an approximation to δ couplings on graphs we discuss a general parametrization for vertices of quantum graphs and show in particular how the s and couplings at an n edge vertex can be approximated by means of n 1 couplings of the type provided the latter are properly scaled

approximation of quantum graph vertex couplings by scaled schrodinger operators on thin branched manifolds we discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds we show that if a neumann type laplacian on such manifolds is amended by suitable potentials the resulting schrodinger operators can approximate non trivial vertex couplings the latter include not only the couplings but also those with wavefunctions discontinuous at the vertex we work out the example of the symmetric couplings and make a conjecture that the same method can be applied to all couplings invariant with respect to the time reversal we conclude with a result that certain vertex couplings cannot be approximated by a pure laplacian

self diffusion and visited surface in the droplet condensation problem breath figures abstract we study experimentally and theoretically new aspects of condensation growth and coalescence of droplets on a substrate we address in particular the dynamics of a marked droplet which undergoes coalescences with neighbouring droplets we find that the number of coalescences of the droplet grows as log t t is time and that the traveled distance increases as r t the average radius of the droplets at time t the fraction of the surface which was never covered by any droplet an important quantity for applications such as drug spreading or surface decontamination decays as t showing that completion occurs only slowly heuristic arguments accurate numerical simulations on simplified models which neglect polydispersity and an exact solution reported elsewhere a bray b derrida and c godreche europhys lett 27 1994 175 strongly support these findings and show that this power law decay is a generic feature common to many different situations finally the contour of the ensemble of dry sites appears fractal with dimension d 1 22 an experimental result not reproduced by the simplified models

persistence properties of a system of coagulating and annihilating random walkers we study a d dimensional system of diffusing particles that on contact either annihilate with probability 1 q ensuremath 1 or coagulate with probability q ensuremath 2 q ensuremath 1 in one dimension the system models the zero temperature glauber dynamics of domain walls in the q state potts model we calculate overline p m t the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly m ensuremath 1 coagulations using perturbative renormalization group analysis for dl2 we show that if the number of coagulations m is much less than the typical number m t then overline p m t ensuremath sim m ensuremath zeta d t ensuremath ensuremath theta with ensuremath theta dq q q ensuremath 1 2 ensuremath epsilon o ensuremath epsilon 2 ensuremath zeta 2q ensuremath 1 ensuremath epsilon 2q ensuremath 1 q ensuremath 1 1 2 aq ensuremath epsilon 2 o ensuremath epsilon 3 where q q ensuremath 1 q ensuremath epsilon 2 ensuremath d and a ensuremath 0 006 dots m t is shown to scale as m t ensuremath sim t d 2 ensuremath ensuremath delta where ensuremath delta d 1 ensuremath q q ensuremath 1 q ensuremath 1 2 ensuremath epsilon o ensuremath epsilon 2 in two dimensions we show that overline p m t ensuremath sim mathrm ln t q 3 ensuremath 2q mathrm ln m 2q ensuremath 1 2 t ensuremath 2q for m ensuremath ll t 2q ensuremath 1 we also derive an exact nonperturbative relation between the exponents namely ensuremath delta q ensuremath theta 1 ensuremath q the one dimensional results corresponding to ensuremath epsilon 1 are compared with results from monte carlo simulations

potential approximations to δ an inverse klauder phenomenon with norm resolvent convergence we show that there is a family schrodinger operators with scaled potentials which approximates the interaction hamiltonian in the norm resolvent sense this approximation based on a formal scheme proposed by cheon and shigehara has nontrivial convergence properties which are in several respects opposite to those of the klauder phenomenon

persistence of a continuous stochastic process with discrete time sampling non markov processes we consider the problem of discrete time persistence which deals with the zero crossings of a continuous stochastic process x t measured at discrete times t ndeltat for a gaussian stationary process the persistence no crossing probability decays as exp theta d t rho a n for large n where a exp deltat 2 and the discrete persistence exponent theta d is given by theta d ln rho 2 ln a using the independent interval approximation we show how theta d varies with deltat for small deltat and conclude that experimental measurements of persistence for smooth processes such as diffusion are less sensitive to the effects of discrete sampling than measurements of a randomly accelerated particle or random walker we extend the matrix method developed by us previously phys rev e 64 015101 r 2001 to determine rho a for a two dimensional random walk and the one dimensional random acceleration problem we also consider alternating persistence which corresponds to a 0 and calculate rho a for this case

