Graph Models of Mesoscopic Systems, WaveGuides and NanoStructures
Tuesday 10th April 2007 to Friday 13th April 2007
08:30 to 09:55  Registration  
10:00 to 11:00 
Quantum swimming Chair: P Exner I shall describe a theory of quantum swimming developed jointly with Boris Gutkin and David Oaknin. The theory has the remarkable feature that swimming in a one dimensional Fermi sea at T=0 is quantized. The theory is closely related to quantum pumping and to adiabatic, time dependent scattering on graphs. 
INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:30 
Quantum fields and scale invariance on star graph Chair: P Exner We construct quantum fields and vertex algebras on star graphs. Our construction uses a specific deformation of the canonical algebra, encoding the interaction at the vertex of the graph. Special attention is devoted to the scale invariant interactions, which determine the critical properties of the system. We classify the critical points and investigate their features. The physical observables, we focus on, are the Casimir energy density and the conductance. Among the examples illustrating nontrivial interactions in the bulk of the graph, we consider the nonlinear Schrodinger and the massless Thirring models. 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
N March ([Oxford]) Electronic states in ordered and disordered quantum networks: with applications to graphene and to C and B nanotubes Chair: U Smilansky The idea behing the quantum network (QN) model is simple enough. One joins each atom to its neighbours, and then treat electron (though quantum mechanically of course) as though they flowed through onedimensional wires as in an electrical circuit obeying Kirfhhoff's Laws at every node. Here we will begin with two periodic systems: namely an infinite graphene layer and a twodimensional sheet of boson atoms. This will be followed by a discussion of C and especially B nanotubes. Finally a brief summary will be given on the nature of the electronic states in a disorderd network, via an approximate treatment related to Boltzmann's equation. 
INI 1  
15:00 to 15:30  Tea  
15:30 to 16:30 
Quantum network model of zigzag carbon nanotube Chair: U Smilansky We consider the quantum network which is a model for a zigzag carbon nanotube under assumption that the confinement potential created by sigma electrons restricts the pi electrons to the network. A parallel magnetic field and a wide range of periodic scalar potentials are taken into account. The spectral properties of the corresponding Schroedinger operator are analyzed using the Lyapunov function technique. We explicitly calculate the point spectrum and all localized eigenstates. We prove that there are stable spectral gaps, where the endpoints are periodic and antiperiodic eigenvalues, and resonance spectral gaps, where the endpoints are resonances; the asymptotics of all endpoints at high energies are provided. All finite gap potentials are described. The dependence of the spectrum on magnetic field is also investigated. 
INI 1  
16:30 to 17:30 
On the spectra of carbon nanostructures Chair: U Smilansky An explicit derivation of dispersion relations and spectra for periodic Schrödinger operators on carbon nanostructures (including graphene and all types of singlewall nanotubes) is provided. 
INI 1  
17:30 to 18:30  Welcome Wine Reception  
18:45 to 19:30  Dinner at Wolfson Court (Residents only) 
09:00 to 10:00 
Excitonic influence on transport coefficients in low dimensional quantum systems Chair: JE Avron This is joint work with P. Duclos and B. Ricaud. Consider a manybody fermionic Hamiltonian defined with periodic boundary conditions on a two dimensional torus (which models a very long and thin nanoring). We will study the low lying spectrum of this operator, and show that in the HartreeFock approximation one can obtain effective oneparticle models describing these particular states. Even though it sounds more physically oriented, the main purpose of this talk is to formulate clear mathematical problems related to PDEs, unbounded linear selfadjoint operators, and integral equations. 
INI 1  
10:00 to 11:00 
Quantum transport in networks of weakly disordered metallic wires Chair: JE Avron I will consider the quantum transport in networks of weakly disordered metallic wires. Quantum interferences of reversed trajectories are responsible for a small contribution to the conductance, known as the "weak localization correction" (WL). From the experimental point of view the study of the WL provides an efficient tool to probe phase coherence in weakly disordered metals. The WL is identified through its magnetic field dependence: for example, the conductance of a ring presents oscillations as a function of the flux with period $h/2e$, known as Al'tshulerAronovSpivak (AAS) oscillations. The contributions of interfering reversed trajectories are encoded in the socalled "Cooperon". I will show how the Cooperon must be properly integrated into a multiterminal network connected to reservoirs and will emphasize the role on nonlocality of quantum transport. In a second part I will discuss the effect of decoherence due to electronelectron interaction and more specifically how the AAS oscillations are affected by electronelectron interaction in several networks. 
INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:30 
Localization under external interactions in periodic quantum graphs Chair: JE Avron We discuss the creation of eigenvalues in the T3shaped quantum graph using magnetic field and Rashba interaction. For some combinations of parameters the whole spectrum degenerates into a series of isolated flat bands. 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
15:00 to 16:00 
Graphs which sound the same Chair: M Levitin After a short review of the conditions for unique spectral inversion for quantum graphs, I shall describe a method for constructing families of isospectral yet not isometric garphs: "graphs which sound the same". I shall then discuss the conjecture that graphs which sound the same can be resolved by the difference between their sequences of counts of nodal domains, and will present a proof that this is indeed the case for a simple yet non trivial example. 
INI 1  
16:00 to 17:00 
V Kostrykin ([FraunhoferInstitute]) Inverse scattering problems for quantum graphs Chair: M Levitin The talk is devoted to inverse scattering problems for Laplace operators on metric graphs. Some possible applications to network design will also be discussed. The talk is based on a joint work with R. Schrader. 
INI 1  
17:00 to 18:00 
On the skeleton method Chair: M Levitin In the spectral analysis of few one dimensional quantum particles interacting through delta potentials it is well known that one can recast the problem into the spectral analysis of an integral operator (the skeleton) living on the submanifold which supports the delta interactions. We shall present several tools which allow direct insights into the spectral structure of this skeleton. Application to effective models of excitons in nanotubes as well as some nets of quantum wires will be given. This is a work in progress with H. Cornean and B. Ricaud, see e.g. Three quantum charged particles interacting through delta potentials}, FewBody Systems 38(24), 125131, 2006, ArXiv mathph/0604003 
INI 1  
18:45 to 19:30  Dinner at Wolfson Court (Residents only) 
09:00 to 10:00 
Perfect transfer of quantum information across graphs Chair: A Figotin Quantum information is encoded in quantum mechanical states of physical systems. Hence, reliable transmission of quantum information from one location to another entails the perfect transfer of quantum mechanical states between these locations. We consider the situation in which the system used for this information transmission consists of N interacting spins and we address the problem of arranging the spins in a network in a manner which would allow perfect state transfer over the largest possible distance. The network is described by a graph G, with the vertices representing the locations of the spins and the edges connecting spins which interact with each other. State transfer is achieved by the time evolution of the spin system under a suitable Hamiltonian. This can be equivalently viewed as a continuous time quantum walk on the graph G. We find the maximal distance of perfect state transfer and prove that the corresponding quantum walk exhibits an exponential speedup over its classical counterpart. 
INI 1  
10:00 to 11:00 
Optimal computation with noisy quantum walks Chair: A Figotin Quantum versions of random walks on the line and cycle show a quadratic improvement in their spreading rate and mixing times respectively. The addition of decoherence to the quantum walk produces a more uniform distribution on the line, and even faster mixing on the cycle by removing the need for timeaveraging to obtain a uniform distribution. By calculating the entanglement between the coin and the position of the quantum walker, the optimal decoherence rates are found to be such that all the entanglement is just removed by the time the final measurement is made. This requires only O(log T) random bits for a quantum walk of T steps. 
INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:30 
Finding eigenvalues and resonances of the Laplacian on domains with regular ends Chair: A Figotin In this joint work with Marco Marletta (Cardiff), we present a simple uniform algorithm for finding eigenvalues (if they exist) lying below or embedded into the continuous spectrum, as well as complex resonances, of the Laplace operator on infinite domains with regular ends  e.g. cylindrical. 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
D Grieser ([Carl von Ossietzky, Oldenburg]) Spectral asymptotics of the Dirichlet Laplacian on fat graphs Chair: N Datta and V Kendon We investigate the behavior of the eigenvalues of the Laplacian, or a similar operator, on a family of Riemannian manifolds with boundary, called fat graphs, obtained by associating to the edges of a given finite graph crosssectional Riemannian manifolds with boundary, and also to the vertices certain Riemannian manifolds with boundary, glueing them according to the graph structure, and scaling them by a factor of $\varepsilon$ while keeping the lengths of the edges fixed. The simplest model of this is the $\varepsilon$neighborhood of a the graph embedded with straight edges in $R^n$. We determine the asymptotics of the eigenvalues with various boundary conditions as $\varepsilon\to 0$ in terms of combinatorial and scattering data. 
INI 1  
15:00 to 15:30  Tea  
15:30 to 16:30 
Dirichlet eigenvalues in a narrow strip Chair: N Datta and V Kendon
We study the Dirichlet Laplacian. We also show that convergence of eigenvalues here is a consequence of some ‘generalized’ version of the convergence in norm of the resolvents. A modificaton of the standard resolvent convergence is necessary, since the operators ?e for different ?, as well as the operator H, act in different spaces.

