09:00 to 10:00 Excitonic influence on transport coefficients in low dimensional quantum systemsChair: JE Avron This is joint work with P. Duclos and B. Ricaud. Consider a many-body fermionic Hamiltonian defined with periodic boundary conditions on a two dimensional torus (which models a very long and thin nano-ring). We will study the low lying spectrum of this operator, and show that in the Hartree-Fock approximation one can obtain effective one-particle 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 self-adjoint operators, and integral equations. INI 1 10:00 to 11:00 Quantum transport in networks of weakly disordered metallic wiresChair: 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'tshuler-Aronov-Spivak (AAS) oscillations. The contributions of interfering reversed trajectories are encoded in the so-called "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 electron-electron interaction and more specifically how the AAS oscillations are affected by electron-electron interaction in several networks. INI 1 11:00 to 11:30 Coffee 11:30 to 12:30 Localization under external interactions in periodic quantum graphsChair: JE Avron We discuss the creation of eigenvalues in the T3-shaped 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 sameChair: 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 ([Fraunhofer-Institute])Inverse scattering problems for quantum graphsChair: 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 methodChair: 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}, Few-Body Systems 38(2-4), 125-131, 2006, ArXiv math-ph/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 graphsChair: 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 speed-up over its classical counterpart. INI 1 10:00 to 11:00 Optimal computation with noisy quantum walksChair: 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 time-averaging 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 endsChair: 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 graphsChair: 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 cross-sectional 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 stripChair: 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