Quantum Graphs, Their Spectra and Applications
Monday 2nd April 2007 to Thursday 5th April 2007
08:30 to 09:55  Registration  
09:55 to 10:00  Welcome Address  INI 1  
10:00 to 11:00 
Spectral properties and trace formulae for quantum graphs with general selfadjoint boundary conditions We consider general selfadjoint 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. 
INI 1  
11:00 to 11:30  Coffee  
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 SelbergGutzwiller type formula for the trace and extends earlier results by other authors in the context of quantum graphs. 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
A lower bound for nodal count on discrete and metric graphs According to a wellknow theorem by Sturm, a vibrating string is divided into exactly N nodal intervals by zeros of its Nth eigenfunction. Courant showed that one half of Sturm's theorem for the strings can be carried over to the theory of membranes: Nth 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 nontrivial 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 chaingraph. 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 Nth 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 nontrees). The result that we derive applies to both types of graphs: the number of nodal domains of the Nth eigenfunction is bounded below by NL, 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. 
INI 1  
15:00 to 15:30  Tea  
15:30 to 16:30 
Spectra of selfadjoint extensions related to the duality between discrete graphs and quantum networks Due to the de GennesAlexander 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 tightbinding Hamiltonian arises. We consider this question in the framework of the Krein selfadjoint 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. 
INI 1  
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. 
INI 1  
17:00 to 18:00  Welcome Wine Reception  
18:45 to 19:30  Dinner at Wolfson Court (Residents only) 
09:00 to 10:00 
A Klein ([California, Irvine]) 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. 
INI 1  
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. 
INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:30 
Spectral statistics of a pseudointegrable 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 semiPoisson 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. 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
S Gnutzmann ([Nottingham]) Spectral correlations of individual quantum graphs The spectral correlations of large wellconnected quantum graphs are shown to behave according to the predictions of randommatrix theory by using a supersymmentry method. In a first (generally applicable) step the energyaverage over the spectrum of individual graphs can be traded for the functional average over a supersymmetric nonlinear sigmamodel action. Reducing the full sigmamodel to its mean field theory is equivalent to the randommatrix theory of the WignerDyson 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. 
INI 1  
15:00 to 15:30  Tea  
15:30 to 16:00 
D Lenz ([Chemnitz]) 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 ShubinPastur 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

INI 1  
16:10 to 16:40 
Spectral asymptotics of percolation Laplacians on amenable Cayley graphs We study spectral properties of subcritical edgepercolation 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. 
INI 1  
16:50 to 17:20 
Reduction of quantum graphs to tightbinding 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 spinorbit coupling is also demonstrated. 
INI 1  
18:45 to 19:30  Dinner at Wolfson Court (Residents only)  
20:00 to 21:00  Musical event  INI 1 
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 nonpositive and delocalized and strongly localized eigenstates are present. 
INI 1  
10:00 to 11:00 
T Kottos ([Wesleyan]) Network models for BoseEinstein condensates We report our progress in studying ultracold atoms loaded in onedimensional waveguides with complicated topology. We discuss the structure and of the resulting stationary wavefunctions and analyze the spectral properties of the corresponding chemical potentials. 
INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:30 
Contractive semigroups on metric graphs The talk is devoted to the study of general (not necessarily selfadjoint) 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. 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
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), 183191, 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, 504520 (2004) 
INI 1  
16:30 to 17:00 
P Kurasov ([Lund Institute of Technology]) 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 noncompact 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. 
INI 1  
17:10 to 17:40 
On control and Inverse problems for the wave equation on graphs The inverse problem for the SturmLiouville 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. 
INI 1  
17:50 to 18:20 
Eigenvalue estimates for Schroedinger operators on regular metric trees We consider LiebThirring and CwikelLiebRozenblum 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. 
INI 1  
20:00 to 18:00  Conference Dinner at Christ's College 
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. 
INI 1  
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 DirichlettoNeumann map and subsequent rational approximation of the DNmap. The model admits complete fitting in terms of the relevant scattering matrix. 
INI 1  
11:00 to 11:30  Coffee  
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. 2scale convergence on lowdimensional manifolds in Rn . As an analogy of the previous topic, we shall discuss 2scale 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). 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
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 selfadjoint 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 KottosSmilansky 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 frequencydependent) 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 KirchhoffNeumann 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. PHY0554849. 
INI 1  
15:00 to 15:30  Tea  
15:30 to 16:00 
J Harrison ([Texas A and M]) 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. 
INI 1  
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. 
INI 1  
16:50 to 17:20 
Inverse spectral problems on compact and noncompact trees Secondorder 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 socalled Weyl vector. For this inverse problem we prove a uniqueness theorem and provide a procedure for constructing the solution. 
INI 1  
17:30 to 18:00  Conference closing  INI 1  
18:45 to 19:30  Dinner at Wolfson Court (Residents only) 