spectral determinant of schrodinger operators on graphs we study the spectral properties of the operator v x on a graph is the laplacian and v x is some potential defined on the graph in particular we derive an expression for the spectral determinant that generalizes one previously obtained for the laplacian operator

functionals of brownian motion localization and metric graphs we review several results related to the problem of a quantum particle in a random environment in an introductory part we recall how several functionals of brownian motion arise in the study of electronic transport in weakly disordered metals weak localization two aspects of the physics of the one dimensional strong localization are reviewed some properties of the scattering by a random potential time delay distribution and a study of the spectrum of a random potential on a bounded domain the extreme value statistics of the eigenvalues then we mention several results concerning the diffusion on graphs and more generally the spectral properties of the schrodinger operator on graphs the interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated finally we consider a two dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices we recall the connection between spectral properties of this model and winding functionals of planar brownian motion

the local time distribution of a particle diffusing on a graph we study the local time distribution of a brownian particle diffusing along the links on a graph in particular we derive an analytic expression of its laplace transform in terms of the green function on the graph we show that the asymptotic behaviour of this distribution has non gaussian tails characterized by a nontrivial large deviation function

distribution theory for discontinuous test functions and differential operators with generalized coefficients investigation of the differential operators with the generalized coefficients having singular support on a disjoint set of points requires the consideration of the distribution theory with the set of discontinuous test functions such a distribution theory for test functions having discontinuity at one point is developed a four parameter family of schrodinger operators formed by the operators with singular potential singular metrics and singular gauge field is considered it is proved that this family of singular interactions describes all possible selfadjoint extensions of the second derivative operator defined on the functions vanishing in a neighbourhood of the point approximation by operators with smooth coefficients is discussed c 1996 academic press inc

zeta functions of quantum graphs in this paper we construct zeta functions of quantum graphs using a contour integral technique based on the argument principle we start by considering the special case of the star graph with neumann matching conditions at the center of the star we then extend the technique to allow any matching conditions at the center for which the laplace operator is self adjoint and finally obtain an expression for the zeta function of any graph with general vertex matching conditions in the process it is convenient to work with new forms for the secular equation of a quantum graph that extend the well known secular equation of the neumann star graph in the second half of this paper we apply the zeta function to obtain new results for the spectral determinant vacuum energy and heat kernel coefficients of quantum graphs these have all been topics of current research in their own right and in each case this unified approach significantly expands results in the literature

maximum entropy and spectral estimation a review the method of maximum entropy is reviewed with emphasis on its relationship to entropy rate wiener filters autoregressive processes extrapolation the levinson algorithm lattice all pole and all pass filters and stability

spectral determinants and zeta functions of schrodinger operators on metric graphs a derivation of the spectral determinant of the schrodinger operator on a metric graph is presented where the local matching conditions at the vertices are of the general form classified according to the scheme of kostrykin and schrader to formulate the spectral determinant we first derive the spectral zeta function of the schrodinger operator using an appropriate secular equation the result obtained for the spectral determinant is along the lines of the recent conjecture texier 2010 j phys a math theor 43 425203

asymptotic behavior of jost functions near resonance points for wigner von neumann type potentials in this work are considered radial schrodinger operators v r e where v r a sin br r w r with w r bounded w r o r 2 at infinity a b real the asymptotic behavior of the jost function and the scattering matrix near the resonance point e0 b2 4 are studied if a b then this point may be an eigenvalue embedded in the continuous spectrum the leading behavior of the jost function for all values of a and b were determined somewhat surprisingly situations were found where the jost function becomes singular as e e0 even if e0 is an embedded eigenvalue moreover it was found that the scattering matrix is always discontinuous at e0 except in a few special cases it is also shown that the asymptotics for the jost function and the scattering matrix hold under weaker assumptions on w r in particular an angular momentum term l l 1 r 2 may be incorporated into w r the results were also applied to a whole line problem with a potential v x such that v x 0 for x 0 and v x of wigner von neumann

recovering differential pencils on compact graphs abstract we study boundary value problems on compact graphs without circles i e on trees for second order ordinary differential equations with nonlinear dependence on the spectral parameter we establish properties of the spectral characteristics and investigate the inverse spectral problem of recovering the coefficients of the differential equation from the so called weyl vector which is a generalization of the weyl function m function for the classical sturm liouville operator for this inverse problem we prove the uniqueness theorem and obtain a procedure for constructing the solution by the method of spectral mappings