the balian low theorem for the symplectic form on r2d in this paper we extend the balian low theorem which is a version of the uncertainty principle for gabor weyl heisenberg systems to functions of several variables in particular we first prove the balian low theorem for arbitrary quadratic forms then we generalize further and prove the balian low theorem for differential operators associated with a symplectic basis for the symplectic form on r2d

factorization of j unitary matrix polynomials on the line and a schur algorithm for generalized nevanlinna functions we prove that a 2 2 matrix polynomial which is j unitary on the real line can be written as a product of normalized elementary j unitary factors and a j unitary constant in the second part we give an algorithm for this factorization using an analog of the schur transformation

structured invariant spaces of vector valued rational functions hermitian matrices and a generalization of the lohvidov laws abstract finite dimensional indefinite inner product spaces of vector valued rational functions which are 1 invariant under the generalized backward shift and 2 subject to a structural identity and subspaces and superspaces thereof are studied the theory of these spaces is then applied to deduce a generalization of a pair of rules due to lohvidov for evaluating the inertia of certain subblocks of hermitian toeplitz and hermitian hankel matrices the connecting link rests on the identification of a hermitian matrix as the gram matrix of a space of vector valued functions of the type considered in the first part of the paper corresponding generalizations of another pair of theorems by lohvidov on the rank of certain subblocks of non hermitian teoplitz and non hermitian hankel matrices are also stated but the proofs will be presented elsewhere

on the effective size of a non weyl graph we show how to find the coefficient of the leading term of the resonance asymptotics using the method of pseudo orbit expansion for quantum graphs which do not obey weyl asymptotics for a non weyl graph we develop a method to reduce the number of edges of a corresponding directed graph through this method we prove bounds on the above coefficient depending on the structure of the graph for graphs with the same lengths of internal edges we explicitly find the positions of the resolvent resonances

persistence exponents for fluctuating interfaces numerical and analytic results for the exponent ensuremath theta describing the decay of the first return probability of an interface to its initial height are obtained for a large class of linear langevin equations the models are parametrized by the dynamic roughness exponent ensuremath beta with 0l ensuremath beta l1 for ensuremath beta frac 1 2 the time evolution is markovian using simulations of solid on solid models of the discretized continuum equations as well as of the associated zero dimensional stationary gaussian process we address two problems the return of an initially flat interface and the return to an initial state with fully developed steady state roughness the two problems are shown to be governed by different exponents for the steady state case we point out the equivalence to fractional brownian motion which has a return exponent ensuremath theta s 1 ensuremath ensuremath beta the exponent ensuremath theta 0 for the flat initial condition appears to be nontrivial we prove that ensuremath theta 0 ensuremath rightarrow ensuremath infty for ensuremath beta ensuremath rightarrow 0 ensuremath theta 0 g ensuremath theta s for ensuremath beta l frac 1 2 and ensuremath theta 0 l ensuremath theta s for ensuremath beta g frac 1 2 and calculate ensuremath theta 0 s perturbatively to first order in an expansion around the markovian case ensuremath beta frac 1 2 using the exact result ensuremath theta s 1 ensuremath ensuremath beta accurate upper and lower bounds on ensuremath theta 0 can be derived which show in particular that ensuremath theta 0 g 1 ensuremath ensuremath beta 2 ensuremath beta for small ensuremath beta

a remark on krein s resolvent formula and boundary conditions we prove an analogue of krein s resolvent formula expressing the resolvents of self adjoint extensions in terms of boundary conditions applications to quantum graphs and systems with point interactions are discussed

jl unitary factorization and the schur algorithm for nevanlinna functions in an indefinite setting abstract we introduce a schur transformation for generalized nevanlinna functions and show that it can be used in obtaining the unique minimal factorization of a class of rational j l unitary 2 2 matrix functions into elementary factors from the same class

semiflexible polymer in the half plane and statistics of the integral of a brownian curve a continuum model of a polymer with non zero bending energy fluctuating without overhangs in the half plane is considered the exact partition function is obtained from the marshall watson solution of the klein kramers equation for brownian motion in the half space the partition function contains information on probabilities associated with the integral of a brownian curve and reproduces sinai s t 5 4 result for the asymptotic first passage time density the t 5 2 dependence of a different passage probability implies a first order polymer adsorption transition for short range pinning potentials

