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Seminars (AGA)

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Event When Speaker Title Presentation Material
AGA 18th January 2007
14:30 to 15:30
What is a Discrete Complex Analysis on the Equilateral Triangle Lattice?
AGA 18th January 2007
16:00 to 17:00
A Zuk Spectra of groups
AGA 23rd January 2007
14:30 to 15:30
T Sunada On the K$_4$ crystal
AGA 23rd January 2007
16:00 to 17:00
P Kuchment Liouville theorems for equations on coverings of graphs and manifolds

Analogs of the classical Liouville theorem concerning analytic or harmonic functions of polynomial growth have been studied in the recent decades in the settings of Riemannian manifolds, complex manifolds, discrete groups, and graphs. The talk will describe the recent progress for the case of periodic equations on abelian coverings, which works simultaneously in all these situations.

AGA 26th January 2007
11:00 to 12:00
Asymptotic aspects of Schreier graphs of self-similar groups

We consider Schreier graphs of actions of self-similar groups and present relations between their asymptotic properties, such as volume growth, growth of diameters, rate of vanishing of the spectral gap, etc. Among the examples illustrating the possible types of behavior we consider Hanoi Towers groups, Baumslag-Solitar groups, lamplighters, Basilica group, Bellaterra group, etc.

AGA 1st February 2007
14:30 to 15:30
YV Kurylev An approximate boundary distance representation and an approximate rec
AGA 7th February 2007
16:00 to 17:00
Eigenvalue estimates for Schroedinger operators on regular metric trees
AGA 8th February 2007
14:30 to 15:30
J Harrison Quantum chaos on graphs
AGA 15th February 2007
14:30 to 15:30
Scattering on a decorated star-graph as a toy model for the spectral theory of automorphic functions

A compact Riemannian manifold of dimension less than 4 with a finite number of semi-lines attached to the manifold is considered. It is shown that there is a deep analogy between the scattering and spectral properties of Schrodinger operators for this hybrid manifold and those for the automorphic Laplacian on Riemann surfaces with cusps. As an application, a relation between the scattering amplitude for hybrid manifolds with underlying compact Riemann surfaces of constant negative curvature and the Selberg zeta-function for this surface is obtained.

AGA 15th February 2007
16:00 to 17:00
Matrix valued orthogonal polynomials and random walks

There is a classical result going at least as far back as Karlin and McGregor (around 1950) giving the transition probability p(x,y,t) or the N-step transition probability for a birth-and-death process in terms of some scalar valued orthogonal polynomials arising from the corresponding tridiagonal matrix. I will introduce the theory and some examples of matrix valued orthogonal polynomials, review the results of Karlin-McGregor, and show how these matrix valued polynomails allow one to deal with Markov chains that go beyond birth-and-death processes.

AGA 16th February 2007
11:30 to 12:30
Hyperbolic polynomials and lower bounds in combinatorics

The Van der Waerden conjecture states that the permanent of an nxn doubly stochastic matrix A satisfies the inequality Per(A) >= n! / n^n (VDW bound). It had been for many years the most important conjecture about permanents, till it was finally proven independently by D.I. Falikman and by G.P.Egorychev in 1981. The VDW bound is the simplest and most powerful bound on permanents and therefore is among the most useful general purpose bounds in combinatorics. Its worthy successor , the Erdos-Schrijver-Valiant conjecture on the lower bound on the number of perfect matchings in k-regular bipartite graphs was posed in 1980 and resolved by A.Schrijver in 1998. The Schrijver's proof is one the most remarkable (and "highly complicated") results in the graph theory .

We introduce and describe the proof of a vast algebraic generalization of both Van der Waerden and Schrijver-Valiant conjectures. Besides being much more general, our proof is much shorter and conceptually simpler than the original proofs; it proves Van der Waerden/Schrijver-Valiant conjectures in "one shot". The main tool is the concept of hyperbolic polynomials, which were originally introduced and studied in PDEs. In this language, Van der Waerden and Schrijver-Valiant Conjectures correspond to the case of hyperbolic polynomials that are products of linear forms. Our proof is relatively simple and "noncomputational"; it in fact slightly improves the Schrijver's lower bound, and uses only basic properties of hyperbolic polynomials .

Time permitting, I will talk about non-hyperbolic results, such as analogues of Van der Waerden/Schrijver-Valiant conjectures for mixed volumes on convex sets.

AGA 21st February 2007
16:00 to 17:00
Spectral edges for periodic operators - some quantum graph examples

The Floquet-Bloch theory for a periodic self-adjoint operator allows one to compute the spectrum as a union of discrete spectra of the Bloch Hamiltonian for quasimomenta in the Brillouin zone. In many practical cases one can, in fact, obtain the correct spectrum by running over only the boundary of the reduced Brillouin zone (in other words, the edges of the spectrum occur at symmetry points of the zone). In dimensions 2 and higher there seems to be no analytical reason for why this should be the case, yet counterexamples are difficult to come by or are numerically unconvincing. In collaboration with J. Harrison, P. Kuchment, and A. Sobolev we find examples for which the spectral edges are strictly and unambiguously found in the interior of the Brillouin zone - firmly resolving this question. The examples are first constructed for the graph and quantum graph cases and then bootstrapped to the higher dimensions.

AGA 27th February 2007
14:30 to 15:30
Is there an interesting index theory for quantum graphs?

The classic paper of Roth shows that, for a graph with the Kirchhoff boundary conditions, the constant term in the heat-kernel expansion equals half the difference between the vertex number and the edge number. An index interpretation is achieved by comparing with another operator on the same graph with different boundary conditions. The construction is a generalization of Peter Gilkey's treatment of Neumann and Dirichlet boundary conditions on the interval as the simplest case of the de Rham complex on a manifold with boundary. It uses Justin Wilson's generalization of Roth's formula to any boundary conditions that yield a real bond-to-bond S-matrix.

AGA 1st March 2007
16:00 to 17:00
Another Martini for quantum graphs

I will report on joint work with V. Geyler and K. Pankrashkin. As one might guess from the title, the topic of the talk refers to the "Ten Martini Problem" which consisted in determining the set theoretic nature - band or Cantor structure - of the Harper operator in terms of number theoretic properties of a certain parameter. We treat the analogous problem for a periodic square quantum graph with magnetic field and suitable boundary conditions. We achieve the desired characterization of the spectrum essentially by reduction to a discrete periodic case for which the answer was given recently by Avila and Jitomirskaya.

AGA 6th March 2007
16:00 to 17:00
A Terras Zeta functions of weighted graphs and covering graphs

First we review the basic facts about Ihara zeta functions. Then we discuss two topics. 1)A condition on edge weights will be given that insures that the natural Ihara zeta function associated to a weighted graph has a 3-term determinant formula. 2)The distribution of poles of zeta functions of abelian graph coverings of a fixed small irregular graph will be compared with with those for a random covering.

This is joint work with H. Stark and M. Horton.

AGA 7th March 2007
16:00 to 17:00
Zeta and L-functions of graphs

Zeta functions of regular graphs are closely connected with the adjacency matrices of regular graphs with knowledge of the one immediately giving knowledge of the other. For irregular graphs, although there is a relationship, going back and forth is less easy. In this talk we will explore the various types of zeta functions and give examples of their uses.

AGA 13th March 2007
14:30 to 15:30
A computer-assisted existence proof for photonic band gaps

Authors: V. Hoang, M. Plum, Ch. Wieners (Karlsruhe, Germany)

The investigation of monochromatic waves in a periodic dielectric medium ("photonic crystal") leads to a spectral problem for a Maxwell operator. It is well known that the spectrum is characterized as a countable union of compact real intervals (``bands'') which may or may not be separated by gaps, and the occurrence of such gaps is of great practical interest but difficult to prove analytically. In this talk, we will attack this problem, for the 2D case, by computer-assisted means. First we reduce the problem, using an analytical perturbation type argument, to the computation of enclosures for finitely many eigenvalues of finitely many periodic eigenvalue problems. This task is then carried out by computer-assisted variational methods.

