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Pavel Exner

Publications and source records attributed to Pavel Exner.

At least 145 records · Page 8Linked to original sources
arXiv2007

Quantum networks modelled by graphs

Quantum networks are often modelled using Schroedinger operators on metric graphs. To give meaning to such models one has to know how to interpret the boundary conditions which match the wave functions at the graph vertices. In this article we give a survey, technically not too heavy, of several recent results which serve this purpose. Specifically, we consider approximations by means of ``fat graphs'' -- in other words, suitable families of shrinking manifolds -- and discuss convergence of the spectra and resonances in such a setting.

math-ph↗
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↗
arXiv2007

On the optimization of the principal eigenvalue for single-centre point-interaction operators in a bounded region

We investigate relations between spectral properties of a single-centre point-interaction Hamiltonian describing a particle confined to a bounded domain $Ω\subset\mathbb{R}^{d},\: d=2,3$, with Dirichlet boundary, and the geometry of $Ω$. For this class of operators Krein's formula yields an explicit representation of the resolvent in terms of the integral kernel of the unperturbed one, $(-Δ_Ω^{D}+z) ^{-1}$. We use a moving plane analysis to characterize the behaviour of the ground-state energy of the Hamiltonian with respect to the point-interaction position and the shape of $Ω$, in particular, we establish some conditions showing how to place the interaction to optimize the principal eigenvalue.

math-ph↗
arXiv2007

A "hybrid plane" with spin-orbit interaction

In this paper we attempt to reconstruct one of the last projects of Volodya Geyler which remained unfinished. We study motion of a quantum particle in the plane to which a halfline lead is attached assuming that the particle has spin $\frac12$ and the plane component of the Hamiltonian contains a spin-orbit interaction of either Rashba or Dresselhaus type. We construct the class of admissible Hamiltonians and derive an explicit expression for the Green function applying it to the scattering in such a system.

math-ph↗
arXiv2007

On the dense point and absolutely continuous spectrum for Hamiltonians with concentric $δ$ shells

We consider Schrödinger operator in dimension $d\ge 2$ with a singular interaction supported by an infinite family of concentric spheres, analogous to a system studied by Hempel and coauthors for regular potentials. The essential spectrum covers a halfline determined by the appropriate one-dimensional comparison operator; it is dense pure point in the gaps of the latter. If the interaction is radially periodic, there are absolutely continuous bands; in contrast to the regular case the measure of the p.p. segments does not vanish in the high-energy limit.

math-ph↗
arXiv2007

Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent waveguide

In distinction to the Neumann case the squeezing limit of a Dirichlet network leads in the threshold region generically to a quantum graph with disconnected edges, exceptions may come from threshold resonances. Our main point in this paper is to show that modifying locally the geometry we can achieve in the limit a nontrivial coupling between the edges including, in particular, the class of $δ$-type boundary conditions. We work out an illustration of this claim in the simplest case when a bent waveguide is squeezed.

math-ph↗
arXiv2007

A remark on helical waveguides

Motivated by a proposal to create an optical helix-shaped waveguides for cold atoms and molecules, we discuss local perturbations which can create bound states in such a setting. This is known about a local slowdown of the twist; we show that a similar effect can result from a local tube protrusion or a change of the helix radius in correlation with its pitch angle.

quant-ph↗
arXiv2007

Approximations of singular vertex couplings in quantum graphs

We discuss approximations of the vertex coupling on a star-shaped quantum graph of $n$ edges in the singular case when the wave functions are not continuous at the vertex and no edge-permutation symmetry is present. It is shown that the Cheon-Shigehara technique using $δ$ interactions with nonlinearly scaled couplings yields a $2n$-parameter family of boundary conditions in the sense of norm resolvent topology. Moreover, using graphs with additional edges one can approximate the ${n+1\choose 2}$-parameter family of all time-reversal invariant couplings.

math-ph↗
arXiv2007

On the critical exponent in an isoperimetric inequality for chords

The problem of maximizing the $L^p$ norms of chords connecting points on a closed curve separated by arclength $u$ arises in electrostatic and quantum--mechanical problems. It is known that among all closed curves of fixed length, the unique maximizing shape is the circle for $1 \le p \le 2$, but this is not the case for sufficiently large values of $p$. Here we determine the critical value $p_c(u)$ of $p$ above which the circle is not a local maximizer finding, in particular, that $p_c(\frac12 L)=\frac52$. This corrects a claim made in \cite{EHL}.