cluster persistence of two dimensional soap froth we report on the persistence decay of cell clusters in dynamical simulation of two dimensional soap froth as a function of the cluster volume fraction both slow gas diffusion and fast dynamical topological processes between cells are incorporated in the simulation while the simulation yields decay rate in good agreement with two dimensional soap froth experiment it shows a faster decay in comparison to measurements and calculations of persistence in other curvature driven systems the fraction of individual cells that contain persistent area also decays due to significant cell movements confirming that the neighbor swapping t1 topological transition is important in froth coarsening

index theorems for quantum graphs in geometric analysis an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned sometimes using the heat kernels of two higher order differential operators as an intermediary in this paper the case of quantum graphs is addressed a quantum graph is a graph considered as a singular one dimensional variety and equipped with a second order differential hamiltonian h a laplacian with suitable conditions at vertices for the case of scale invariant vertex conditions i e conditions that do not mix the values of functions and of their derivatives the constant term of the heat kernel expansion is shown to be proportional to the trace of the internal scattering matrix of the graph this observation is placed into the index theory context by factoring the laplacian into two first order operators h a a and relating the constant term to the index of a an independent consideration provides an index formula for any differential operator on a finite quantum graph in terms of the vertex conditions it is also found that the algebraic multiplicity of 0 as a root of the secular determinant of h is the sum of the nullities of a and a

on δ like potential scattering on star graphs we discuss the potential scattering on the noncompact star graph the schrodinger operator with the short range potential localized in a neighborhood of the graph vertex is considered we study the asymptotic behavior of the corresponding scattering matrix in the zero range limit it has been known for a long time that in dimension 1 there is no non trivial hamiltonian with the distributional potential i e the potential acts as a totally reflecting wall several authors have in recent years studied the scattering properties of the regularizing potentials e 2q x e approximating the first derivative of the dirac delta function a non zero transmission through the regularized potential has been shown to exist as e 0 we extend these results to star graphs with the point interaction which is an analog of the potential on the line we prove that generically such a potential on the graph is opaque we also show that there exists a countable set of resonant intensities for which a partial transmission through the potential occurs this set of resonances is referred to as the resonant set and is determined as the spectrum of an auxiliary sturm liouville problem associated with q on the graph

experimental measurement of the persistence exponent of the planar ising model using a twisted nematic liquid crystal system exhibiting planar ising model dynamics we have measured the scaling exponent ensuremath theta which characterizes the time evolution p t ensuremath sim t ensuremath ensuremath theta of the probability p t that the local order parameter has not switched its state by the time t for 0 4 sec 200 sec following the phase quench the system exhibits scaling behavior and measured over this interval ensuremath theta 0 19 ifmmode pm else textpm fi 0 03 in good agreement with theoretical analysis and numerical simulations

the spectrum of the hilbert space valued second derivative with general self adjoint boundary conditions abstract we consider a large class of self adjoint elliptic problems associated with the second derivative acting on a space of vector valued functions we present and survey several results that can be obtained by means of two different approaches to the study of the associated eigenvalues problems the first more general one allows to replace a secular equation which is well known in some special cases by an abstract rank condition the second one though available in general seems to apply particularly well to a specific boundary condition the sometimes dubbed anti kirchhoff condition in the literature that arises in the theory of differential operators on graphs it also permits to discuss interesting and more direct connections between the spectrum of the differential operator and some graph theoretical quantities in particular some results on the symmetry of the spectrum in either case

mobius structure of the spectral space of schrodinger operators with point interaction the schrodinger operator with point interaction in one dimension has a u 2 family of self adjoint extensions we study the spectrum of the operator and show that i the spectrum is uniquely determined by the eigenvalues of the matrix u u 2 that characterizes the extension and that ii the space of distinct spectra is given by the orbifold t2 z2 which is a mobius strip with boundary we employ a parametrization of u 2 that admits a direct physical interpretation and furnishes a coherent framework to realize the spectral duality and anholonomy recently found this allows us to find that iii physically distinct point interactions form a three parameter quotient space of the u 2 family

persistence of a continuous stochastic process with discrete time sampling we introduce the concept of discrete time persistence which deals with zero crossings of a continuous stochastic process x t measured at discrete times t n delta t for a gaussian markov process with relaxation rate mu we show that the persistence no crossing probability decays as rho a n for large n where a exp mu delta t and we compute rho a to high precision we also define the concept of alternating persistence which corresponds to a 1 corresponding to motion in an unstable potential mu 0 there is a nonzero probability of having no zero crossings in infinite time and we show how to calculate it

uniform existence of the integrated density of states for random schrodinger operators on metric graphs over zd abstract we consider ergodic random schrodinger operators on the metric graph 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 random magnetic fields and percolation models

persistence of non markovian gaussian stationary processes in discrete time the persistence of a stochastic variable is the probability that it does not cross a given level during a fixed time interval although persistence is a simple concept to understand it is in gener

the schur algorithm for generalized schur functions iii j unitary matrix polynomials on the circle the main result is that for j 1 0 0 1 every j unitary 2 x 2 matrix polynomial on the unit circle is an essentially unique product of elementary j unitary 2 x 2 matrix polynomials which are either of degree 1 or 2k this is shown by means of the generalized schur transformation introduced in ann inst fourier 8 1958 211 ann acad sci fenn ser a i 250 9 1958 1 7 and studied in pisot and salem numbers birkhauser verlag basel 1992 philips j res 41 1 1986 1 54 and also in the first two parts operator theory adv appl 129 birkhauser verlag basel 2000 p 1 monatshefte fur mathematik in press of this series the essential tool in this paper are the reproducing kernel pontryagin spaces associated with generalized schur functions c 2003 elsevier science inc all fights reserved

spectral asymptotics for quantum graphs with equal edge lengths if q is smooth on the edges of a finite metric graph with all edges of length one the eigenvalues of the schrodinger operator d2 q with standard vertex conditions have generalized asymptotic expansions to all orders

statistics of multiple sign changes in a discrete non markovian sequence we study analytically the statistics of multiple sign changes in a discrete non markovian sequence psi i phi i phi i 1 i 1 2 em leader n where phi i s are independent and identically distributed random variables each drawn from a symmetric and continuous distribution rho phi we show that the probability p m n of m sign changes up to n steps is universal i e independent of the distribution rho phi the mean and variance of the number of sign changes are computed exactly for all n 0 we show that the generating function tilde p p n summation operator infinity m 0 p m n p m approximately exp theta d p n for large n where the discrete partial survival exponent theta d p is given by a nontrivial formula theta d p ln sin 1 square root of 1 p 2 square root of 1 p 2 for 0 infinity n infinity but keeping x m n fixed p m n approximately exp n phi x where the large deviation function phi x is computed the implications of these results for ising spin glasses are discussed

persistence in a stationary time series we study the persistence in a class of continuous stochastic processes that are stationary only under integer shifts of time we show that under certain conditions the persistence of such a continuous process reduces to the persistence of a corresponding discrete sequence obtained from the measurement of the process only at integer times we then construct a specific sequence for which the persistence can be computed even though the sequence is non markovian we show that this may be considered as a limiting case of persistence in the diffusion process on a hierarchical lattice

a generalization of schur functions applications to nevanlinna functions orthogonal polynomials random walks and unitary and open quantum walks abstract recent work on return properties of quantum walks the quantum analogue of random walks has identified their generating functions of first returns as schur functions this is connected with a representation of schur functions in terms of the operators governing the evolution of quantum walks i e the unitary operators on hilbert spaces in this paper we propose a generalization of schur functions by extending the above operator representation to arbitrary closed operators on banach spaces such generalized schur functions meet the formal structure of first return generating functions thus we call them fr functions we derive some general properties of fr functions among them a simple relation with an operator version of stieltjes functions which generalizes the renewal equation already known for random and quantum walks we also prove that fr functions satisfy splitting properties which extend useful factorizations of schur functions when specialized to self adjoint operators on hilbert spaces we show that fr functions become nevanlinna functions this allows us to obtain properties of nevanlinna functions which as far as we know seem to be new the fr function structure leads to a new operator representation of nevanlinna functions in terms of self adjoint operators whose spectral measures provide also new integral representations of such functions this allows us to characterize each nevanlinna function by a measure on the real line which we refer to as the measure of the nevanlinna function in contrast to standard operator and integral representations of nevanlinna functions these new ones are exact analogues of those already known for schur functions the above results are also the source of a very simple schur algorithm for nevanlinna functions based on interpolations at points on the real line which we refer to as the schur algorithm on the real line the paper is completed with several applications of fr functions to orthogonal polynomials and random and quantum walks which illustrate their wide interest an analogue for orthogonal polynomials on the real line of the khrushchev formula for orthogonal polynomials on the unit circle and the use of fr functions to study recurrence in random walks quantum walks and open quantum walks these applications provide numerous explicit examples of fr functions clarifying the meaning of these functions as first return generating functions and their splittings which become recurrence splitting rules they also show that these new tools despite being extensions of very classical ones play an important role in the study of physical problems of a highly topical nature