AGA 13th March 2007
16:00 to 17:00
Validated computation tool for Perron-Frobenius eigenvalue using graph theory

Any nonnegative matrix has a special eigenvalue, so called Perron-Frobenius eigenvalue, which plays an important role in dynamical systems. In this talk we present a numerical tool to compute rigorous (upper and lower) bounds of Perron-Frobenius eigenvalue of nonnegative matrices. Since a non-negative matrix corresponds to a directed graph, we make use of Tarjan's algorithm which founds all strongly connected components of a directed graph very efficiently. By applying Tarjan's algorithm to the directed graph which is derived from the original matrix, we decompose the matrix into (possibly small) irreducible components, and then enclose the aimed Perron-Frobenius eigenvalue. We show some numerical examples which demonstrate the efficiency of our tool.

AGA 15th March 2007
14:30 to 15:30
The Method of Images on a Quantum Graph

We consider an arbitrary quantum graph with the Laplacian H = -d^2/dx^2 and the corresponding eigenvalue problem on the graph. Taking the free space solution to an associated PDE (such as non-stationary Schroedinger equation), we apply the method of images to obtain the corresponding solution for our graph. This method brings in periodic and non-periodic (bounce) orbits and gives insight into both types of orbits and the roles they play. Applying this method to different kernels, we obtain interesting spectral information including the trace formula for the density of the states and the regularized vacuum energy.

AGA 20th March 2007
11:30 to 12:30
Internet tomography

We'll describe the relationship between my earlier work on Electrical Impedance Tomography and some cognate work I did with Casadio Tarabusi about the Radon transform on trees, and IT. The question we address is how to use the input-output map in a finite network to decide whether the traffic may be too large to be generated only by non malicious sources.

AGA 22nd March 2007
11:30 to 12:30
Quantum ergodicity on graphs

Quantum ergodicity is a generic property of the eigenstates of quantum systems whose classical counterparts are ergodic. It has been shown for many different systems that, after excluding an insignificant proportion of the exceptional eigenstates, the remaining eigenstates become equidistributed in the high energy limit.

Despite being one of the few questions in quantum chaos that have been mathematically proved in very general settings, the answer to the question of quantum ergodicity remains elusive for quantum graphs. In fact, the question itself is not entirely clear.

In this talk we will attempt to review what is known about the question. The first fundamental fact is that the eignestates of any fixed quantum graph are not quantum ergodic: one needs to take sequences of graphs to see equidistribution. Then we discuss a negative result for a sequence of graphs that is well understood: the star graphs. We show that the eigenstates of the star graphs do not equidistribute.

Finally, we discuss a positive result for a family of quantum graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakonski et al. As observables we take the L_2 functions on the interval. The proof is based on the periodic orbit expansion of a majorant of the quantum variance.

More precisely, given a one-dimensional, Lebesgue measure-preserving map of an interval, we consider an increasingly refined sequence of partitions of the interval. To this sequence we associate a sequence of graphs whose vertices correspond to elements of the partitions and whose classical analogues (in the sense of Kottos and Smilansky) are approximating the Perron-Frobenius operator corresponding to the above map. We show that, except possibly for a subsequence of density 0, the eigenstates of the quantum graphs equidistribute in the limit of large graphs.

This talk is based on the joint works with Jon Keating, Uzy Smilansky, Brian Winn.

AGA 22nd March 2007
14:30 to 15:30
V Kostrykin On the negative spectrum of quantum graphs

In this talk we will discuss some results on the negative spectrum of quantum graphs. In particular, we present estimates on the number of negative eigenvalues and bounds on the lowest eigenvalue.

AGA 27th March 2007
11:30 to 12:30
U Smilansky Quantum Chaos on Discrete Graphs

Archaeological research can profit from methods and ideas taken from theoretical physics and computer science. I shall show how this general statement was tested in a few examples of archaeological relevance. Using these examples, I shall try to draw some conclusions on future prospects, and discuss possible pitfalls in this kind of interdisciplinary research.

AGA 27th March 2007
15:30 to 16:30
S Avdonin Controllability of Dynamical Systems on Graphs

Boundary exact and spectral controllability for the wave, heat and Schroedinger equations on graphs will be discussed. We will describe also connections between controllability and identification problems.

AGA 29th March 2007
10:30 to 11:30
On Approximation of the Eigenvalues of Perturbed Periodic Schrodinger Operators

In this talk we will address the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic Schrodinger operator perturbed by a fast decreasing potential. We will discuss a geometrically motivated technique, the so called quadratic projection method, in order to achieve convergence free from spectral pollution. The theoretical foundations of the method will be described in detail, and its effectiveness will be illustrated by several examples.

AGA 29th March 2007
14:00 to 15:00
Solvability of the Dirichlet problem on stratified sets
AGAW02 2nd April 2007
10:00 to 11:00
Spectral properties and trace formulae for quantum graphs with general self-adjoint boundary conditions

We consider general self-adjoint realisations of the Laplacian on a compact metric graph and study their spectral properties. We prove a trace formula that expresses the spectral density in terms of a sum over periodic orbits on the graph.

AGAW02 2nd April 2007
11:30 to 12:30
Heat kernels on metric graphs and a trace formula

We report on joint work with V. Kostrykin and J. Potthoff. On metric graphs and for a certain class of Laplace operators a representation for the heat kernel in terms of walks is given. This representation is obtained from a corresponding one for the resolvent derived previously by two of the authors. This results in a Selberg-Gutzwiller type formula for the trace and extends earlier results by other authors in the context of quantum graphs.

AGAW02 2nd April 2007
14:00 to 15:00
A lower bound for nodal count on discrete and metric graphs

According to a well-know theorem by Sturm, a vibrating string is divided into exactly N nodal intervals by zeros of its N-th eigenfunction. Courant showed that one half of Sturm's theorem for the strings can be carried over to the theory of membranes: N-th eigenfunction cannot have more than N domains. He also gave an example of a eigenfunction high in the spectrum with a minimal number of nodal domains, thus excluding the existence of a non-trivial lower bound.

An analogue of Sturm's result for discretizations of the interval was discussed by Gantmacher and Krein. The discretization of an interval is a graph of a simple form, a chain-graph. But what can be said about more complicated graphs? It has been known since the early 90s that the nodal count for the Schrodinger operator on quantum trees (where each edge is identified with an interval of the real line and some matching conditions are enforced on the vertices) is exact too: zeros of the N-th eigenfunction divide the tree into exactly N subtrees.

We discuss two extensions of this result. One deals with the same continuous Schroedinger operator but on general graphs and another deals with discrete Schroedinger operator on combinatorial graphs (both trees and non-trees). The result that we derive applies to both types of graphs: the number of nodal domains of the N-th eigenfunction is bounded below by N-L, where L is the number of links that distinguish the graph from a tree (defined as the dimension of the cycle space or the rank of the fundamental group of the graph). This existence of a lower bound is a reminder of the differences between the graphs and domains in R^d.

AGAW02 2nd April 2007
15:30 to 16:30
Spectra of self-adjoint extensions related to the duality between discrete graphs and quantum networks

Due to the de Gennes–Alexander correspondence, at least for the equilateral quantum graphs the finding of the spectrum is reduced to the same problem for the underlying discrete graph. Therefore, the question concerning the correspondence between various parts of the spectra of a quantum graph and the corresponding tight-binding Hamiltonian arises. We consider this question in the framework of the Krein self-adjoint extension theory and give an affirmative answer on the question of the correspondence between classical parts of the spectrum: essential, discrete, pure point, absolutely continuous, and singular continuous ones. In the case of the pure point spectrum, the correspondence between eigenvectors is described.

AGAW02 2nd April 2007
16:30 to 17:00
Convergence of resonances on thin branched quantum wave guides

We consider convergence results of a family of noncompact, thin branched quantum waveguides (QWG) to the associated quantum graph. The branched quantum waveguide can either be a thin neighbourhood of the (embedded) quantum graph or be defined as a manifold without boundary (like the surface of a pipeline network approaching the metric graph). On the QWG has boundary, we consider the (Neumann) Laplacian; on the metric graph we consider the Laplacian with free boundary conditions. Our main result is a convergence result for the spectrum and resonances under some natural uniformity conditions on the spaces.