INI 1  
16:30 to 17:30 
M Harmer ([Australian National]) Spin filtering and the Rashba effect We discuss quantum graphs with the Rashba Hamiltonian with application to spin filtering 
INI 1  
20:00 to 18:00  Conference Dinner at Magdalene College 
09:00 to 10:00 
Phase transitions on lattice graphs Chair: H Cornean We explain the randomcluster (or FK) representation of the classical Potts/Ising models, when the underlying graph is a lattice. There is a continuum version of the randomcluster model that may be used to study the quantum Ising model with transverse field. A selection of open problems will be included. 
INI 1  
10:00 to 11:00 
Slow light in photonic crystals Chair: H Cornean The problem of slowing down light by orders of magnitude has been extensively discussed in the literature. Such a possibility can be useful in a variety of optical and microwave applications. Many qualitatively different approaches have been explored. Here we discuss how this goal can be achieved in linear dispersive media, such as photonic crystals. The existence of slowly propagating electromagnetic waves in photonic crystals is quite obvious and well known. The main problem, though, has been how to convert the input radiation into the slow mode without losing a significant portion of the incident light energy to absorption, reflection, etc. We show that the socalled frozen mode regime offers a unique solution to the above problem. Under the frozen mode regime, the incident light enters the photonic crystal with little reflection and, subsequently, is completely converted into the frozen mode with huge amplitude and almost zero group velocity. The linearity of the above effect allows the slowing of light regardless of its intensity. An additional advantage of photonic crystals over other methods of slowing down light is that photonic crystals can preserve both time and space coherence of the input electromagnetic wave. 
INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:30 
Simulation of quantum graphs by microwave networks Chair: H Cornean Experimental and theoretical study of irregular microwave graphs (networks) consisting of coaxial cables connected by the joints are presented. The spectra of the microwave networksare measured for bidirectional and directional microwave networks consisting of coaxial cables and Faraday isolators for which the time reversal symmetry is broken. In this way the statistical properties of the graphs such as the integrated nearest neighbor spacing distribution and the spectral rigidity are obtained.We show that microwave irregular microwave networks with absorption can be used to experimental studies of the distributions of Wigner's reaction K matrix. The distributions of the imaginary and real parts of K matrix, P(v) and P(u), respectively, are obtained from the measurements and numerical calculations of the scattering matrix S of the networks. We demonstrate that the experimental and numerical results are in good agreement with the theoretical predictions. Furthermore, we present the results of studies of parameter dependent velocity correlation functions in the graphs. 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
Localized shelf waves on a curved coast  existence of eigenvalues of a linear operator pencil in a curved waveguide Chair: P Kuchment The study of the possibility of the nonpropagating, trapped continental shelf waves along curved coasts reduces mathematically to a spectral problem for a selfadjoint operator pencil in a curved strip. Using the methods developed in the setting of the waveguide trapped mode problem, we show that such continental shelf waves exist for a wide class of coast curvature and depth profiles. This is joint work with Ted Johnson (UCL) and Michael Levitin (HeriotWatt) 
INI 1  
15:00 to 15:30  Tea  
15:30 to 16:30 
O Turek ([Czech Technical]) Approximations of strongly singular couplings at quantum graph vertices Chair: P Kuchment Approximation of a general strongly singular coupling in the center of a star graph by means of a deltacoupling and deltainteractions will be presented. We will (i) formulate necessary conditions under which the approximation is possible; (ii) show a way how to realize such approximation; (iii) explain its character: we will demonstrate that the coupling is approximated in the normresolvent sense. 
INI 1  
16:30 to 17:30 
Twisting versus bending in quantum waveguides Chair: P Kuchment The Dirichlet Laplacian in tubular domains is a simple but remarkably successful model for the quantum Hamiltonian in mesoscopic waveguide systems. We make an overview of mathematical results established so far about the spectrum of the Dirichlet Laplacian in infinite curved threedimensional tubes with arbitrary crosssection and mention consequences for the electronic transport. We focus on the interplay between bending and twisting as regards the existence of quantum bound states, associated with the discrete spectrum of the Laplacian. As the most recent result, we show that twisting of an infinite straight threedimensional tube with noncircular crosssection gives rise to a Hardytype inequality, with important consequences for the stability of the spectrum. We also discuss similar effects induced by curvature of the ambient space or switch of boundary conditions. Related Links

INI 1  
18:45 to 19:30  Dinner at Wolfson Court (Residents only) 