high energy analysis and levinson s theorem for the selfadjoint matrix schrodinger operator on the half line the matrix schrodinger equation with a selfadjoint matrix potential is considered on the half line with the general selfadjoint boundary condition at the origin when the matrix potential is integrable the high energy asymptotics are established for the related jost matrix the inverse of the jost matrix and the scattering matrix under the additional assumption that the matrix potential has a first moment levinson s theorem is derived relating the number of bound states to the change in the argument of the determinant of the scattering matrix

the free dirac operator on compact and noncompact graphs the free dirac operator defined on composite one dimensional structures consisting of finitely many half lines and intervals is investigated the influence of the connection points between the constituents is modeled by transition conditions for the wave functions or equivalently by different self adjoint extensions of the dirac operator general relations between the parameters of the extensions and the eigenvalues resp the scattering coefficients are derived and then applied to the cases of a bundle of half lines a point defect a branching line and an eye shaped structure

four parameter point interaction in 1d quantum systems the author constructs a four parameter point interaction for a nonrelativistic particle moving on a line as the limit of a short range interaction with a range tending toward zero for particular choices of the parameters one can obtain a delta interaction or the so called delta interaction the hamiltonian corresponding to the four parameter point interaction is shown to correspond to the four parameter self adjoint hamiltonian of the free particle moving on the line with the origin excluded

scattering theory on graphs we consider the scattering theory for the schrodinger operator dx2 v x on the graphs made of one dimensional wires connected to external leads we derive two expressions for the scattering matrix on arbitrary graphs one involves matrices that couple arcs oriented bonds the other involves matrices that couple vertices we discuss a simple way to tune the coupling between the graph and the leads the efficiency of the formalism is demonstrated with a few known examples

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

a factorization of the scattering matrix for the schrodinger equation and for the wave equation in one dimension for the one dimensional schrodinger equation with a potential decaying at both ends of the real axis the scattering matrix of the potential is given explicitly in terms of the scattering matrices corresponding to the fragments of this potential a similar result also holds for the wave equation in a nonhomogeneous nondispersive medium where the wave speed has the same asymptotics at both ends of the real line

comment on factorization of scattering matrices due to partitioning of potentials in one dimensional schrodinger type equations j math phys 37 5897 1996

appendix resonances on a simple graph we study the scattering problem on a graph consisting of a line with a finite length appendix the two parts are coupled through boundary conditions depending on three parameters the motion on the line is free while the appendix supports a potential the appendix bound states give rise to a ladder of resonances we construct the resolvent and solve the corresponding pole condition for a weak coupling in general the condition only admits an analytic solution in particular cases we find the pole positions numerically for a linear potential and show that the poles eventually return to the real axis when the coupling strength increases

green function approach for general quantum graphs we present a recursive prescription to calculate the exact green function for general quantum graphs for the closed case the expression for the poles of g which gives the individual eigenstates has the structure of a semiclassical formula where the sum over the periodic orbit is already performed as applications we discuss eigenstate localization for a three arm closed star and filter like mechanisms for transmission throughout an open trident graph

ζ regularized spectral determinants on metric graphs several general results for the spectral determinant of the schrodinger operator on metric graphs are reviewed then a simple derivation for the regularized spectral determinant is proposed based on the roth trace formula two types of boundary conditions are studied functions continuous at the vertices and functions whose derivative is continuous at the vertices the regularized spectral determinant of the schrodinger operator acting on functions with the most general boundary conditions is conjectured in conclusion the relation to the ihara bass and bartholdi formulae obtained for combinatorial graphs is also discussed

maximum likelihood sequence estimation of cpm signals transmitted over rayleigh flat fading channels a method for the sequential updating of log likelihood functions for maximum likelihood sequence estimation is presented it is shown that in a general case this method can be implemented using kalman filtering techniques for the special case of rayleigh flat fading and continuous phase modulation cpm signaling this approach is shown to lead to an attractive receiver structure this type of receiver called the linear predictive receiver can be implemented in the form of the viterbi algorithm with the trellis updates being computed using a bank of finite pulse response fir filter and square operations simulation results are presented that demonstrate the superiority of the linear predictive receiver over receivers employing differential detection in the presence of fast fading it is shown that the optimal linear predictive receiver does not possess an irreducible error rate for a class of rayleigh fading channels used to model terrestrial mobile channels