traces and inverse nodal problems for dirac type integro differential operators on a graph abstract in this paper dirac type integro differential operators on the star graph are studied first we derive the asymptotic expressions for the large eigenvalues second the regularized trace formulae are obtained for these operators we also provide asymptotic expressions for the nodal points and give a constructive procedure for solving an inverse nodal problem

reducible boundary conditions in coupled channels we study hamiltonians with point interactions in spaces of vector valued functions using some information from the theory of quantum graphs we describe a class of the operators which can be reduced to the direct sum of several one dimensional problems it shown that such a reduction is closely connected with the invariance under channel permutations examples are provided by some model interactions in particular the so called and the kirchhoff couplings

zero crossing statistics for non markovian time series in applications spanning from image analysis and speech recognition to energy dissipation in turbulence and time to failure of fatigued materials researchers and engineers want to calculate how often a stochastic observable crosses a specific level such as zero at first glance this problem looks simple but it is in fact theoretically very challenging and therefore few exact results exist one exception is the celebrated rice formula that gives the mean number of zero crossings in a fixed time interval of a zero mean gaussian stationary process in this study we use the so called independent interval approximation to go beyond rice s result and derive analytic expressions for all higher order zero crossing cumulants and moments our results agree well with simulations for the non markovian autoregressive model

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

asymptotically isospectral quantum graphs and generalised trigonometric polynomials abstract the theory of almost periodic functions is used to investigate spectral properties of schrodinger operators on metric graphs also known as quantum graphs in particular we prove that two schrodinger operators may have asymptotically close spectra if and only if the corresponding reference laplacians are isospectral our result implies that a schrodinger operator is isospectral to the standard laplacian on a may be different metric graph only if the potential is identically equal to zero

unitary dimension reduction for a class of self adjoint extensions with applications to graph like structures abstract we consider a class of self adjoint extensions using the boundary triplet technique assuming that the associated weyl function has the special form m z m z id t n z 1 with a bounded self adjoint operator t and scalar functions m n we show that there exists a class of boundary conditions such that the spectral problem for the associated self adjoint extensions in gaps of a certain reference operator admits a unitary reduction to the spectral problem for t as a motivating example we consider differential operators on equilateral metric graphs and we describe a class of boundary conditions that admit a unitary reduction to generalized discrete laplacians

hermitian symplectic geometry and extension theory here we give brief account of hermitian symplectic spaces showing that they are intimately connected to symmetric as well as self adjoint extensions of a symmetric operator furthermore we find an explicit parametrization of the lagrange grassmannian in terms of the unitary matrices this allows us to explicitly describe all self adjoint boundary conditions for the schrodinger operator on the graph in terms of a unitary matrix we show that the asymptotics of the scattering matrix can be expressed simply in terms of this unitary matrix

persistence exponents and the statistics of crossings and occupation times for gaussian stationary processes we consider the persistence probability the occupation time distribution and the distribution of the number of zero crossings for discrete or equivalently discretely sampled gaussian stationary processes gsps of zero mean we first consider the ornstein uhlenbeck process finding expressions for the mean and variance of the number of crossings and the partial survival probability we then elaborate on the correlator expansion developed in an earlier paper g c m a ehrhardt and a j bray phys rev lett 88 070602 2002 to calculate discretely sampled persistence exponents of gsps of known correlator by means of a series expansion in the correlator we apply this method to the processes d n x dt n ensuremath eta t with ng 3 incorporating an extrapolation of the series to the limit of continuous sampling we then extend the correlator method to calculate the occupation time and crossing number distributions as well as their partial survival distributions and the means and variances of the occupation time and number of crossings we apply these general methods to the d n x dt n ensuremath eta t processes for n 1 random walk n 2 random acceleration and larger n and to simple diffusion from random initial conditions in one to three dimensions the results for discrete sampling are extrapolated to the continuum limit where possible