AGAW02 3rd April 2007
09:00 to 10:00
A Klein Universal occurrence of localization in continuum Anderson Hamiltonians

We will discuss Anderson and dynamical localization for continuum random Schrodinger operators and present a proof of localization for the continuum Anderson model with arbitrary single- site probability distribution.

AGAW02 3rd April 2007
10:00 to 11:00
Singular spectral types for certain spherically homogeneous trees

The talk will describe examples of trees for which the Laplacian exhibits `exotic' spectral phenomena. These examples are constructed via a decomposition of the Laplacian as a direct sum of Jacobi matrices.

AGAW02 3rd April 2007
11:30 to 12:30
Spectral statistics of a pseudo-integrable map

Different curious spectral properties of a quantum interval exchange map are discussed. In particular, it is demonstrated that when the matrix dimension obeys a certain congruence property the spectral statistics of the map coincides with the semi-Poisson statistics with integer and half integer level repulsion. Special attention is given to the general case where the spectral statistics is calculated by the transfer matrix approach.

AGAW02 3rd April 2007
14:00 to 15:00
S Gnutzmann Spectral correlations of individual quantum graphs

The spectral correlations of large well-connected quantum graphs are shown to behave according to the predictions of random-matrix theory by using a supersymmentry method. In a first (generally applicable) step the energy-average over the spectrum of individual graphs can be traded for the functional average over a supersymmetric nonlinear sigma-model action. Reducing the full sigma-model to its mean field theory is equivalent to the random-matrix theory of the Wigner-Dyson ensembles. Conditions for the validity of a mean field description will be discussed along with the stability of the universal random matrix behavior with regard to perturbations.

AGAW02 3rd April 2007
15:30 to 16:00
D Lenz Uniform existence of the integrated density of states for random Schrodinger operators on metric graphs over Z$^d$

We consider ergodic random magnetic Schr\"odinger operators on the metric graph $\mathbb{Z}^d$ with random potentials and random boundary conditions taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable. This limit, the integrated density of states, can be expressed by a closed Shubin-Pastur type trace formula. It supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. Among other examples we discuss percolation models.

Related Links

AGAW02 3rd April 2007
16:10 to 16:40
Spectral asymptotics of percolation Laplacians on amenable Cayley graphs

We study spectral properties of subcritical edge-percolation subgraphs of Cayley graphs of finitely generated, amenable groups. More precisely, we consider Laplace operators which are normalised in such a way that zero is the infimum of the spectrum and analyse the asymptotic behaviour of the integrated density of states near zero.

AGAW02 3rd April 2007
16:50 to 17:20
Reduction of quantum graphs to tight-binding Hamiltonians

There are several works which reduce the solution of the Schroedinger equation on quantum graphs to the study of some discrete operators, see e.g. S. Alexander, Phys. Rev. B27 (1983) 1541; J. von Below, Linear Alg. Appl. 71 (1985) 309; P. Exner, Ann. Inst. H. Poincare 66 (1997) 359; C. Cattaneo, Mh. Math. 124 (1997) 215; K. Pankrashkin, Lett. Math. Phys. 77 (2006) 139, etc. Nevertheless, they all assume some specific boundary conditions at the points of gluing, like Kirchhof, delta or delta' couplings. We show that a similar reduction can be done for a much larger class of boundary conditions and even for structures which are more general than quantum graphs, i.e. hybrid manifolds etc. The contribution of external interactions like magnetic field or spin-orbit coupling is also demonstrated.

AGAW02 4th April 2007
09:00 to 10:00
Some properties of distance matrices

Distance matrices are matrices whose entries are the relative distances between points located on a certain manifold. One central problem consists in isometric embedding, namely to find the conditions that a distance matrix must fulfill in order that one can find points in the euclidean space such that the euclidean distance between each pair of points coincide with the given distance matrix. One can investigate the spectral properties of distance matrices when the points are uncorrelated and uniformly distributed on (hyper)cubes and (hyper)spheres. The spectrum exhibits characteristic features, in particular all eigenvalues except one are non-positive and delocalized and strongly localized eigenstates are present.

AGAW02 4th April 2007
10:00 to 11:00
T Kottos Network models for Bose-Einstein condensates

We report our progress in studying ultra-cold atoms loaded in one-dimensional waveguides with complicated topology. We discuss the structure and of the resulting stationary wavefunctions and analyze the spectral properties of the corresponding chemical potentials.

AGAW02 4th April 2007
11:30 to 12:30
Contractive semigroups on metric graphs

The talk is devoted to the study of general (not necessarily self-adjoint) Laplace operators on metric graphs generating contractive semigroups. We present necessary and sufficient conditions on the boundary conditions at the vertices of the graph, ensuring the contractivity of generated semigroups. Preservation of continuity and positivity under such semigroups will also be discussed. The talk is based on a joint work with J.~Potthoff and R.~Schrader.

AGAW02 4th April 2007
15:30 to 16:30
Maximum principle for elliptic inequalities on the stratified sets

A concept of an elliptic equation (and inequality) on the stratified set was described in details in [P] (see also [NP]). Roughly speaking, a stratified set $\Omega$ is a connected set in $\mathbb{R}^n$, consisting of a finite number of smooth manifolds (strata). One can imagine a simplicial complex as an example. Using a special "stratified" measure we define an analogue of a divergence operator (acting on tangent vector fields) as a density of the "flow" of that field. Finally, we define an analogue of the Laplacian on the stratified set. Among other results we give a following strong maximum principle (jointly with S.N. Oshepkova). Theorem. A solution of inequality $\Delta u\geq 0$ cannot have a point of local nontrivial maximum on $\Omega_0$.

Here $\Omega_0$ is a connected part of $\Omega$, consisting of strata in them and such that $\overline\Omega_0=\Omega$. Our proof is based on the following lemma.

Lemma. Let $f_i$ be a continuous function $(i=0,\dots,k)$ on $[0;a]$ which is differentiable on $(0;a]$. Let us also assume $f_i$ be nonpositive and $f_i(0)=0$. Then an inequality $ r^kf_k'(r)+\dots+rf_1'(r)+f_0'(r)\geq 0\ (r\in(0;a]) $ follows $f_i\equiv 0$.

We give also some applications. For example, a following analogue of so called Bochner's lemma is an easy consequence of the strong maximum principle:

Theorem. Let $\Omega_0=\Omega$ and $\Delta u\geq 0$. Then $u\equiv{\rm const}$.

Bibliography [P] Penkin, O. M. About a geometrical approach to multistructures and some qualitative properties of solutions. Partial differential equations on multistructures (Luminy, 1999), 183--191, Lecture Notes in Pure and Appl. Math., 219, Dekker, New York, 2001

[NP] Nicaise, Serge; Penkin, Oleg M. Poincar\'e -- Perron's method for the Dirichlet problem on stratified sets. J. Math. Anal. Appl. 296, No.2, 504-520 (2004)

AGAW02 4th April 2007
16:30 to 17:00
P Kurasov Quantum graphs and Topology

Laplace operators on metric graphs are considered. It is proven that for compact graphs the spectrum of the Laplace operator determines the total length, the number of connected components, and the Euler characteristic. For a class of non-compact graphs the same characteristics are determined by the scattering data consisting of the scattering matrix and the discrete eigenvalues. This result is generalized for Schr\"odinger operators on metric graphs.

AGAW02 4th April 2007
17:10 to 17:40
On control and Inverse problems for the wave equation on graphs

The inverse problem for the Sturm-Liouville operator on a graph is considered. We suppose that the graph is a tree (i.e., it does not contain cycles), and on each edge the Schr\"odinger equation (with a variable potential) is defined. The Weyl matrix function is introduced through all but one boundary vertices. We prove that, the Weyl matrix function uniquely determines the graph (its connectivity and the lengths of the edges together with potentials on them). If the connectivity of the graph is known, the lengths of the edges and potentials on them are uniquely determined by the diagonal terms of either the Weyl matrix function, the response operator or by the back scattering coefficients.