on the spectrum of the laplace operator of metric graphs attached at a vertex spectral determinant approach we consider a metric graph made of two graphs and attached at one point we derive a formula relating the spectral determinant of the laplace operator in terms of the spectral determinants of the two subgraphs the result is generalized to describe the attachment of n graphs the formulae are also valid for the spectral determinant of the schrodinger operator

skew group algebras in the representation theory of artin algebras then the corresponding crossed product algebra i g or i x g for short has an elements cgieg gi a li i addition is componentwise and multiplication is given by g a g and g g2 y g g2 m in this paper we assume that the values of y lie in the center z a of i hence the action of g on i is given by a group homomorphism g p aut i and 3 can be left out in the special case that y is the trivial map we write 1g instead of i g and the elements as c c li gi ag is then called a skew group ring there is a lot of literature on skew group algebras and crossed product algebras and on the relationship with the ring ac whose elements are those elements of a left fixed by g much work has been done on which properties of li are inherited by n g or ac some of the work on the relationship between these rings has its roots in trying to develop a galois theory for noncommutative rings we refer to 3 7 13 15 19 21 23 25 27 281 and their references in this paper we study these constructions when i is an artin algebra and g a finite group of order n such that n is invertible in 1 under these assumptions the construction preserves central properties of interest in the

on the number of states bound by one dimensional finite periodic potentials bound states and zero energy resonances of one dimensional finite periodic potentials are investigated by means of levinson s theorem for finite range potentials supporting no bound states a lower bound for the reduced time delay at threshold is derived

value distribution of the eigenfunctions and spectral determinants of quantum star graphs we compute the value distributions of the eigenfunctions and spectral determinant of the schrodinger operator on families of star graphs the values of the spectral determinant are shown to have a cauchy distribution with respect both to averages over bond lengths in the limit as the wavenumber tends to infinity and to averages over wavenumber when the bond lengths are fixed and not rationally related this is in contrast to the spectral determinants of random matrices for which the logarithm is known to satisfy a gaussian limit distribution the value distribution of the eigenfunctions also differs from the corresponding random matrix result we argue that the value distributions of the spectral determinant and of the eigenfunctions should coincide with those of seba type billiards

factorization of scattering matrices due to partitioning of potentials in one dimensional schrodinger type equations the one dimensional schrodinger equation and two of its generalizations are considered as they arise in quantum mechanics wave propagation in a nonhomogeneous medium and wave propagation in a nonconservative medium where energy may be absorbed or generated generically the zero energy transmission coefficient vanishes when the potential is nontrivial but in the exceptional case this coefficient is nonzero resulting in tunneling through the potential it is shown that any nontrivial exceptional potential can always be fragmented into two generic pieces furthermore any nontrivial potential generic or exceptional can be fragmented into generic pieces in infinitely many ways the results remain valid when dirac delta functions are included in the potential and other coefficients are added to the schrodinger equation for such schrodinger equations factorization formulas are obtained that relate the scattering matrices of the fragments to the scattering matrix of the full problem

some properties of prediction and interpolation errors linear prediction and interpolation are widely used in signal processing usually they are obtained by linear filters of finite memory and the prediction and interpolation errors decrease when this memory increases if the asymptotic values of these errors are null the signal is singular some theoretical aspects of finite and infinite prediction and interpolation are discussed and a numerical method to compute the prediction and interpolation errors for signals with a given power spectrum is presented a large number of results obtained with this method are discussed especially the difference between regular and singular signals this method also allows the behavior of prediction and interpolation errors when the memory of the filter increases to be studied

boundary conditions at the derivative of a delta function the author derives the joining conditions on psi at a point where the potential is the nth derivative of a dirac delta function

on the inverse scattering problem on branching graphs the inverse scattering problem on branching graphs is studied the definition of the schrodinger operator on such graphs is discussed the operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices it is shown that in general the scattering matrix does not determine the topology of the graph the potentials on the edges and the boundary conditions uniquely

scattering theory on graphs ii the friedel sum rule we consider the friedel sum rule fsr in the context of the scattering theory for the schrodinger operator dx2 v x on graphs made of one dimensional wires connected to external leads we generalize the smith formula for graphs we give several examples of graphs where the state counting method given by the fsr does not work the reason for the failure of the fsr to count the states is the existence of states localized in the graph and not coupled to the leads which occurs if the spectrum is degenerate and the number of leads too small

