arXiv subjects

Pavel Exner

Publications and source records attributed to Pavel Exner.

At least 181 records · Page 10Linked to original sources

arXiv2002

Multiple bound states in scissor-shaped waveguides

We study bound states of the two-dimensional Helmholtz equations with Dirichlet boundary conditions in an open geometry given by two straight leads of the same width which cross at an angle $θ$. Such a four-terminal junction with a tunable $θ$ can realized experimentally if a right-angle structure is filled by a ferrite. It is known that for $θ=90^o$ there is one proper bound state and one eigenvalue embedded in the continuum. We show that the number of eigenvalues becomes larger with increasing asymmetry and the bound-state energies are increasing as functions of $θ$ in the interval $(0,90^o)$. Moreover, states which are sufficiently strongly bent exist in pairs with a small energy difference and opposite parities. Finally, we discuss how with increasing $θ$ the bound states transform into the quasi-bound states with a complex wave vector.

cond-mat↗

arXiv2002

Curvature-induced bound states for a $δ$ interaction supported by a curve in $\mathbb{R}^3$

We study the Laplacian in $L^2(\mathbb{R}^3)$ perturbed on an infinite curve $Γ$ by a $δ$ interaction defined through boundary conditions which relate the corresponding generalized boundary values. We show that if $Γ$ is smooth and not a straight line but it is asymptotically straight in a suitable sense, and if the interaction does not vary along the curve, the perturbed operator has at least one isolated eigenvalue below the threshold of the essential spectrum.

math-ph↗

arXiv2002

Extended Standard Map with Spatio-Temporal Asymmetry

We analyze the transport properties of a set of symmetry-breaking extensions %, both spatial and temporal, of the Chirikov--Taylor Map. The spatial and temporal asymmetries result in the loss of periodicity in momentum direction in the phase space dynamics, enabling the asymmetric diffusion which is the origin of the unidirectional motion. The simplicity of the model makes the calculation of the global dynamical properties of the system feasible both in phase space and in controlling-parameter space. We present the results of numerical experiments which show the intricate dependence of the asymmetric diffusion to the controlling parameters.

cond-mat.stat-mech↗

arXiv2002

Persistent currents due to point obstacles

We discuss properties of the two-dimensional Landau Hamiltonian perturbed by a family of identical $δ$ potentials arranged equidistantly along a closed loop. It is demonstrated that for the loop size exceeding the effective size of the point obstacles and the cyclotronic radius such a system exhibits persistent currents at the bottom of the spectrum. We also show that the effect is sensitive to a small disorder.

quant-ph↗

arXiv2002

Persistent currents for 2D Schrödinger operator with a strong $δ$-interaction on a loop

We investigate the two-dimensional magnetic Schrödinger operator $H_{B,β}=(-i\nabla-A)^2 -βδ(\cdot-Γ)$, where $Γ$ is a smooth loop and the vector potential $A$ corresponds to a homogeneous magnetic field $B$ perpendicular to the plane. The asymptotics of negative eigenvalues of $H_{B,β}$ for $β\to\infty$ is found. It shows, in particular, that for large enough positive $β$ the system exhibits persistent currents.

math-ph↗

arXiv2001

Generalised boundary conditions for the Aharonov-Bohm effect combined with a homogeneous magnetic field

The most general admissible boundary conditions are derived for an idealised Aharonov-Bohm flux intersecting the plane at the origin on the background of a homogeneous magnetic field. A standard technique based on self-adjoint extensions yields a four-parameter family of boundary conditions; other two parameters of the model are the Aharonov-Bohm flux and the homogeneous magnetic field. The generalised boundary conditions may be regarded as a combination of the Aharonov-Bohm effect with a point interaction. Spectral properties of the derived Hamiltonians are studied in detail.

quant-ph↗

arXiv2001

Band gap of the Schroedinger operator with a strong delta-interaction on a periodic curve

In this paper we study the operator $H_β=-Δ-βδ(\cdot-Γ)$ in $L^{2}(\mathbb{R}^{2})$, where $Γ$ is a smooth periodic curve in $\mathbb{R}^{2}$. We obtain the asymptotic form of the band spectrum of $H_β$ as $β$ tends to infinity. Furthermore, we prove the existence of the band gap of $σ(H_β)$ for sufficiently large $β>0$. Finally, we also derive the spectral behaviour for $β\to\infty$ in the case when $Γ$ is non-periodic and asymptotically straight.

math-ph↗

arXiv2001

Potential approximations to $δ'$: an inverse Klauder phenomenon with norm-resolvent convergence

We show that there is a family Schroedinger operators with scaled potentials which approximates the $δ'$-interaction Hamiltonian in the norm-resolvent sense. This approximation, based on a formal scheme proposed by Cheon and Shigehara, has nontrivial convergence properties which are in several respects opposite to those of the Klauder phenomenon.

math-ph↗

arXiv2001

Bound states in point-interaction star-graphs

We discuss the discrete spectrum of the Hamiltonian describing a two-dimensional quantum particle interacting with an infinite family of point interactions. We suppose that the latter are arranged into a star-shaped graph with N arms and a fixed spacing between the interaction sites. We prove that the essential spectrum of this system is the same as that of the infinite straight "polymer", but in addition there are isolated eigenvalues unless N=2 and the graph is a straight line. We also show that the system has many strongly bound states if at least one of the angles between the star arms is small enough. Examples of eigenfunctions and eigenvalues are computed numerically.