math-ph↗
arXiv2007

Convergence of resonances on thin branched quantum wave guides

We prove an abstract criterion stating resolvent convergence in the case of operators acting in different Hilbert spaces. This result is then applied to the case of Laplacians on a family $X_\eps$ of branched quantum waveguides. Combining it with an exterior complex scaling we show, in particular, that the resonances on $X_\eps$ approximate those of the Laplacian with ``free'' boundary conditions on $X_0$, the skeleton graph of $X_\eps$.

math-ph↗
arXiv2006

Equivalence of resolvent and scattering resonances on quantum graphs

We discuss resonances for Schrödinger operators on metric graphs which consists of a finite compact part and a finite number of halflines attached to it; the vertex coupling is assumed to be of the $δ$-type or certain modifications of it. Using exterior complex scaling on the graph we show that the resolvent and scattering resonances coincide in this case.

math-ph↗
arXiv2006

Resonance asymptotics in the generalized Winter model

We consider a modification of the Winter model describing a quantum particle in presence of a spherical barrier given by a fixed generalized point interaction. It is shown that the three classes of such interactions correspond to three different types of asymptotic behaviour of resonances of the model at high energies.

quant-ph↗
arXiv2006

Zeno product formula revisited

We introduce a new product formula which combines an orthogonal projection with a complex function of a non-negative operator. Under certain assumptions on the complex function the strong convergence of the product formula is shown. Under more restrictive assumptions even operator-norm convergence is verified. The mentioned formula can be used to describe Zeno dynamics in the situation when the usual non-decay measurement is replaced by a particular generalized observables in the sense of Davies.

math-ph↗
arXiv2006

The decay law can have an irregular character

Within a well-known decay model describing a particle confined initially within a spherical $δ$ potential shell, we consider the situation when the undecayed state has an unusual energy distribution decaying slowly as $k\to\infty$; the simplest example corresponds to a wave function constant within the shell. We show that the non-decay probability as a function of time behaves then in a highly irregular, most likely fractal way.

quant-ph↗
arXiv2005

Necklaces with interacting beads: isoperimetric problems

We discuss a pair of isoperimetric problems which at a glance seem to be unrelated. The first one is classical: one places $N$ identical point charges at a closed curve $Γ$ at the same arc-length distances and asks about the energy minimum, i.e. which shape does the loop take if left by itself. The second problem comes from quantum mechanics: we take a Schrödinger operator in $L^2(\mathbb{R}^d), d=2,3,$ with $N$ identical point interaction placed at a loop in the described way, and ask about the configuration which \emph{maximizes} the ground state energy. We reduce both of them to geometric inequalities which involve chords of $Γ$; it will be shown that a sharp local extremum is in both cases reached by $Γ$ in the form of a regular (planar) polygon and that such a $Γ$ solves the two problems also globally.

math-ph↗
arXiv2005

Inequalities for means of chords, with application to isoperimetric problems

We consider a pair of isoperimetric problems arising in physics. The first concerns a Schrödinger operator in $L^2(\mathbb{R}^2)$ with an attractive interaction supported on a closed curve $Γ$, formally given by $-Δ-αδ(x-Γ)$; we ask which curve of a given length maximizes the ground state energy. In the second problem we have a loop-shaped thread $Γ$ in $\mathbb{R}^3$, homogeneously charged but not conducting, and we ask about the (renormalized) potential-energy minimizer. Both problems reduce to purely geometric questions about inequalities for mean values of chords of $Γ$. We prove an isoperimetric theorem for $p$-means of chords of curves when $p \leq 2$, which implies in particular that the global extrema for the physical problems are always attained when $Γ$ is a circle. The article finishes with a discussion of the $p$--means of chords when $p > 2$.

math-ph↗
arXiv2005

Approximations by graphs and emergence of global structures

We study approximations of billiard systems by lattice graphs. It is demonstrated that under natural assumptions the graph wavefunctions approximate solutions of the Schroedinger equation with energy rescaled by the billiard dimension. As an example, we analyze a Sinai billiard with attached leads. The results illustrate emergence of global structures in large quantum graphs and offer interesting comparisons with patterns observed in complex networks of a different nature.

quant-ph↗