which a new spectral branch opens. The result is generalized to situations with multiple potential `channels'.

math-ph↗
arXiv2013

Schrödinger operators with δ- and δ'-interactions on Lipschitz surfaces and chromatic numbers of associated partitions

We investigate Schrödinger operators with δ- and δ'-interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial properties of the partition and the spectral properties of the corresponding operators are related. As the main result we prove an operator inequality for the Schrödinger operators with δ- and δ'-interactions which is based on an optimal colouring and involves the chromatic number of the partition. This inequality implies various relations for the spectra of the Schrödinger operators and, in particular, it allows to transform known results for Schrödinger operators with δ-interactions to Schrödinger operators with δ'-interactions.

math-ph↗
arXiv2013

Approximations of quantum-graph vertex couplings by singularly scaled potentials

We investigate the limit properties of a family of Schrödinger operators of the form $H_\varepsilon= -\frac{\mathrm{d}^2}{\mathrm{d}x^2}+ \frac{λ(\varepsilon)}{\varepsilon^2}Q \big(\frac{x}{\varepsilon}\big)$ acting on $n$-edge star graphs with Kirchhoff conditions imposed at the vertex. The real-valued potential $Q$ is supposed to have compact support and $λ(\cdot)$ to be analytic around $\varepsilon=0$ with $λ(0)=1$. We show that if the operator has a zero-energy resonance of order $m$ for $\varepsilon=1$ and $λ(1)=1$, in the limit $\varepsilon\to 0$ one obtains the Laplacian with a vertex coupling depending on $1+\frac12 m(2n-m+1)$ parameters. We prove the norm-resolvent convergence as well as the convergence of the corresponding on-shell scattering matrices. The obtained vertex couplings are of scale-invariant type provided $λ'(0)=0$; otherwise the scattering matrix depends on energy and the scaled potential becomes asymptotically opaque in the low-energy limit.

math-ph↗
arXiv2013

Absence of absolutely continuous spectrum for the Kirchhoff Laplacian on radial trees

In this paper we prove that the existence of absolutely continuous spectrum of the Kirchhoff Laplacian on a radial metric tree graph together with a finite complexity of the geometry of the tree implies that the tree is in fact eventually periodic. This complements the results by Breuer and Frank in \cite{BreuerFrank2009} in the discrete case as well as for sparse trees in the metric case.

math.SP↗
arXiv2013

Spectral asymptotics of a strong $δ'$ interaction on a planar loop

We consider a generalized Schrödinger operator in $L^2(\R^2)$ with an attractive strongly singular interaction of $δ'$ type characterized by the coupling parameter $β>0$ and supported by a $C^4$-smooth closed curve $Γ$ of length $L$ without self-intersections. It is shown that in the strong coupling limit, $β\to 0_+$, the number of eigenvalues behaves as $\frac{2L}{πβ} + \OO(|\lnβ|)$, and furthermore, that the asymptotic behaviour of the $j$-th eigenvalue in the same limit is $-\frac{4}{β^2} +μ_j+\OO(β|\lnβ|)$, where $μ_j$ is the $j$-th eigenvalue of the Schrödinger operator on $L^2(0,L)$ with periodic boundary conditions and the potential $-\frac14 γ^2$ where $γ$ is the signed curvature of $Γ$.

math-ph↗
arXiv2013

Resonances on hedgehog manifolds

We discuss resonances for a nonrelativistic and spinless quantum particle confined to a two- or three-dimensional Riemannian manifold to which a finite number of semiinfinite leads is attached. Resolvent and scattering resonances are shown to coincide in this situation. Next we consider the resonances together with embedded eigenvalues and ask about the high-energy asymptotics of such a family. For the case when all the halflines are attached at a single point we prove that all resonances are in the momentum plane confined to a strip parallel to the real axis, in contrast to the analogous asymptotics in some metric quantum graphs; we illustrate it on several simple examples. On the other hand, the resonance behaviour can be influenced by a magnetic field. We provide an example of such a `hedgehog' manifold at which a suitable Aharonov-Bohm flux leads to absence of any true resonance, i.e. that corresponding to a pole outside the real axis.

math-ph↗
arXiv2013

On some sharp spectral inequalities for Schrödinger operators on semi-axis

In this paper we obtain sharp Lieb-Thirring inequalities for a Schrödinger operator on semi-axis with a matrix potential and show how they can be used to other related problems. Among them are spectral inequalities on star graphs and spectral inequalities for Schrödinger operators on half-spaces with Robin boundary conditions.