convergence of spectra of mesoscopic systems collapsing onto a graph abstract let m be a finite graph in the plane and let me be a domain that looks like the e fattened graph m exact conditions on the domain are given it is shown that the spectrum of the neumann laplacian on me converges when e 0 to the spectrum of an ode problem on m the presence of an electromagnetic field is also allowed considerations of this kind arise naturally in mesoscopic physics and other areas of physics and chemistry the results of the paper extend the ones previously obtained by j rubinstein and m schatzman

crossing intervals of non markovian gaussian processes we review the properties of time intervals between the crossings at a level m of a smooth stationary gaussian temporal signal the distribution of these intervals and the persistence are derived within the independent interval approximation iia these results grant access to the distribution of extrema of a general gaussian process exact results are obtained for the persistence exponents and the crossing interval distributions in the limit of large m in addition the small time behavior of the interval distributions and the persistence is calculated analytically for any m the iia is found to reproduce most of these exact results and its accuracy is also illustrated by extensive numerical simulations applied to non markovian gaussian processes appearing in various physical contexts

classifying the general family of 1d point interactions a scattering approach we propose a classification scheme for the complete family of 1d point interactions to do so we first review the solutions of the wavefunctions and green functions of the problem second we derive the exact time dependent propagators in such a way that we can write the expressions for the k s in a very compact form as they should such expressions do reduce to the known results in the literature according to the potential parameter values then we analyse in general terms how the different point interactions scatter off arbitrary initially localized wave packets finally we show that the physical features associated with the scattering process can be used to establish a classification procedure moreover these physical characteristics are directly related to the potential parameters leading to the many formulae for the k s as an application we present numerical calculations for gaussian wave packets

on the boundary conditions in self adjoint multi interval sturm liouville problems abstract in this paper we completely solve a problem whose two interval case has been studied recently by wang et al in a wang j sun a zettl two interval sturm liouville operators in modified hilbert spaces j math anal appl 328 2007 390 399 the problem is to give an explicit description of all boundary conditions defining the self adjoint differential operators associated with several sturm liouville equations together i e the so called multi interval sturm liouville problems the explicit description obtained in this paper only uses hermitian matrices as an application we then show that a generic self adjoint boundary condition can not come from any direct sum of self adjoint operators associated with the individual sturm liouville equations and it depends non trivially on the multiples used in the definition of the so called maximum operator

non weyl asymptotics for quantum graphs with general coupling conditions inspired by a recent result of davies and pushnitski we study resonance asymptotics of quantum graphs with general coupling conditions at the vertices we derive a criterion for the asymptotics to be of a non weyl character we show that for balanced vertices with permutation invariant couplings the asymptotics is non weyl only in the case of kirchhoff or anti kirchhoff conditions while for graphs without permutation symmetry numerous examples of non weyl behaviour can be constructed furthermore we present an insight into what makes the kirchhoff anti kirchhoff coupling particular from the resonance point of view finally we demonstrate a generalization to quantum graphs with unequal edge weights

non trivial exponents in the zero temperature dynamics of the 1d ising and potts models we consider the glauber dynamics of the q state potts model in one dimension at zero temperature starting with a random initial configuration we measure the density r t of spins which have never flipped from the beginning of the simulation until time t we find that for large t the density r t has a power law decay left r t sim t theta right where the exponent theta varies with q our simulations lead to theta simeq 37 for q 2 theta simeq 53 for q 3 and theta longrightarrow 1 as q longrightarrow infty

variational problems on multiply connected thin strips i basic estimates and convergence of the laplacian spectrum let m be a planar embedded graph whose arcs meet transversally at the vertices let be a strip shaped domain around m of width except in a neighborhood of the singular points assume that the boundary of is smooth we define comparison operators between functions on and on m and we derive energy estimates for the compared functions we define a laplace operator on m which is in a certain sense the limit of the laplace operator on with neumann boundary conditions in particular we show that the p th eigenvalue of the laplacian on converges to the p th eigenvalue of the laplacian on m as tends to 0 a similar result holds for the magnetic schrodinger operator