On Relations Between Stable and Zeno Dynamics in a Leaky Graph Decay Model

Exner, Pavel; Ichinose, Takashi; Kondej, Sylwia

doi:10.1007/978-3-7643-8135-6_3

Cited by 4 publications

(7 citation statements)

References 13 publications

(37 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Using these formulae one has to find zeros of det D(·) (−1) ; we shall sketch the argument referring to [EK04,EIK07] for details. We have mentioned that resonances are caused by tunneling between the parts of the interaction support, hence it is convenient to introduce the following reparametrization, .…”

Section: Leaky Graphs: a Caricature Of Quantum Wire And Dotsmentioning

confidence: 99%

“…We are particularly interested in the weak-coupling situation which in the present case means that the distance between Σ and Π is a large at the scale given by (− m ) −1/2 . Let us denote by E j the one-dimensional projection associated with the eigenfunction ψ j , the one can make the following claim [EIK07].…”

Section: Leaky Graphs: a Caricature Of Quantum Wire And Dotsmentioning

confidence: 99%

See 1 more Smart Citation

Solvable Models of Resonances and Decays

Exner

2013

Mathematical Physics, Spectral Theory and Stochastic Analysis

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Leaky Graphs: a Caricature Of Quantum Wire And Dotsmentioning

confidence: 99%

Section: Leaky Graphs: a Caricature Of Quantum Wire And Dotsmentioning

confidence: 99%

Solvable Models of Resonances and Decays

Exner

2013

Mathematical Physics, Spectral Theory and Stochastic Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

“…Another interesting aproach which has been discussed mainly by Exner and his coworkers is based on leaky graphs [101,100,102,103,105,104,107,108,109]. Here, a finite attractive potential in the Schrödinger equation is centered on a metric graph.…”

Section: Introductionmentioning

confidence: 99%

Quantum graphs: Applications to quantum chaos and universal spectral statistics

Gnutzmann

Smilansky

2006

Advances in Physics

313

450

View full text Add to dashboard Cite

During the last years quantum graphs have become a paradigm of quantum chaos with applications from spectral statistics to chaotic scattering and wave function statistics. In the first part of this review we give a detailed introduction to the spectral theory of quantum graphs and discuss exact trace formulae for the spectrum and the quantum-to-classical correspondence. The second part of this review is devoted to the spectral statistics of quantum graphs as an application to quantum chaos. Especially, we summarise recent developments on the spectral statistics of generic large quantum graphs based on two approaches: the periodic-orbit approach and the supersymmetry approach. The latter provides a condition and a proof for universal spectral statistics as predicted by random-matrix theory. Chapter 1 IntroductionThe general mathematical concept of a graph (network) as a set of elements which are connected by some relation has found applications in many branches of science, engineering, and also social science. A street network of a traffic engineer, the network of neurons studied by a neuroscientist, the structure of databases in computer science can all be described by graphs.Recently the Laplacian on a metric graph has gained a lot of attention in physics and mathematics in terms of the diffusion equation or Schrödinger equation. They have now become known as quantum graphs but different aspects are studied under various names such as quantum networks or quantum wires. They have a long history in mathematics and physics.In physics, the first application has probably been in the context of free electron models for organic molecules about seventy years ago by Pauling [198], an approach which has been further developed in subsequent years [172,173,201,209,77,182,204]. Quantum graphs have also been applied successfully to superconductivity in granular and artificial materials [12], acoustic and electromagnetic waveguide networks [112,181], the Anderson transition in a disordered wire [18,222], quantum Hall systems [71,151,150], fracton excitations in fractal structures [22,190], and mesoscopic quantum systems [142,166,242,243,241]. Quantum graphs have also been simulated experimentally [141].The construction of self-adjoint operators, or wave equations with appropriate boundary conditions on graphs has first been addressed by Ruedenberg and Scherr [209] (see also [204]). They considered graphs as an idealisation of networks of wires or wave guides of finite cross-section in the limit where the diameter of the wire is much smaller than any other length scale. Similar approaches to graphs as networks of thin wires with a finite diameter, or fat quantum graphs as they are now called, have been a topic in mathematical physics recently [211,212,208,167,106,202].Another interesting aproach which has been discussed mainly by Exner and his coworkers is based on leaky graphs [101,100,102,103,105,104 , 108, 109]. Here, a finite attractive potential in the Schrödinger equation is centered on a metric graph. A leaky g...

show abstract

“…In this paper also other spectral and scattering properties of this system are discussed as well as an extension to the three-dimensional situation with the interaction supported by a plane and a finite number of points. Furthermore, resonances in the case n > 1 are discussed in [EIK07] where one analyzes also the decay of an eigenstate of H 0,β due to the presence of the leaky line -see also Sec. 7.18.…”

Section: Edge Currents In the Absence Of Edgesmentioning

confidence: 99%

“…For the caricature model of Sec. 6.2 it can be done [EIK07]; one would welcome to learn more about the decay processes in physically more interesting cases.…”

mentioning

confidence: 99%

Leaky quantum graphs: A review

Exner

2008

Analysis on Graphs and Its Applications

Self Cite

View full text Add to dashboard Cite

Abstract. The aim of this review is to provide an overview of a recent work concerning "leaky" quantum graphs described by Hamiltonians given formally by the expression −∆ − αδ(x − Γ) with a singular attractive interaction supported by a graph-like set in R ν , ν = 2, 3. We will explain how such singular Schrödinger operators can be properly defined for different codimensions of Γ. Furthermore, we are going to discuss their properties, in particular, the way in which the geometry of Γ influences their spectra and the scattering, strong-coupling asymptotic behavior, and a discrete counterpart to leakygraph Hamiltonians using point interactions. The subject cannot be regarded as closed at present, and we will add a list of open problems hoping that the reader will take some of them as a challenge.

show abstract

On Relations Between Stable and Zeno Dynamics in a Leaky Graph Decay Model

Cited by 4 publications

References 13 publications

Solvable Models of Resonances and Decays

Solvable Models of Resonances and Decays

Quantum graphs: Applications to quantum chaos and universal spectral statistics

Leaky quantum graphs: A review

Contact Info

Product

Resources

About