math-ph↗
arXiv2012

Spectral estimates for Dirichlet Laplacians on perturbed twisted tubes

We investigate Dirichlet Laplacian in a straight twisted tube of a non-circular cross section, in particular, its discrete spectrum coming from a local slowdown of the twist. We prove a Lieb-Thirring-type estimate for the spectral moments and present two examples illustrating how the bound depends on the tube cross section.

math-ph↗
arXiv2012

Spectral estimates for Dirichlet Laplacians and Schroedinger operators on geometrically nontrivial cusps

The goal of this paper is to derive estimates of eigenvalue moments for Dirichlet Laplacians and Schrödinger operators in regions having infinite cusps which are geometrically nontrivial being either curved or twisted; we are going to show how those geometric properties enter the eigenvalue bounds. The obtained inequalities reflect the essentially one-dimensional character of the cusps and we give an example showing that in an intermediate energy region they can be much stronger than the usual semiclassical bounds.

math-ph↗
arXiv2012

A general approximation of quantum graph vertex couplings by scaled Schroedinger operators on thin branched manifolds

We demonstrate that any self-adjoint coupling in a quantum graph vertex can be approximated by a family of magnetic Schroedinger operators on a tubular network built over the graph. If such a manifold has a boundary, Neumann conditions are imposed at it. The procedure involves a local change of graph topology in the vicinity of the vertex; the approximation scheme constructed on the graph is subsequently `lifted' to the manifold. For the corresponding operator a norm-resolvent convergence is proved, with the natural identification map, as the tube diameters tend to zero.

math-ph↗
arXiv2012

Momentum operators on graphs

We discuss ways in which momentum operators can be introduced on an oriented metric graph. A necessary condition appears to the balanced property, or a matching between the numbers of incoming and outgoing edges; we show that a graph without an orientation, locally finite and at most countably infinite, can made balanced oriented \emph{iff} the degree of each vertex is even. On such graphs we construct families of momentum operators; we analyze their spectra and associated unitary groups. We also show that the unique continuation principle does not hold here.

math-ph↗
arXiv2012

Solvable models of resonances and decays

Resonance and decay phenomena are ubiquitous in the quantum world. To understand them in their complexity it is useful to study solvable models in a wide sense, that is, systems which can be treated by analytical means. The present review offers a survey of such models starting the classical Friedrichs result and carrying further to recent developments in the theory of quantum graphs. Our attention concentrates on dynamical mechanism underlying resonance effects and at time evolution of the related unstable systems.

math-ph↗
arXiv2011

On the ground state of quantum graphs with attractive $δ$-coupling

We study relations between the ground-state energy of a quantum graph Hamiltonian with attractive $δ$ coupling at the vertices and the graph geometry. We derive a necessary and sufficient condition under which the energy increases with the increase of graph edge lengths. We show that this is always the case if the graph has no branchings while both change signs are possible for graphs with a more complicated topology.

math-ph↗
arXiv2011

Spectral estimates for a class of Schrödinger operators with infinite phase space and potential unbounded from below

We analyze two-dimensional Schrödinger operators with the potential $|xy|^p - λ(x^2+y^2)^{p/(p+2)}$ where $p\ge 1$ and $λ\ge 0$. We show that there is a critical value of $λ$ such that the spectrum for $λ<λ_\mathrm{crit}$ is below bounded and purely discrete, while for $λ>λ_\mathrm{crit}$ it is unbounded from below. In the subcritical case we prove upper and lower bounds for the eigenvalue sums.

math-ph↗
arXiv2011

Built-up structure criticality

The built-up land represents an important type of overall landscape. In this paper the built-up land structure in the largest cities in the Czech Republic and selected cities in the U.S.A. is analysed using the framework of statistical physics. We calculate the variance of the built-up area and the number variance of built-up landed plots in discs. In both cases the variance as a function of a disc radius follows a power law. The obtained values of power law exponents are comparable through different cities. The study is based on cadastral data from the Czech Republic and building footprints from GIS data in the U.S.A.

physics.soc-ph↗
arXiv2011

Inverse scattering problem for quantum graph vertices

We demonstrate how the inverse scattering problem of a quantum star graph can be solved by means of diagonalization of Hermitian unitary matrix when the vertex coupling is of the scale invariant (or Fülőp-Tsutsui) form. This enables the construction of quantum graphs with desired properties in a tailor-made fashion. The procedure is illustrated on the example of quantum vertices with equal transmission probabilities.

quant-ph↗