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

charge and current distribution in graphs we consider graphs made of one dimensional wires connected at vertices and on which may live a scalar potential we are interested in a scattering situation where such a network is connected to infinite leads we study the correlations of the charge in such graphs out of equilibrium as well as the distribution of the currents in the wires inside the graph these quantities are related to the scattering matrix of the graph we discuss the case where the graph is weakly connected to the wires

characteristic functions under series and parallel connection of quantum graphs for a graph consisting of parallel connected subgraphs we express the characteristic function of the boundary value problem with generalized neumann conditions at both joining points via characteristic functions of different boundary problems on the subgraphs

classification and recognition of dynamical models the role of phase independent components kernels and optimal transport we address the problem of performing decision tasks and in particular classification and recognition in the space of dynamical models in order to compare time series of data motivated by the application of recognition of human motion in image sequences we consider a class of models that include linear dynamics both stable and marginally stable periodic both minimum and nonminimum phases driven by non gaussian processes this requires extending existing learning and system identification algorithms to handle periodic modes and nonminimum phase behavior while taking into account higher order statistics of the data once a model is identified we define a kernel based cord distance between models which includes their dynamics their initial conditions and input distribution this is made possible by a novel kernel defined between two arbitrary non gaussian distributions which is computed by efficiently solving an optimal transport problem we validate our choice of models inference algorithm and distance on the tasks of human motion synthesis sample paths of the learned models and recognition nearest neighbor classification in the computed distance however our work can be applied more broadly where one needs to compare historical data while taking into account periodic trends nonminimum phase behavior and non gaussian input distributions

predictable processes and wold s decomposition a review the concepts of predictability and band limitedness are reexamined and a simple proof of wold s decomposition is presented in the context of mean square estimation

two body scattering on a graph and application to simple nanoelectronic devices the two body system on a graph with one junction is considered the effective three body scattering problem turns out to be exactly solvable for pointwise interactions additional degrees of freedom corresponding to a dynamics of some structure e g an atomic cluster located in the junction point of common contact of three thin electrodes are considered these degrees of freedom bring effective energy dependent interaction into the effective schrodinger equation in the scattering channel the wave function of the system is constructed in the explicit form using the extension theory methods the obtained results are applied to the qualitative description of a simple three electrode nanoelectronic device the perturbation theory approach based on the analysis of the liouville equation is suggested for calculation of the conductivity for such a device in terms of the obtained wave function

sequential reconstruction of driving forces from nonlinear nonstationary dynamics abstract this paper describes a functional analysis based method for the estimation of driving forces from nonlinear dynamic systems the driving forces account for the perturbation inputs induced by the external environment or the secular variations in the internal variables of the system the proposed algorithm is applicable to the problems for which there is too little or no prior knowledge to build a rigorous mathematical model of the unknown dynamics we derive the estimator conditioned on the differentiability of the unknown system s mapping and smoothness of the driving force the proposed algorithm is an adaptive sequential realization of the blind prediction error method where the basic idea is to predict the observables and retrieve the driving force from the prediction error our realization of this idea is embodied by predicting the observables one step into the future using a bank of echo state networks esn in an online fashion and then extracting the raw estimates from the prediction error and smoothing these estimates in two adaptive filtering stages the adaptive nature of the algorithm enables to retrieve both slowly and rapidly varying driving forces accurately which are illustrated by simulations logistic and moran ricker maps are studied in controlled experiments exemplifying chaotic state and stochastic measurement models the algorithm is also applied to the estimation of a driving force from another nonlinear dynamic system that is stochastic in both state and measurement equations the results are judged by the posterior cramer rao lower bounds the method is finally put into test on a real world application extracting sun s magnetic flux from the sunspot time series