AGAW02 4th April 2007
17:50 to 18:20
Eigenvalue estimates for Schroedinger operators on regular metric trees

We consider Lieb-Thirring and Cwikel-Lieb-Rozenblum inequalities for Schroedinger operators on regular metric trees. We give a necessary and sufficient condition for the validity of these inequalities in terms of the global growth of the tree. The behavior of these inequalities in the weak and strong coupling regime is discussed. The talk is based on joint work with T. Ekholm and H. Kovarik.

AGAW02 5th April 2007
09:00 to 10:00
Boundary value problems for infinite metric graphs

The second derivative operator with standard interior vertex conditions is considered on infinite metric graphs with finite volume or other smallness conditions. A large collection of self adjoint domains defined by local boundary conditions on the metric space completion of the graph is constructed for a rich family of metric graphs that are 'weakly connected'. The developed techniques also provide the solution of the Dirichlet problem for harmonic functions.

AGAW02 5th April 2007
10:00 to 11:00
Solvable model of a junction

For any junction (without any additional condition on the geometry) the corresponding solvable model is suggested based on the representation of the scattering matrix of it in terms of the intermediate Dirichlet-to-Neumann map and subsequent rational approximation of the DN-map. The model admits complete fitting in terms of the relevant scattering matrix.

AGAW02 5th April 2007
11:30 to 12:30
Some differential operators on trees and manifolds

The following 3 topics will be presented:

1. Image of the Neumann Laplacian on a tree. We shall discuss how these two operators are related based on a joint work with W.D.Evans (Cardiff, U.K.).

2. Limit of the Neumann Laplacians on shrinking domains. It will be shown that the solution of the limiting equation on ? satisfies the Kirchhoff boundary condition on each vertex of the tree.

3. 2-scale convergence on low-dimensional manifolds in Rn . As an analogy of the previous topic, we shall discuss 2-scale convergence of a function, its derivatives and solutions of differential equations when the domain shrinks to a lower dimensional manifold based by a recent joint work with Willi Jager (Heidelberg, Germany).

AGAW02 5th April 2007
14:00 to 15:00
Vacuum energy and closed orbits in quantum graphs

The vacuum (Casimir) energy of a quantized scalar field in a given geometrical situation is a certain moment of the eigenvalue density of an associated self-adjoint differential operator. For various classes of quantum graphs it has been calculated by several methods: (1) Direct calculation from the explicitly known spectrum is feasible only in simple cases. (2) Analysis of the secular equation determining the spectrum, as in the Kottos-Smilansky derivation of the trace formula, yields a sum over periodic orbits in the graph. (3) Construction of an associated integral kernel by the method of images yields a sum over closed (not necessarily periodic) orbits. We show that for the Kirchhoff boundary condition the sum over nonperiodic orbits in fact makes no contribution to the total energy, whereas for more general vertex scattering matrices (complex or frequency-dependent) it can make a nonvanishing contribution, which, however, is localized near vertices and hence can be "indexed" a posteriori by truly periodic orbits. Physically, the question of greatest interest is the sign of the Casimir effect in a given situation; for graphs it is being studied by both analytical and numerical methods. For example, in a star graph with 4 or more bonds ("pistons") of equal length and the standard Kirchhoff-Neumann boundary conditions, the vacuum force is repulsive (expanding), whereas the electromagnetic Casimir forces of laboratory interest are usually attractive. This work is done in collaboration with Justin H. Wilson, with contributions from Lev Kaplan, Gregory Berkolaiko, Jonathan Harrison, Melanie Pivarski, and Brian Winn and support from National Science Foundation Grant No. PHY-0554849.

AGAW02 5th April 2007
15:30 to 16:00
J Harrison Vacuum energy calculations for quantum graphs

We consider a quantum graph with spectrum $k_0^2 \le k_1^2 \le \dots$. The vacuum energy may be defined as the constant term in the asymptotic series expansion of $\frac{1}{2} \sum_{j=1}^{\infty} k_j \exp (-t k_j)$ as $t$ goes to zero. If the spectrum was instead that of the electromagnetic field between two parallel plates this would correspond to a renormalized energy of the vacuum whose derivative with respect to the plate separation is the attractive Casimir force measurable by experiment.

Using the trace formula for the density of states of a quantum graph a simple expression for the vacuum energy as a sum over periodic orbits has been derived. In the talk I will explain some calculations used to deduce properties of the vacuum energy. Of particular physical interest is the effect of changes in the lengths of the edges on the vacuum energy. To address this we show that the periodic orbit sum is convergent and differentiable with respect to the edge lengths. This work is part of a joint project with G. Berkolaiko, S. Fulling, M. Pivarski, J. H. Wilson and B. Winn.

AGAW02 5th April 2007
16:10 to 16:40
Quantum star graphs and related systems

This talk will survey a number of recent results on the spectra of quantum star graphs. From the point of view of quantum chaos these objects are quite interesting since they provide a tractable model for quantum systems with intermediate spectral statistics. Some implications will be discussed.

AGAW02 5th April 2007
16:50 to 17:20
Inverse spectral problems on compact and noncompact trees

Second-order differential operators on compact or noncompact trees with standard or general mathching conditions in internal vertices are studied. We establish properties of the spectral characteristics and investigate the inverse problem of recovering the operator from the so-called Weyl vector. For this inverse problem we prove a uniqueness theorem and provide a procedure for constructing the solution.

AGAW03 10th April 2007
10:00 to 11:00
Quantum swimming

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.

AGAW03 10th April 2007
11:30 to 12:30
Quantum fields and scale invariance on star graph

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.

AGAW03 10th April 2007
14:00 to 15:00
N March Electronic states in ordered and disordered quantum networks: with applications to graphene and to C and B nanotubes

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 one-dimensional 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 two-dimensional 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.

AGAW03 10th April 2007
15:30 to 16:30
Quantum network model of zigzag carbon nanotube

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 anti-periodic 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.

AGAW03 10th April 2007
16:30 to 17:30
On the spectra of carbon nano-structures

An explicit derivation of dispersion relations and spectra for periodic Schrödinger operators on carbon nano-structures (including graphene and all types of single-wall nano-tubes) is provided.

AGAW03 11th April 2007
09:00 to 10:00
Excitonic influence on transport coefficients in low dimensional quantum systems

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.

AGAW03 11th April 2007
10:00 to 11:00
Quantum transport in networks of weakly disordered metallic wires

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.

AGAW03 11th April 2007
11:30 to 12:30
Localization under external interactions in periodic quantum graphs

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.

AGAW03 11th April 2007
15:00 to 16:00
Graphs which sound the same

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.

AGAW03 11th April 2007
16:00 to 17:00
V Kostrykin Inverse scattering problems for quantum graphs

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.

AGAW03 11th April 2007
17:00 to 18:00
On the skeleton method

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

AGAW03 12th April 2007
09:00 to 10:00
Perfect transfer of quantum information across graphs

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.

AGAW03 12th April 2007
10:00 to 11:00
Optimal computation with noisy quantum walks

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.

AGAW03 12th April 2007
11:30 to 12:30
Finding eigenvalues and resonances of the Laplacian on domains with regular ends

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.

AGAW03 12th April 2007
14:00 to 15:00
D Grieser Spectral asymptotics of the Dirichlet Laplacian on fat graphs

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.

AGAW03 12th April 2007
15:30 to 16:30
Dirichlet eigenvalues in a narrow strip
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.
AGAW03 12th April 2007
16:30 to 17:30
M Harmer Spin filtering and the Rashba effect

We discuss quantum graphs with the Rashba Hamiltonian with application to spin filtering

AGAW03 13th April 2007
09:00 to 10:00
Phase transitions on lattice graphs

We explain the random-cluster (or FK) representation of the classical Potts/Ising models, when the underlying graph is a lattice. There is a continuum version of the random-cluster model that may be used to study the quantum Ising model with transverse field. A selection of open problems will be included.

AGAW03 13th April 2007
10:00 to 11:00
Slow light in photonic crystals

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 so-called 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.

AGAW03 13th April 2007
11:30 to 12:30
Simulation of quantum graphs by microwave networks

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.