quant-ph↗

arXiv2001

Bound States in Mildly Curved Layers

It has been shown recently that a nonrelativistic quantum particle constrained to a hard-wall layer of constant width built over a geodesically complete simply connected noncompact curved surface can have bound states provided the surface is not a plane. In this paper we study the weak-coupling asymptotics of these bound states, i.e. the situation when the surface is a mildly curved plane. Under suitable assumptions about regularity and decay of surface curvatures we derive the leading order in the ground-state eigenvalue expansion. The argument is based on Birman-Schwinger analysis of Schroedinger operators in a planar hard-wall layer.

math-ph↗

arXiv2001

Asymptotics of eigenvalues of the Schroedinger operator with a strong delta-interaction on a loop

In this paper we investigate the operator $H_β=-Δ-βδ(\cdot-Γ)$ in $L^{2}({\Bbb R}^{2})$, where $β>0$ and $Γ$ is a closed $C^{4}$ Jordan curve in ${\Bbb R}^{2}$. We obtain the asymptotic form of each eigenvalue of $H_β$ as $β$ tends to infinity. We also get the asymptotic form of the number of negative eigenvalues of $H_β$ in the strong coupling asymptotic regime.

math-ph↗

arXiv2001

Quantum mechanics of layers with a finite number of point perturbations

We study spectral and scattering properties of a spinless quantum particle confined to an infinite planar layer with hard walls containing a finite number of point perturbations. A solvable character of the model follows from the explicit form of the Hamiltonian resolvent obtained by means of Krein's formula. We prove the existence of bound states, demonstrate their properties, and find the on-shell scattering operator. Furthermore, we analyze the situation when the system is put into a homogeneous magnetic field perpendicular to the layer; in that case the point interactions generate eigenvalues of a finite multiplicity in the gaps of the free Hamiltonian essential spectrum.

math-ph↗

arXiv2001

A single-mode quantum transport in serial-structure geometric scatterers

We study transport in quantum systems consisting of a finite array of N identical single-channel scatterers. A general expression of the S matrix in terms of the individual-element data obtained recently for potential scattering is rederived in this wider context. It shows in particular how the band spectrum of the infinite periodic system arises in the limit $N\to\infty$. We illustrate the result on two kinds of examples. The first are serial graphs obtained by chaining loops or T-junctions. A detailed discussion is presented for a finite-periodic "comb"; we show how the resonance poles can be computed within the Krein formula approach. Another example concerns geometric scatterers where the individual element consists of a surface with a pair of leads; we show that apart of the resonances coming from the decoupled-surface eigenvalues such scatterers exhibit the high-energy behavior typical for the delta' interaction for the physically interesting couplings.

quant-ph↗

arXiv2001

Bound States in Curved Quantum Layers

We consider a nonrelativistic quantum particle constrained to a curved layer of constant width built over a non-compact surface embedded in $R^3$. We suppose that the latter is endowed with the geodesic polar coordinates and that the layer has the hard-wall boundary. Under the assumption that the surface curvatures vanish at infinity we find sufficient conditions which guarantee the existence of geometrically induced bound states.

math-ph↗

arXiv2000

Open quantum dots: resonances from perturbed symmetry and bound states in strong magnetic fields

We discuss the Noeckel model of an open quantum dot, i.e., a straight hard-wall channel with a potential well. If this potential depends on the longitudinal variable only, there are embedded eigenvalues which turn into resonances if the symmetry is violated, either by applying a magnetic field or by deformation of the well. For a weak symmetry breaking we derive the perturbative expansion of these resonances. We also deduce a sufficient condition under which the discrete spectrum of such a system (without any symmetry) survives the presence of a strong magnetic field. It is satisfied, in particular, if the dot potential is purely attractive.

math-ph↗

arXiv2000

Geometrically induced spectrum in curved leaky wires

We study measure perturbations of the Laplacian in $L^2(\R^2)$ supported by an infinite curve $Γ$ in the plane which is asymptotically straight in a suitable sense. We show that if $Γ$ is not a straight line, such a ``leaky quantum wire'' has at least one bound state below the threshold of the essential spectrum.

math-ph↗

arXiv2000

Bound states of infinite curved polymer chains

We investigate an infinite array of point interactions of the same strength in R^d, d=2,3, situated at vertices of a polygonal curve with a fixed edge length. We demonstrate that if the curve is not a line, but it is asymptotically straight in a suitable sense, the corresponding Hamiltonian has bound states. Example is given in which the number of these bound states can exceed any positive integer.

math-ph↗

arXiv2000

Avoided crossings in mesoscopic systems: electron propagation on a non-uniform magnetic cylinder

We consider an electron constrained to move on a surface with revolution symmetry in the presence of a constant magnetic field $B$ parallel to the surface axis. Depending on $B$ and the surface geometry the transverse part of the spectrum typically exhibits many crossings which change to avoided crossings if a weak symmetry breaking interaction is introduced. We study the effect of such perturbations on the quantum propagation. This problem admits a natural reformulation to which tools from molecular dynamics can be applied. In turn, this leads to the study of a perturbation theory for the time dependent Born-Oppenheimer approximation.

math-ph↗