AGAW03 13th April 2007
14:00 to 15:00
Localized shelf waves on a curved coast - existence of eigenvalues of a linear operator pencil in a curved waveguide

The study of the possibility of the non-propagating, trapped continental shelf waves along curved coasts reduces mathematically to a spectral problem for a self-adjoint 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 (Heriot-Watt)

AGAW03 13th April 2007
15:30 to 16:30
O Turek Approximations of strongly singular couplings at quantum graph vertices

Approximation of a general strongly singular coupling in the center of a star graph by means of a delta-coupling and delta-interactions 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 norm-resolvent sense.

AGAW03 13th April 2007
16:30 to 17:30
Twisting versus bending in quantum waveguides

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 three-dimensional tubes with arbitrary cross-section 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 three-dimensional tube with non-circular cross-section gives rise to a Hardy-type 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

AGA 16th April 2007
14:30 to 15:30
Y Colin de Verdiere Multiplicities of eigenvalues and topology
AGA 18th April 2007
14:30 to 15:30
Y Colin de Verdiere Lower bounds of the entropy of quantum limits (after N. Anantharaman and S. Nonnemacher)
AGA 19th April 2007
14:30 to 15:30
J von Below Travelling fronts on networks for reaction-diffusion-equations
AGA 24th April 2007
14:30 to 15:30
Kirchhoff Constants for Helmholtz Resonator
AGA 26th April 2007
14:30 to 15:30
On Inverse Problems for Trees

The uniqueness question for the inverse problem for a Schr\"odinger equation on a compact interval was answered by theorems due to Borg, Levinson, and Marchenko for various sets of data (two spectra, one spectrum and boundary data of eigenfunctions, and one spectrum and norming constants of eigenfunctions, respectively). These theorems have generalizations to finite trees which will be presented and discussed.

This is joint work with Malcolm Brown, Cardiff.

AGA 26th April 2007
16:00 to 17:00
J von Below Can one hear the shape of a network?
AGA 1st May 2007
16:00 to 17:00
Spectral inequalities for a class of hypo-elliptic operators

We obtain sharp inequalities of Berezin-Li-Yua type for the eigenvalues of the Dirichlet boundary value problem for Hörmander Laplacians in domains of finite measure.

AGA 15th May 2007
14:30 to 15:30
A Teplyaev Schur complement, Drichlet-to-Neumann map, and eigenfunctions on self-similar graphs

We study eigenvalues and eigenfunctions on the class of self-similar symmetric finitely ramified graphs. We consider such examples as the graphs modeled on the Sierpinski gasket, a non-p.c.f. analog of the Sierpinski gasket, the Level-3 Sierpinski gasket, a fractal 3-tree, the Hexagasket, and one dimensional fractal graphs. We develop a matrix analysis, including analysis of singularities, which allows us to compute eigenvalues, eigenfunctions and their multiplicities exactly.

AGA 17th May 2007
14:30 to 15:30
Differential operators with singularities and generalised Nevanlinna functions
AGA 23rd May 2007
11:00 to 12:00
Pollution-free methods for finding eigenvalues in the gaps of the continuous spectrum

We describe a universal non-variational approach allowing to find numerically the eigenvalues of a self-adjoint operator in the gaps of essential spectrum. The method does not produce spurious eigenvalues and gives two-sided a posteriori error estimates. As an illustration, we discuss a periodic Schroedinger operator perturbed by a compact potential.

AGA 23rd May 2007
16:00 to 17:00
Structural constraints in complex networks

We present a link-rewiring mechanism to produce surrogates of a network where both the degree distribution and the rich-club connectivity are preserved. We consider three real networks, the AS-Internet, the protein interaction and the scientific collaboration. We show the degree-degree correlation in a network is constrained by both the degree distribution and the rich-club connectivity. We also comment on the suitability of using the maximally randomized network as a null model to assess network properties in real networks. This talk is based on our recent paper to appear in the New Journal of Physics: http://www.iop.org/EJ/journal/-page=forthart/1367-2630/8

AGA 24th May 2007
16:00 to 17:00
Self-similar graphs, algebra and fractals

We will discuss operator algebras associated with self-similar objects such as graphs, groups and dynamical systems. In particular, we will show how to reconstruct the Julia set of a rational function from its iterated monodromy group (or from the associated countable graph)using C*-algebras. We will also present a formula for the Hausdorff dimension of the Julia set, which suggests that this approach might be useful for constructing (quantized?) calculus on Julia sets.

AGA 5th June 2007
14:30 to 15:30
Wave propagation in networks of thin fibres: small diameter asymptotics

We consider the Laplace equation in systems of wave guides with arbitrary boundary conditions, when the diameter of the wave guides is vanishing. The asymptotic behavior of scattering solutions and the resolvent on the bulk of the spectrum and near the treshold will be discussed, as well as asymptotic behavior of the solutions of some non-stationary problems. This is a joint work with S. Molchanov (UNCC)

AGA 7th June 2007
14:30 to 15:30
Asymptotical properties of random trees on manifolds
AGA 8th June 2007
11:00 to 12:00
Transition density of Markov processes on fractal-like spaces

We study the off-diagonal estimates for transition densities of diffusions and jump processes in a setting when they depend essentially only on the time and distance. We state the dichotomy for the tail of the transition density and provide necessary and sufficient conditions for each type of the estimates.

AGA 12th June 2007
14:30 to 15:30
On one hydrostatic problem

The talk will present several results related to a recent surprising theorem by A. Cherny and P. Grigoriev originating from the economic risk theory. The theorem has various interesting formulations, including a hydrostatic one.

AGA 13th June 2007
10:30 to 11:30
Self-similar groups and Schur complement

The talk will contain a discussion of the technique based on the matrix transfomation known as Schur complement. It enables one to compute in certain situations the spectra of discrete Laplace operators on self-similar groups and associated Schreier graphs (which usually inherit the self-similar structure), and of elements in associated C*- algebras. This technique will be applied to show usefullnes of Schur complement in study of random walks on groups and in proving amenability. KNS spectral measures will be mentioned and for a torsion group of intermediate growth a result will be formulated about the relation between this measure and Kesten spectral measure. This relation will be used to compute Jacobi parameters of the corresponding Markov operator.

AGA 13th June 2007
16:00 to 17:00
U Smilansky Applications of computer methods and mathematical ideas in archaeology

Archaeological research can profit from methods and ideas taken from theoretical physics and computer science. I shall show how this general statement was tested in a few examples of archaeological relevance. Using these examples, I shall try to draw some conclusions on future prospects, and discuss possible pitfalls in this kind of interdisciplinary research.

AGA 18th June 2007
18:00 to 19:00
On the spectra and dynamics of operators with disorder
AGA 19th June 2007
14:30 to 15:30
The spectral theory of some directed graphs
AGA 19th June 2007
16:00 to 17:00
On the random analytic functions

The talk will present some results on random analytic functions, including the random Taylor series and random zeta-functions by H. Cramer. The main topic is the analytic continuation of such functions and related properties.

AGA 21st June 2007
11:00 to 12:00
Some spectral problems for non-selfadjoint operators

We consider abstract Dirichlet-to-Neumann operators (M-functions) for non-selfadjoint operators in an abstract setting. We prove that isolated eigenvalues of an operator correspond to poles of the associated M-function and show that second order elliptic PDEs in smooth domains satisfying fairly general boundary conditions fit into our framework.

AGA 21st June 2007
14:30 to 15:30
Bernoulli decompositions for random variables and applications

We present a decomposition which highlights the presence of a Bernoulli component in any random variable. Two applications are discussed: 1. A concentration inequality in the spirit of Littlewood-Offord for a class of functions of independent random variables; 2. A proof, based on the Bernoulli case, of spectral localization for random Schroedinger operators with arbitrary probability distributions for the single site coupling constants. (This is joint work with M. Aizenman, F. Germinet and A. Klein)

AGA 22nd June 2007
14:30 to 15:30
P Kuchment Analysis on Graphs and its Applications
AGA 26th June 2007
14:30 to 15:30
Diffusion in narrow randomly perturbed tubes

I will discuss diffusion and reaction-diffusion in narrow tubes with periodic or random boundaries. A homogenization type result for the effective diffusion coefficient and for the asymptotic wavefront speed are calculated. Reaction-diffusion equations in a class of random nets of narrow tubes will be considered.

AGA 26th June 2007
16:00 to 17:00
The spectral minimum of the random displacement model
AGAW06 26th July 2010
09:00 to 10:00
Non-Weyl Asymptotics for Resonances of Quantum Graphs
Consider a compact quantum graph ${\cal G}_0$ consisting of finitely many edges of finite length joined in some manner at certain vertices. Let ${\cal G}$ be obtained from ${\cal G}_0$ by attaching a finite number of semi-infinite leads to ${\cal G}_0$, possibly with more than one lead attached to some vertices.

Let $H_0$ ( resp. $H$) $=-\frac{{\rm d}^2}{{\rm d} x^2}$ acting in $L^2({\cal G}_0)$ ( resp. $L^2({\cal G})$ ) subject to continuity and Kirchhoff boundary conditions at each vertex. The spectrum of $H$ is $[0,\infty)$, but unlike the normal case for Schrödinger operators $H$ may possess many $L^2$ eigenvalues corresponding to eigenfunctions that have compact support. However some eigenvalues of $H_0$ turn into resonances of $H$, and when defining the resonance counting function \[ N(r)=\#\{ \mbox{ resonances $\lambda=k^2$ of $H$ such that $|k|<r$}\} \] one should regard eigenvalues of $H$ as special kinds of resonance.

One might hope that $N(r)$ obeys the same leading order asymptotics as $r\to\infty$ as in the case of ${\cal G}_0$, but this is not always the case. A Pushnitski and EBD have proved the following theorem, whose proof will be outlined in the lecture.

Theorem 1 The resonances of $H$ obey the Weyl asymptotic law if and only if the graph ${\cal G}$ does not have any balanced vertex. If there is a balanced vertex then one still has a Weyl law, but the effective volume is smaller than the volume of ${\cal G}_0$.

AGAW06 26th July 2010
10:30 to 11:15
R Band Scattering from isospectral graphs.
In 1966 Marc Kac asked 'Can one hear the shape of a drum?'. The answer was given only in 1992, when Gordon et al. found a pair of drums with the same spectrum. The study of isospectrality and inverse problems is obviously not limited to drums and treats various objects such as molecules, quantum dots and graphs. In 2005 Okada et al. conjectured that isospectral drums can be distinguished by their scattering poles (resonances). We prove that this is not the case for isospectral quantum graphs, i.e., isospectral quantum graphs share the same resonance distribution. This is a joint work with Adam Sawicki and Uzy Smilansky.
AGAW06 26th July 2010
11:15 to 12:00
G Berkolaiko Nodal domains and spectral critical partitions on graphs
The $k$-th eigenfunction of a Schrodinger operator on a bounded regular domain $\Omega$ with Dirichlet boundary conditions defines a partition of $\Omega$ into $n$ nodal subdomains. A famous result by Courant establishes that $n \leq k$; the number $k-n$ will be referred to as the nodal deficiency. The nodal subdomains, when endowed with Dirichlet boundary conditions, have equal first eigenvalue, which coincides with the $k$-th eigenvalue of the original Schrodinger problem. Additionally, the partition is bipartite, i.e. it consists of positive and negative subdomains (corresponding to the sign of the eigenfunction), with two domains of the same sign not sharing a boundary. Conversely, for a given partition, define the energy of the partition to be the largest of the first Dirichlet eigenvalues of its subdomains. An $n$-partition with the minimal energy is called the minimal $n$-partition. It is interesting to relate the extremal properties of the partitions to the eigenstates of the operator on $\Omega$. Recently, Helffer, Hoffmann-Ostenhof and Terracini proved that $n$-th minimal partition is bipartite if and only if it corresponds to a Courant-sharp eigenfunction (an eigenfunction with nodal deficiency zero). We study partitions on quantum graphs and discover a complete characterization of eigenfunctions as critical equipartitions. More precisely, equipartitions are partitions with all first eigenvalues equal. We parameterize the manifold of all equipartitions and consider the energy of an $n$-equipartition as a function on this manifold. For a generic graph and large enough $n$ we establish the following theorem: a critical point of the energy function with $b$ unstable directions is a bipartite equipartition if and only if it corresponds to an eigenfunction with nodal deficiency $b$. Since by constructions it has $n$ nodal domains it is therefore the $n+b$-th eigenfunction in the spectral sequence. Since at a minimum the number of unstable directions is $b=0$, our results include the quantum graph analogue of the results of Helffer et al. They also provide a new formulation of known bounds on the number of nodal domains on generic graphs. This is joint work with Rami Band, Hillel Raz and Uzy Smilansky.
AGAW06 26th July 2010
14:00 to 14:45
Quantum statistics on Graphs
I will discuss possible quantum exchange statistics in the case of graphs
AGAW06 26th July 2010
14:45 to 15:30
J Bolte Many-particle systems on quantum graphs with singular interactions
Single quantum particles on graphs have proven to provide interesting models of complex quantum systems; their spectral properties have been studied in great detail. In this talk we discuss extensions to quantum many-particle systems on graphs with singular interactions. We focus on two-particle interactions that are either localised at the vertices, or are of Dirac-delta type on the edges. In both cases the interactions are realised in terms of self-adjoint extensions of suitable Laplacians in two variables. These extensions can be characterised in terms of boundary conditions, and given particular boundary conditions the type of interactions can be identified. (This talk is based on joint work with Joachim Kerner.)
AGAW06 26th July 2010
16:00 to 16:45
S Avdonin Recursive Algorithms Solving Inverse Problems on Quantum Graphs
In this talk we describe a new approach to solving boundary inverse problems on quantum graphs. This approach is based on the Boundary Control method and combines the spectral and dynamical approaches to inverse problems on graphs. It was proposed in [1] for the Schr\"odinger equation with standard matching conditions and was extended in [2] to the two-velocity wave equation. Since the number of edges of graphs arising in applications is typically very big, we propose a recursive procedure which may serve as a base for developing effective numerical algorithms. For trees, this procedure allows recalculating efficiently the inverse data from the original tree to the smaller trees, `removing' leaves step by step up to the rooted edge. Numerical tests for inverse problems are impossible without producing accurate inverse data. This means that we have to have reliable numerical algorithms for solving the direct Problems --- given the coefficients of equations and the graph topology find its spectral (and dynamical) data. Even for the simplest graph --- a finite interval or the semi-axis --- this is a rather difficult problem from the numerical point of view. The surprising fact is that to solve numerically, say, the Gelfand--Levitan equation and find the potential from the given spectral function is much easier than to find the spectral function from the given potential. For graphs with many edges these difficulties increase dramatically. Therefore, at the moment there are no efficient algorithms for, or numerical experiments in, solving inverse problems on graphs. Based on the results of [3], we propose a way to reduces the `direct' problem to solving second kind Volterra integral equations. 1. S. Avdonin and P. Kurasov, Inverse problems for quantum trees,} Inverse Problems and Imaging, {2} (2008), 1--21. 2. S. Avdonin, G. Leugering and V. Mikhaylov, On an inverse problem for tree-like networks of elastic strings}, Zeit. Angew. Math. Mech., {90} (2010), 136--150. 3. S. Avdonin, V. Mikhaylov and A.Rybkin, The boundary control approach to the Titchmarsh-Weyl $m-$function,} Comm. Math. Phys., {275} (2007), 791--803.
AGAW06 26th July 2010
16:45 to 17:30
The HELP Inequality on Trees
This is joint work with BM Brown and M Langer. We establish analogues of Hardy and Littlewood's integro-differential equation for Schroedinger-type operators on metric and discrete trees, based on a generalised strong limit-point property of the graph Laplacian.
AGAW06 26th July 2010
17:30 to 17:45
The Floquet spectrum of a quantum graph
We define the Floquet spectrum of a quantum graph as the collection of all spectra of operators of the form $D=(-i\frac{\partial}{\partial x}+\alpha(\frac{\partial}{\partial x}))^2$ where $\alpha$ is a closed $1$-form. We show that the Floquet spectrum completely determines planar 3-connected graphs (without any genericity assumptions on the graph). It determines whether or not a graph is planar. Given the combinatorial graph, the Floquet spectrum uniquely determines all edge lengths of a quantum graph.
AGAW06 26th July 2010
17:45 to 18:00
The Absence of Absolutely Continuous Spectra for Radial Tree Graphs
We will introduce a family of Schrödinger operators on tree graphs with coupling conditions given by (b_n-1)^2+4 real parameters where b_n is the branching number. We will show the unitary equivalence of the Hamiltonian on the tree graph and the orthogonal sum of the Hamiltonians on the halflines. We will use this unitary equivalence to prove that for a large family of coupling conditions there is no absolutely continuous spectrum of the Hamiltonian on the sparse tree. On the other hand, we will show nontrivial examples of trees with the spectrum which is purely absolutely continuous.
AGAW06 27th July 2010
09:00 to 10:00
Periodic walks on random graphs and random matrix theory
The spectral statistics of the discrete Laplacian of d-regular graphs on V vertices are intimately connected with the distribution of the number of cycles of period t (t-cycles) on the graph. I shall discuss this connection by using a trace formula which expresses the spectral density in terms of the t-cycle counts. The trace formula will be used to write the spectral pair correlations in terms of the properly normalized variance of the t-cycle counts. Based on these results, I would like to propose a conjecture which uses Random Matrix Theory to compute the variance of the t-periodic cycle counts in the limit V,t -> infinity fixed value of with t/V. Numerical computations support this conjecture.
AGAW06 27th July 2010
10:30 to 11:15
J Harrisson Properties of zeta functions of quantum graphs
The Ihara-Selberg zeta function plays a fundamental role in the spectral theory of combinatorial graphs. However, in contrast the spectral zeta function has remained a relatively unstudied area of analysis of quantum graphs. We consider the Laplace operator on a metric graph with general vertex matching conditions that define a self-adjoint realization of the operator. The zeta function can be constructed using a contour integral technique. In the process it is convenient to use new forms for the secular equation that extend the well known secular equation of the Neumann star graph. The zeta function is then expressed in terms of matrices defining the matching conditions at the vertices. The analysis of the zeta function allows us to obtain new results for the spectral determinant, vacuum energy and heat kernel coefficients of quantum graph which are topics of current research in their own right. The zeta function provides a unified approach which obtains general results for such spectral properties.
AGAW06 27th July 2010
11:15 to 12:00
A Terras Explicit Formulas for Zeta Functions of Graphs
Explicit formulas for the Riemann and Dedekind zeta functions, as developed by Andre Weil, have often been compared to the Selberg trace formula. We look at analogs for the Ihara zeta function of a graph.
AGAW06 27th July 2010
14:00 to 14:45
A Teplyaev Uniqueness of Laplacian and Brownian motion on Sierpinski carpets
Up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpinski carpet that is invariant with respect to the local symmetries of the carpet. Consequently for each such fractal the law of the Brownian motion is uniquely determined and the Laplacian is well defined. As a consequence, there are uniquely defined spectral and walk dimensions which determine the behavior of the natural diffusion processes by so called Einstein relation (these dimensions are not directly related to the well known Hausdorff dimension, which describes the distribution of the mass in a fractal).
AGAW06 27th July 2010
14:45 to 15:30
Line integrals of one-forms on the Sierpinski gasket
We give a definition of one-forms on the gasket and of their line integrals, and show that these are compatible with the notion of energy introduced by Kigami. We then introduce a suitable covering of the gasket (which is a projective limit of a sequence of natural finite coverings) and prove that n-exact forms have a primitive which lives on this covering.
AGAW06 27th July 2010
16:00 to 16:45
D Guido Some (noncommutative) geometrical aspects of the Sierpisnki gasket
We present here a 2-parameter family of spectral triples for the Sierpinski gasket, based on spectral triples for the circle. Any hole (lacuna) of the gasket is suitably identified with a circle, and the triple for the gasket is defined as the direct sum of the triples for the lacunas. The first parameter is a scaling parameter for the correspondence between circles and lacunas, the second describes the metric on the circle, which is, roughly, a power of the euclidean metric. We study for which parameters the following features of the gasket can be recovered by the corresponding triple: the integration on the gasket (w.r.t. the Hausdorff measure), a non-trivial distance on the gasket, a non-trivial Dirichlet form (the Kigami energy).
AGAW06 27th July 2010
16:45 to 17:30
Relationship between scattering matrix and spectrum of quantum graphs
We investigate the equivalence between spectral characteristics of the Laplace operator on a metric graph, and the associated unitary scattering operator. We prove that the statistics of level spacings, and moments of observations in the eigenbases coincide in the limit that all bond lengths approach a positive constant value.
AGAW06 27th July 2010
17:30 to 17:45
V Chernyshev Statistical properties of semiclasssical solutions of the non-stationary Schrödinger equation on metric graphs
The talk is devoted to the development of the semiclassical theory on quantum graphs. For the non-stationary Schrödinger equation, propagation of the Gaussian packets initially localized in one point on an edge of the graph is described. Emphasis is placed on statistics behavior of asymptotic solutions with increasing time. It is proven that determination of the number of quantum packets on the graph is associated with a well-known number-theoretical problem of counting the number of integer points in an expanding polyhedron. An explicit formula for the leading term of the asymptotics is presented. It is proven that for almost all incommensurable passing times Gaussian packets are distributed asymptotically uniformly in the time of passage of edges on a finite compact graph. Distribution of the energy on infinite regular trees is also studied. The presentation is based on the joint work with A.I. Shafarevich.
AGAW06 27th July 2010
17:45 to 18:00
T Petrillo On the Splitting of Primes in Coverings
An explicit formula for zeta functions on graphs will be introduced. The result will be extended to L-functions on graphs. An example and possible extensions will be discussed.
AGAW06 28th July 2010
09:00 to 10:00
G Milton Complete characterization and synthesis of the response function of elastodynamic networks
In order to characterize what exotic properties elastodynamic composite materials with high contrast constituents can have in the continuum it makes sense to first understand what behaviors discrete networks of springs and masses can exhibit. The response function of a network of springs and masses, an elastodynamic network, is the matrix valued function W(omega), depending on the frequency omega, mapping the displacements of some accessible or terminal nodes to the net forces at the terminals. We give necessary and sufficient conditions for a given function W(omega) to be the response function of an elastodynamic network assuming there is no damping. In particular we construct an elastodynamic network that can mimic any achievable response in the frequency or time domain. It builds upon work of Camar-Eddine and Seppecher, who characterized the possible response matrices of static three-dimensional spring networks. Authors: F. Guevara Vasquez (University of Utah), G.W. Milton (University of Utah), D.Onofrei (University of Utah)
AGAW06 28th July 2010
10:30 to 11:15
Unbounded Laplacians on graphs: Basic spectral properties and the heat equation
We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on phenomena related to unboundedness of the Laplacians. In particular, we provide criteria for essential selfadjointness, empty essential spectrum and stochastic incompleteness. (Joint work with Matthias Keller).
AGAW06 28th July 2010
11:15 to 12:00
Heat kernel estimates for Laplace operators on metric trees
We consider the integral kernel of the semigroup generated by a differential Laplace operator on certain class of infinite metric trees. We will show how the time decay of the heat kernel depends on the geometry of the tree. (This is a joint work with Rupert Frank).
AGAW06 28th July 2010
12:00 to 12:45
A Laptev On some sharp spectral inequalities for Schrödinger operators on graphs
AGAW06 28th July 2010
12:45 to 13:00
Characterization of the static response of a two-dimensional elastic network
We will show that, in the 2D static case , any positive semidefinte and balanced matrix $W$ is the response matrix of a purely elastic planar network. Moreover we will present a constructive proof for the fact that the network we design for a particular response matrix $W$ can fit within an arbitrarily small neighborhood of the convex hull of the terminal nodes, provided the springs and masses occupy an arbitrarily small volume.
AGAW06 29th July 2010
09:00 to 10:00
D Grieser Fat graphs: Variations on a theme
A fat graph is, generally speaking, a family of spaces depending on a parameter $\varepsilon$ which converge metrically to a metric graph as $\varepsilon\to 0$. The problem of studying the behavior of the spectrum of the Laplacian or other geometric operators, under this limit, arises in various contexts and can be approached by a variety of techniques. In this survey talk I will explain some of these techniques and mention some recent results.
AGAW06 29th July 2010
10:30 to 11:15
A Zuk On a problem of Atiyah
We present constructions of closed manifolds with irrational L2 Betti numbers
AGAW06 29th July 2010
11:15 to 12:00
A Solution to an Ambarzumyan Problem on Trees
The classical Ambarzumyan problem states that when the eigenvalues $\lambda_n$ of a Neumann Sturm-Liouville operator defined on $[0,\pi]$ are exactly $n^2$, then the potential function $q=0$. In 2007, Carlson and Pivovarchik showed the Ambarzumyan problem for the Neumann Sturm-Liouville operator defined on trees where the edges are in rational ratio. We shall extend their result to show that for a general tree, if the spectrum $\sigma(q)=\sigma(0)$, then $q=0$. In our proof, we develop a recursive formula for characteristic functions, together with a pigeon hole argument. This is a joint work with Eiji Yanagida of Tokyo Institute of Technology.
AGAW06 29th July 2010
14:00 to 14:45
Hardy inequalities and asymptotics for heat kernels
We will discuss some Hardy inequalities and its consequences on the large time behavior of diffusion processes. Roughly speaking, the Hardy inequality ensures a further and faster decay rate. Two differente situations will be addressed. First we shall consider the heat equation with a singular square potential located both in the interior of the domain and on the boundary, following a joint work with J. L. Vázquez and a more recent one with C. Cazacu. We shall also present the main results of a recent work in collaboration with D. Krejciírk in which we consider the case of twisted domains. In this case the proof of the extra decay rate requires of important analytical developments based on the theory of self-similar scales. As we shall see, asymptotically, the twisting ends up breaking the tube and adds a further Dirichlet condition, wich eventually produces the increase of the decay rate. Some consequences in which concerns the control of these models will also be presented. References: J. L. Vázquez and E. Zuazua The Hardy inequality and the asymptotic behavior of the heat equation with an inverse square potential. J. Functional Analysis, 173 (2000), 103--153.} J. Vancostenoble and E. Zuazua. Hardy inequalities, Observability and Control for the wave and Schr\"odinger equations with singular potentials, SIAM J. Math. Anal., Volume 41, Issue 4, pp. 1508-1532 (2009) D. Krejcirik and E. Zuazua The heat equation in twisted domains, J. Math pures et appl., to appear. C. Cazacu and E. Z. Hardy inequalities with boundary singular potentials, in preparation.
AGAW06 29th July 2010
14:45 to 15:30
Dirichlet to Neumann Maps for Infinite Metric Graphs
Motivated by problems of modeling the human circulatory system, boundary value problems for differential operators -D2 + q are considered on the metric completions of infinite graphs with finite volume, finite diameter, or other smallness conditions. For a large family of graphs, the existence of an ample family of simple test functions permits a generalized definition of the Dirichlet to Neumann map taking boundary functions to their normal derivatives. Properties of this map, problems exhibiting more regular derivatives, and approximation by finite subgraphs will be discussed.
AGAW06 29th July 2010
16:00 to 16:45
M Eastham The continuous and discrete spectrum of an asymptotically straight leaky wire
The approach to quantum graphs developed by Exner and his co-workers is based on a Hamiltonian which contains a singular potential term with a delta-function support on (in two dimensions) a curve C. Here we give conditions on the potential and on the geometry of C under which the associated spectrum is either a semi-infinite interval or the whole real line. The geometry is expressed in terms of a new and simpler concept of asymptotic straightness which does not rely on an asymptotic estimate for the curvature, and which is only imposed on disjoint long sections of C. We also discuss the case where C is a star graph with N rays and the lower spectrum is discrete. We obtain an estimate for the lowest eigenvalue and we contribute to the conjecture that this eigenvalue is maximised for a given N when the star graph is symmetric. A number of open spectral problems related to this work are mentioned. (Joint work with Malcolm Brown and Ian Wood.)
AGAW06 29th July 2010
16:45 to 17:30
Discrete alloy type models: averaging and spectral properties
We discuss recent results on discrete alloy type models, in particular those with non-monotone parameter dependence. Among others Wegner estimates, averaging techniques, appropriate transformations on the probability space and decoupling properties are dicussed. This is related to the exponential decay of the Green's function and localisation properties.
AGAW06 29th July 2010
17:30 to 17:45
Limits of self-similar graphs and criticality of the Abelian Sandpile Model
We consider covering sequences of (Schreier) graphs arising from self-similar actions by automorphisms of rooted trees. The projective limit of such an inverse system corresponds to the action on the boundary of the tree and its connected components are the (infinite) orbital Schreier graphs of the action. They can be approximated by finite rooted graphs using Hausdorff-Gromov convergence. An interesting example is given by the Basilica group acting by automorphisms on the binary rooted tree in a self-similar fashion. We give a topological as well as a measure-theoretical description of the orbital limit Schreier graphs. In particular, it is shown that they are almost all one-ended with respect to the uniform distribution on the boundary of the tree. We study the statistical-physics Abelian Sandpile Model on such sequences of graphs. The main mathematical question about this model is to prove its criticality -- the correlation between sites situated far away each from the other is high -- what is typically done by exhibiting, asymptotically, a power-law decay of various statistics. In spite of many numerical experiments, the criticality of the model was rigorously proven only in the case of the regular tree. We show that the Abelian Sandpile Model on the limit Schreier graphs of the Basilica group is critical almost everywhere with respect to the uniform distribution on the boundary of the tree.
AGAW06 30th July 2010
09:00 to 09:45
Index Theorems for Quantum Graphs
Work in collaboration with P. Kuchment and J. Wilson
AGAW06 30th July 2010
10:30 to 11:15
Variations on discrete Floquet-Bloch Theory in positive characteristic
The classical Floquet theory deals with Floquet-Bloch solutions of periodic PDEs. A discrete version of this theory for difference vector equations on lattices, including the Floquet theory on infinite periodic graphs, was developed by Peter Kuchment. Here we propose a variation of this theory for matrix convolution operators acting on vector functions on lattices with values in a field of positive characteristic.
AGAW06 30th July 2010
11:15 to 12:00
Convergence results for thick graphs
We will give an overview of convergence results for several natural Laplace-like operators on the thick graph to candidates on the underlying metric graph. Of particular interest are the glueing conditions which can be obtained at a vertex by a pure Laplacian. Moreover we show convergence of the Dirichlet-to-Neumann map of a thick graph with boundary to the corresponding operator of the metric graph.
AGAW06 30th July 2010
14:00 to 14:45
Dirichlet-to-Neumann techniques for periodic problems
Spectral analysis in low energy region will be developed for the Schrodinger operator on periodic multi-dimensional lattices and, in particular a method of estimation of mobility of electron/holes in Silicon=Boron sandvich structures will be suggested. I am able to provide a draft of the paper as soon as you need it.
AGAW06 30th July 2010
14:45 to 15:30
Necklace graphs and slowing down of the light
A possible device for slowing down of the light (propagation of wave packets) will be discussed which is based on periodic branching waveguides. Reduction to a quantum graph with a specific boundary conditions at vertices plays a crucial role.
AGAW06 30th July 2010
16:00 to 16:45
Absolutely continuous spectrum for trees of finite forward cone type
We study a class of rooted trees which are not necessarily regular but exhibit a lot of symmetries. The spectrum of the corresponding graph Laplace operator is purely absolutely continuous and consists of finitely many intervals. Moreover for trees of the class which are not regular the absolutely continuous spectrum is stable under small perturbations by radially symmetric potentials. (This is joint work with Daniel Lenz and Simone Warzel.)
AGAW06 30th July 2010
16:45 to 17:30
The parabolic Harnack inequality for quantum graphs
We consider quantum graphs with Kirchhoff boundary conditions. We study the intrinsic metric, volume doubling and a Poincaré inequality. This enables us to prove a parabolic Harnack inequality. The proof involves various techniques from the theory of strongly local Dirichlet forms.
AGA 21st March 2015
11:00 to 12:00
R Higgitt Longitude Found
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons