geometric aspects of sturm liouville problems iii level surfaces of the nth eigenvalue for a given regular sturm liouville equation and any natural number n we explicitly describe the level surfaces of the nth eigenvalue of the equation regarded as a function on the space of self adjoint boundary conditions the description of each level surface uses only a particular value of the transfer matrix of the equation the shapes of these surfaces are also determined

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

spectral hp least squares finite element formulation for the navier stokes equations we consider the application of least squares finite element models combined with spectral hp methods for the numerical solution of viscous flow problems the paper presents the formulation validation and application of a spectral hp algorithm to the numerical solution of the navier stokes equations governing two and three dimensional stationary incompressible and low speed compressible flows the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity or velocity gradients as additional independent variables and the least squares method is used to develop the finite element model high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method spectral convergence of the l2 least squares functional and l2 error norms is verified using smooth solutions to the two dimensional stationary poisson and incompressible navier stokes equations numerical results for flow over a backward facing step steady flow past a circular cylinder three dimensional lid driven cavity flow and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation

liouville theorems and spectral edge behavior on abelian coverings of compact manifolds the paper describes relations between liouville type theorems for solutions of a periodic elliptic equation or a system on an abelian cover of a compact riemannian manifold and the structure of the dispersion relation for this equation at the edges of the spectrum here one says that the liouville theorem holds if the space of solutions of any given polynomial growth is finite dimensional the necessary and sufficient condition for a liouville type theorem to hold is that the real fermi surface of the elliptic operator consists of finitely many points modulo the reciprocal lattice thus such a theorem generically is expected to hold at the edges of the spectrum the precise description of the spaces of polynomially growing solutions depends upon a homogenized constant coefficient operator determined by the analytic structure of the dispersion relation in most cases simple explicit formulas are found for the dimensions of the spaces of polynomially growing solutions in terms of the dispersion curves the role of the base of the covering in particular its dimension is rather limited while the deck group is of the most importance the results are also established for overdetermined elliptic systems which in particular leads to liouville theorems for polynomially growing holomorphic functions on abelian coverings of compact analytic manifolds analogous theorems hold for abelian coverings of compact combinatorial or quantum graphs

matrix representations of sturm liouville problems with transmission conditions we identify a class of sturm liouville equations with transmission conditions such that any sturm liouville problem consisting of such an equation with transmission condition and an arbitrary separated or real coupled self adjoint boundary condition has a representation as an equivalent finite dimensional matrix eigenvalue problem conversely given any matrix eigenvalue problem of certain type and an arbitrary separated or real coupled self adjoint boundary condition and transmission condition we construct a class of sturm liouville problems with this specified boundary condition and transmission condition each of which is equivalent to the given matrix eigenvalue problem

quantum ergodicity for graphs related to interval maps we prove quantum ergodicity for a family of graphs that are obtained from ergodic one dimensional maps of an interval using a procedure introduced by pakonski et al j phys a 34 9303 9317 2001 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 specifically 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 directed edges correspond to elements of the partitions and on which the classical dynamics approximates the perron frobenius operator corresponding to the map we show that except possibly for subsequences of density 0 the eigenstates of the quantum graphs equidistribute in the limit of large graphs

limits of quantum graph operators with shrinking edges abstract we address the question of convergence of schrodinger operators on metric graphs with general self adjoint vertex conditions as lengths of some of graph s edges shrink to zero we determine the limiting operator and study convergence in a suitable norm resolvent sense it is noteworthy that as edge lengths tend to zero standard sobolev type estimates break down making convergence fail for some graphs we use a combination of functional analytic bounds on the edges of the graph and lagrangian geometry considerations for the vertex conditions to establish a sufficient condition for convergence this condition encodes an intricate balance between the topology of the graph and its vertex data in particular it does not depend on the potential on the differences in the rates of convergence of the shrinking edges or on the lengths of the unaffected edges

dependence of eigenvalues on the diffusion operators with random jumps from the boundary abstract this paper deals with a non self adjoint eigenvalue problem a x y x b x y x y x y 0 0 1 y x d 0 x y 1 0 1 y x d 1 x which is associated with the generator of one dimensional diffusions with random jumps from the boundary we focus on the dependence of spectral gap eigenvalues and eigenfunctions on the coefficients a b and the probability distributions 0 1 to prove this we show that all the eigenvalues are confined to a parabolic neighborhood of the real axis moreover we also prove that zero is an algebraically simple eigenvalue of the problem

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

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

quantum graphs ii some spectral properties of quantum and combinatorial graphs the paper deals with some spectral properties of mostly infinite quantum and combinatorial graphs quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics nanotechnology optics and other areas a schnol type theorem is proven that allows one to detect that a point belongs to the spectrum when a generalized eigenfunction with an sub exponential growth integral estimate is available a theorem on spectral gap opening for decorated quantum graphs is established its analogue is known for the combinatorial case it is also shown that if a periodic combinatorial or quantum graph has a point spectrum it is generated by compactly supported eigenfunctions scars

hermitian symplectic geometry and the factorization of the scattering matrix on graphs hermitian symplectic spaces provide a natural framework for the extension theory of symmetric operators here we show that hermitian symplectic spaces may also be used to describe the solution to the factorization problem for the scattering matrix on a graph i e we derive a formula for the scattering matrix of a graph in terms of the scattering matrices of its subgraphs the solution of this problem is shown to be given by the intersection of a lagrange plane and a coisotropic subspace which in an appropriate hermitian symplectic space forms a new lagrange plane the scattering matrix is given by a distinguished basis to the lagrange plane using our construction we are also able to give a simple proof of the unitarity of the scattering matrix as well as provide a characterization of the discrete eigenvalues embedded in the continuous spectrum

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

dependence of eigenvalues of sturm liouville problems with interface conditions this paper investigates sturm liouville problems with interface conditions we obtain that the eigen values depend not only continuously but also smoothly on the coefficient functions boundary conditions and interface conditions moreover the derivative of the eigenvalues with respect to each of these parameters is given

indefinite hardy spaces on finite bordered riemann surfaces abstract it is well known that one may study hardy spaces with the domain a finite bordered riemann surface rather than the unit disk however for domains with more than one boundary component it is natural to consider besides the usual positive definite inner product on h 2 indefinite inner products obtained by picking up different signs or in the vector valued case different signature matrices when integrating over different components in this paper we obtain a necessary and sufficient condition for such an indefinite inner product to be nondegenerate and show that when this condition is satisfied we actually get a krein space furthermore each holomorphic mapping of the finite bordered riemann surface onto the unit disk which maps boundary to boundary determines an explicit isometric isomorphism between this space and a usual vector valued hardy space on the unit disk with an indefinite inner product defined by an appropriate hermitian matrix as is usual when studying hardy spaces on a multiply connected domain the elements of the space are sections of a vector bundle rather than functions the main point is to construct an appropriate extension of this bundle to the double of the finite bordered riemann surface and to use cauchy kernels for certain vector bundles on a compact riemann surface

non self adjoint graphs on finite metric graphs we consider laplace operators subject to various classes of non self adjoint boundary conditions imposed at graph vertices we investigate spectral properties existence of a riesz basis of projectors and similarity transforms to self adjoint laplacians among other things we describe a simple way to relate the similarity transforms between laplacians on certain graphs with elementary similarity transforms between matrices defining the boundary conditions

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

characterization of jacobian varieties in terms of soliton equations pour dans le demi espace superieur de siegel h g soit x c g z g z g la variete abelienne principalement polarisee correspondante et soit z z m e z gexp 2 i t mz i t m m la fonction theta de riemann pour x les deux conditions a et b pour x sont equivalentes a 1 il existe des vecteurs a 1 a 2 a 3 c g a 1 0 et une forme quadratique q t i j 1 3 q ij t i t j q ij c tel que pour c g t t exp q t t 1 a 1 t 2 a 2 t 3 a 3 donne une solution a l equation de kadomtsev petviashvili sous la forme d hirota d 1 4 3d 2 2 4d 1 d 3 0 2 le diviseur de x est irreductible b x est isomorphe a la variete jacobienne d une courbe lisse complete de genre g sur c

the number of nodal domains on quantum graphs as a stability index of graph partitions the courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of laplacian type operators in particular it holds for generic eigenfunctions of a quantum graph the theorem stipulates that after ordering the eigenvalues as a non decreasing sequence the number of nodal domains n of the n th eigenfunction satisfies n n here we provide a new interpretation for the courant nodal deficiency d n n n in the case of quantum graphs it equals the morse index at a critical point of an energy functional on a suitably defined space of graph partitions thus the nodal deficiency assumes a previously unknown and profound meaning it is the number of unstable directions in the vicinity of the critical point corresponding to the n th eigenfunction to demonstrate this connection the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail

the study of jacobi and cyclic jacobi matrix eigenvalue problems using sturm liouville theory abstract we study the eigenvalues of matrix problems involving jacobi and cyclic jacobi matrices as functions of certain entries of particular interest are the limits of the eigenvalues as these entries approach infinity our approach is to use the recently discovered equivalence between these problems and a class of sturm liouville problems and then to apply the sturm liouville theory

the very unusual properties of the resolvent heat kernel and zeta function for the operator d2 dr2 1 4r2 in this paper we analyze the resolvent the heat kernel and the spectral zeta function of the operator d2 dr2 1 4r2 over the finite interval the structural properties of these spectral functions depend strongly on the chosen self adjoint realization of the operator a choice being made necessary because of the singular potential present only for the friedrichs realization standard properties are reproduced for all other realizations highly nonstandard properties are observed in particular for k n we find terms like logt k in the small t asymptotic expansion of the heat kernel furthermore the zeta function has s 0 as a logarithmic branch point

how to count zeroes of polynomials on quadrature domains using the bezout matrix abstract classically the bezout matrix or simply bezoutian of two polynomials is used to locate the roots of the polynomial and in particular test for stability in this paper we develop the theory of bezoutians on real riemann surfaces of dividing type the main result connects the signature of the bezoutian of two real meromorphic functions to the topological data of their quotient which can be seen as the generalization of the classical cauchy index as an application we propose a method to count the number of zeroes of a polynomial in a quadrature domain using the inertia of the bezoutian we provide examples of our method in the case of simply connected quadrature domains

representations of lie algebras by non skewselfadjoint operators in hilbert space we study non skewselfadjoint representations of a finite dimensional real lie algebra gg to this end we embed a non skewselfadjoint representation of gg into a more complicated structure that we call a gg operator vessel and that is associated to an overdetermined linear conservative input state output system on the corresponding simply connected lie group gg we develop the frequency domain theory of the system in terms of representations of gg and introduce the joint characteristic function of a gg operator vessel which is the analogue of the classical notion of the characteristic function of a single non selfadjoint operator as the first non commutative example we apply the theory to the lie algebra of the ax bax b group the group of affine transformations of the line

spectral functions for the schrodinger operator on r with a singular potential in this article we analyze the spectral zeta function the heat kernel and the resolvent of the operator d2 dr2 r2 r2 over the interval 0 for 1 4 depending on the self adjoint extension chosen nonstandard properties of the zeta function and of asymptotic properties of the heat kernel and resolvent are observed in particular for the zeta function nonstandard locations of poles as well as logarithmic branch cuts at s k k n0 do occur this implies that the small t asymptotic expansion of the heat kernel can have nonstandard powers as well as terms such as tk ln t l 1 for k l n0 the corresponding statements for the resolvent are also shown furthermore we evaluate the zeta determinant of the operator for all values of and any self adjoint extension

analyticity of the spectrum and dirichlet to neumann operator technique for quantum graphs in some previous works the analytic structure of the spectrum of a quantum graph operator as a function of the vertex conditions and other parameters of the graph were established however a specific local coordinate chart on the grassmannian of all possible vertex conditions was used thus creating an erroneous impression that something wrong can happen at the boundaries of the chart here we show that the analyticity of the corresponding dispersion relation holds over the whole grassmannian as well as over other parameter spaces we also address the dirichlet to neumann technique of relating quantum and discrete graph operators which allows one to transfer some results from the discrete to the quantum graph case but which has issues at the dirichlet spectrum we conclude that this difficulty as in the first part of the paper stems from the use of specific coordinates in a grassmannian and show how to avoid it to extend some of the consequent results to the general situation in some previous works the analytic structure of the spectrum of a quantum graph operator as a function of the vertex conditions and other parameters of the graph were established however a specific local coordinate chart on the grassmannian of all possible vertex conditions was used thus creating an erroneous impression that something wrong can happen at the boundaries of the chart here we show that the analyticity of the corresponding dispersion relation holds over the whole grassmannian as well as over other parameter spaces we also address the dirichlet to neumann technique of relating quantum and discrete graph operators which allows one to transfer some results from the discrete to the quantum graph case but which has issues at the dirichlet spectrum we conclude that this difficulty as in the first part of the paper stems from the use of specific coordinates in a grassmannian and show how to avoid it to extend some of the consequent results to the general situation

two interval sturm liouville operators in modified hilbert spaces abstract by modifying the inner product in the direct sum of the hilbert spaces associated with each of two underlying intervals on which the sturm liouville equation is defined we generate self adjoint realizations for boundary conditions with any real coupling matrix whose determinant is positive this contrasts with the usual theory which requires the coupling matrix to have determinant one

livsic type determinantal representations and hyperbolicity abstract hyperbolic homogeneous polynomials with real coefficients i e hyperbolic real projective hypersurfaces and their determinantal representations play a key role in the emerging field of convex algebraic geometry in this paper we consider a natural notion of hyperbolicity for a real subvariety x p d of an arbitrary codimension l with respect to a real l 1 dimensional linear subspace v p d and study its basic properties we also consider a class of determinantal representations that we call livsic type and a nice subclass of these that we call very reasonable much like in the case of hypersurfaces l 1 the existence of a definite hermitian very reasonable livsic type determinantal representation implies hyperbolicity we show that every curve admits a very reasonable livsic type determinantal representation our basic tools are cauchy kernels for line bundles and the notion of the bezoutian for two meromorphic functions on a compact riemann surface that we introduce we then proceed to show that every real curve in p d hyperbolic with respect to some real d 2 dimensional linear subspace admits a definite hermitian or even definite real symmetric very reasonable livsic type determinantal representation

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

dependence of the nth sturm liouville eigenvalue on the problem abstract we consider the nth eigenvalue as a function on the space of self adjoint regular sturm liouville problems with positive leading coefficient and weight functions the discontinuity of the nth eigenvalue is completely characterized

least squares spectral method for solving advective population balance problems the combined cfd pbe population balance models are computationally intensive requiring efficient numerical methods for solving practical problems in this paper a high order method is presented based on the least squares method lsm for the solution of a spatial dependent population balance equation which includes advective processes numerical experiments are performed in order to study the behavior of the proposed method for one dimensional cases using model problems with analytical solutions

on the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators the article is devoted to the following question consider a periodic self adjoint difference differential operator on a graph quantum graph g with a co compact free action of the integer lattice mathbb z n it is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum a feature uncommon for periodic elliptic operators of second order in all known constructions of such examples the corresponding eigenfunction is compactly supported one wonders whether this must always be the case the paper answers this question affirmatively what is more surprising one can estimate that the eigenmode must be localized not far away from the perturbation in a neighborhood of the perturbation s support the width of the neighborhood dependent upon the unperturbed operator only the validity of this result requires the condition of irreducibility of the fermi floquet surface of the periodic operator which is known in some cases and is expected to be satisfied for periodic schrodinger operators

a note on eigenvalues of a class of singular continuous and discrete linear hamiltonian systems abstract in this paper we show that the analytic and geometric multiplicities of an eigenvalue of a class of singular linear hamiltonian systems are equal where both endpoints are in the limit circle cases the proof is fundamental and is given for both continuous and discrete hamiltonian systems the method used in this paper also works for both endpoints are regular or one endpoint is regular and the other is in the limit circle case

almost periodic transformable difference systems abstract we analyse almost periodic homogeneous linear difference systems whose coefficient matrices belong to the so called transformable groups we introduce the notion of weakly transformable groups and we use this notion to obtain generalizations of known results about non almost periodic solutions of considered systems

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

noncommutative convexity arises from linear matrix inequalities this paper concerns polynomials in g noncommutative variables x x1 xg inverses of such polynomials and more generally noncommutative rational expressions with real coefficients which are formally symmetric and analytic near 0 the focus is on rational expressions r r x which are matrix convex near 0 i e those rational expressions r for which there is an 0 such that if x x1 xg is a g tuple of n n symmetric matrices satisfying in x12 xg2 is positive definite and y is also then the symmetric matrix tr x 1 t r y r tx 1 t y is positive semidefinite for all numbers t 0 t 1 this article gives a complete classification of matrix convex rational expressions see theorem 3 3 by representing such r in terms of a symmetric linear pencil l x id jajxj 0d 100 1 r 0 in the noncommuting variables xj where aj are symmetric d d matrices namely for a real number r is a schur complement of the linear pencil l moreover given a matrix convex r the set consisting of g tuples x of n n symmetric matrices 0 1 x r x iis negative definite has component containing 0 which is the same as the negativity set 0 2 x l x is negative definite for l conditions like l x is negative definite are known as linear matrix inequalities lmis in the engineering literature and arguably the main advance in linear systems theory in the 1990s was the introduction of lmi techniques in this language what we have shown in 0 1 vs 0 2 is that the set of solutions to a convex matrix inequality with noncommutative unknowns is the same as the set of solutions to some lmi in many engineering systems problems convexity would have all of the advantages of lmis indeed convexity guarantees that solutions are global and convexity bodes well for reliability of the numerics since lmis have a structure which is seemingly much more rigid than convexity there is the continual hope that a convexity based theory will be more far reaching than lmis but will it there are two natural situations one where the unknowns are scalars and one where the unknowns are matrices appearing in formulas which respect matrix multiplication these latter problems mathematically yield expressions with noncommutative unknowns and they arise in engineering systems problems which are dimensionless in the sense that they scale automatically with dimension as do most of the classics of control theory that is the case we study here and the result stated above suggests the surprising conclusion that for dimensionless systems problems convexity offers no greater generality than lmis indeed the result proves this for a class of model problems furthermore we show that existing algorithms together with algorithms described here construct the lmis above which are equivalent to the matrix inequalities based on the given matrix convex rational function r in a very different direction we prove that a symmetric polynomial p in g noncommutative symmetric variables has a symmetric determinantal representation namely there are symmetric matrices a0 ag in srd d with a0 invertible such that 0 3 detp x det a0 la x for each x a g tuple of symmetric n n matrices of course taking n 1 implies immediately that a commuting variables polynomial p on rg has a symmetric determinantal representation for g 2 much stronger commutative results can be obtained using tools of algebraic geometry but these do not seem to generalize to the higher dimensional case on the other hand a nonsymmetric commutative determinantal representation for any g is due to valiant universality of determinant in algebraic complexity theory our determinantal representation theorem is a bi product of the theory of systems realizations of noncommutative rational functions and can be read independently of much of the rest of the paper while the notion of noncommutative rational functions is standard the equivalence relation we use on rational expressions in our construction based on evaluating rational expressions on matrices is new and gives a new approach to noncommutative rational functions

eigenvalues of regular sturm liouville problems abstract the eigenvalues of sturm liouville sl problems depend not only continuously but smoothly on the problem an expression for the derivative of an eigenvalue with respect to a given parameter an endpoint a boundary condition a coefficient or the weight function is found

on the correlation functions of the vector bundle generalization of the bc system it is shown that the determinants of the correlation functions of the generalized bc system introduced recently are given as pullbacks of the non abelian theta divisor

floquet theory for left definite sturm liouville problems abstract we determine the essential spectrum of left definite sturm liouville problems with periodic coefficients and a weight function which changes sign

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

m matrix asymptotics for sturm liouville problems on graphs we consider a system formulation for sturm liouville operators with formally self adjoint boundary conditions on a graph an m matrix associated with the boundary value problem is defined and related to the matrix prufer angle associated with the system boundary value problem and consequently with the boundary value problem on the graph asymptotics for the m matrix are obtained as the eigenparameter tends to negative infinity we show that the boundary conditions may be recovered up to a unitary equivalence from the m matrix and that the m matrix is a herglotz function this is the first in a series of papers devoted to the reconstruction of the sturm liouville problem on a graph from its m matrix

on the multilevel internal structure of the asymptotic distribution of resonances abstract we prove that the set of resonances h has a multilevel asymptotic structure for the following classes of hamiltonians h schrodinger operators with point interactions y j r 3 quantum graphs and 1 d photonic crystals in the case of n 2 point interactions h consists of a finite number of sequences with logarithmic asymptotics leading parameters n of these sequences are connected with the geometry of the family y y j j 1 n the minimal parameter min corresponds to sequences with more narrow and so more observable resonances the asymptotic density of narrow resonances is expressed via the multiplicity of min and is connected with symmetries of y in the cases of quantum graphs and 1 d photonic crystals the decomposition of h into asymptotic sequences is proved under commensurability conditions to address the general quantum graph we introduce asymptotic density functions for two classes of complex strips the obtained results and effects are compared with obstacle scattering

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

an isoperimetric type inequality for the principal eigenvalue of schrodinger operators depending on the curvature of a loop abstract let be the curvature of a smooth closed curve of length 2 the 2 periodic eigenvalue problems u g 2 u u 0 where g is a real parameter are considered and upper and lower bounds for their principal eigenvalues are obtained the bounds are of isoperimetric type and characterize uniquely the curve in the case where equality occurs

on the absence of absolutely continuous spectra for schrodinger operators on radial tree graphs the subject of the paper is schrodinger operators on tree graphs which are radial having the branching number bn at all the vertices at the distance tn from the root we consider a family of coupling conditions at the vertices characterized by bn 1 2 4 real parameters we prove that if the graph is sparse so that there is a subsequence of tn 1 tn growing to infinity in the absence of the potential the absolutely continuous spectrum is empty for a large subset of these vertex couplings but on the the other hand there are cases when the spectrum of such a schrodinger operator can be purely absolutely continuous

space time coupled spectral hp least squares finite element formulation for the incompressible navier stokes equations we consider least squares finite element models for the numerical solution of the non stationary navier stokes equations governing viscous incompressible fluid flows the paper presents a formulation where the effects of space and time are coupled resulting in a true space time least squares minimization procedure as opposed to a space time decoupled formulation where a least squares minimization procedure is performed in space at each time step the formulation is first presented for the linear advection diffusion equation and then extended to the navier stokes equations the formulation has no time step stability restrictions and is spectrally accurate in both space and time to allow the use of practical c0 element expansions in the resulting finite element model the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity as an additional independent variable and the least squares method is used to develop the finite element model of the governing equations high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method in matrix free form spectral convergence of the l2 least squares functional and l2 error norms in space time is verified using a smooth solution to the two dimensional non stationary incompressible navier stokes equations numerical results are presented for impulsively started lid driven cavity flow oscillatory lid driven cavity flow transient flow over a backward facing step and flow around a circular cylinder the results demonstrate the predictive capability and robustness of the proposed formulation even though the space time coupled formulation is emphasized we also present the formulation and numerical results for least squares space time decoupled finite element models the numerical results show that the space time coupled formulation has superior predictive capabilities for flows demanding high space time resolution exemplified here by the transient flow over a backward facing step

an algebraic geometry study of the b c system with arbitrary twist fields and arbitrary statistics we present an analysis of the generalb c system including the system on a compact riemann surface of arbitrary genusg 0 by postulating that its correlation functions should only have the singularities imposed by the operator product expansion ope of the system studying a very in fact optimally general form of theb c system we prove rigorously that the standard practice of eliminating zero modes and even the standard lagrangian follow from the analyticity structure dictated by the ope alone we extend the analysis to consider the most general case of the presence of twist e g spin fields we then determine all the possible correlation functions of theb c system with statistics unspecified compatible with the ope on imposing fermi and bose statistics we obtain the correlation functions of the fermionicb c and systems respectively

elementary transformations of pfaffian representations of plane curves abstract let c be a smooth curve in p 2 given by an equation f 0 of degree d in this paper we consider elementary transformations of linear pfaffian representations of c elementary transformations can be interpreted as actions on a rank 2 vector bundle on c with canonical determinant and no sections which corresponds to the cokernel of a pfaffian representation every two pfaffian representations of c can be bridged by a finite sequence of elementary transformations pfaffian representations and elementary transformations are constructed explicitly for a smooth quartic applications to aronhold bundles and theta characteristics are given

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

the bc system of higher rank revisited the bc system of higher rank introduced recently is examined further it is shown that the correlation functions are connected with certain non abelian functions and it is discussed how quasi determinants arise

scattering on compact manifolds with infinitely thin horns the quantum mechanical scattering on a compact manifold with semi axes attached to the manifold hedgehog shaped manifold is considered the complete description of the spectral structure of schrodinger operators on such a manifold is done the proof of existence and uniqueness of scattering states is presented an explicit form for the scattering matrix is obtained and unitarity of this matrix is proven it is shown that the positive part of the spectrum of the schrodinger operator on the initial compact manifold as well as the spectrum of a point perturbation of such an operator may be recovered from the scattering amplitude for one attached half line moreover the positive part of the spectrum of the initial schrodinger operator is fully determined by the conductance properties of an electronic device consisting of the initial manifold and two wires attached to it

finite dimensional de branges spaces on riemann surfaces abstract we study certain finite dimensional reproducing kernel indefinite inner product spaces of multiplicative half order differentials on a compact real riemann surface these spaces are analogues of the spaces introduced by l de branges when the riemann sphere is replaced by a compact real riemann surface of a higher genus in de branges theory an important role is played by resolvent like difference quotient operators r here we introduce generalized difference quotient operators r y for any non constant meromorphic function y on the riemann surface the spaces we study are invariant under generalized difference quotient operators and can be characterized as finite dimensional indefinite inner product spaces invariant under two operators r y 1 i and r y 2 2 where y 1 and y 2 generate the field of meromorphic functions on the riemann surface which satisfy a supplementary identity analogous to the de branges identity for difference quotients just as the classical de branges spaces and difference quotient operators appear in the operator model theory for a single nonselfadjoint or nonunitary operator the spaces we consider and generalized difference quotient operators appear in the model theory for commuting nonselfadjoint operators with finite nonhermitian ranks

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

dependence of eigenvalues of a class of fourth order sturm liouville problems on the boundary in this paper we consider the dependence of eigenvalues of a class of fourth order sturm liouville problems on the boundary we show that the eigenvalues depend not only continuously but smoothly on boundary points and that the derivative of the nth eigenvalue as a function of an endpoint satisfies a first order differential equation in addition we prove that as the length of the interval shrinks to zero all higher fourth order dirichlet eigenvalues march off to plus infinity this is also true for the first i e lowest eigenvalue

regular third order boundary value problems abstract in this paper we consider some boundary value problems generated by third order formally symmetric self adjoint regular differential expression and separated real coupled and complex coupled boundary conditions it is shown that these problems generate self adjoint operators moreover the dependence of eigenvalues of these problems on the data are studied and some derivatives of the eigenvalues with respect to some elements of data are introduced

existence of multiple positive solutions for sturm liouville like four point boundary value problem with p laplacian abstract in this paper we consider the sturm liouville like four point boundary value problem with p laplacian p u t f t u t 0 t 0 1 u 0 u 0 u 1 u 0 where p s s p 2 s p 1 by means of a fixed point theorem for operators on a cone the existence of multiple at least three positive solutions to the above boundary value problem is obtained

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

zero pole interpolation for matrix meromorphic functions on a compact riemann surface and a matrix fay trisecant identity this paper presents a new approach to constructing a meromorphic bundle map between flat vector bundles over a compact riemann surface having a prescribed weil divisor i e having prescribed zeros and poles with directional as well as multiplicity information included in the vector case this new formalism unifies the earlier approach of ball clancey in the setting of trivial bundles over an abstract riemann surface with an earlier approach of the authors where the riemann surface was assumed to be the normalizing riemann surface for an algebraic curve embedded in c 2 with determinantal representation and the vector bundles were assumed to be presented as the kernels of linear matrix pencils the main tool is a version of the cauchy kernel appropriate for flat vector bundles over the riemann surface our formula for the interpolating bundle map in the special case of a single zero and a single pole can be viewed as a generalization of the fay trisecant identity from the usual line bundle case to the vector bundle case in terms of cauchy kernels in particular we obtain a new proof of the fay trisecant identity

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

schrodinger operators with nonlocal point interactions schrodinger operators with nonlocal point interactions are considered as new solvable models with point interactions examples in one and three dimensions are discussed corresponding direct and inverse scattering problems in one dimension are also discussed

geometric aspects of sturm liouville problems ii space of boundary conditions for left definiteness for a given regular sturm liouville equation with an indefinite weight function we explicitly describe the space of left definite selfadjoint boundary conditions the description only uses one value of a fundamental solution of the matrix form of the equation as a consequence we show that this space has the shape of a solid consisting of two cones sharing a common base

intermediate statistics for a system with symplectic symmetry the dirac rose graph we study the spectral statistics of the dirac operator on a rose shaped graph a graph with a single vertex and all bonds connected at both ends to the vertex we formulate a secular equation that generically determines the eigenvalues of the dirac rose graph which is seen to generalize the secular equation for a star graph with neumann boundary conditions we derive approximations to the spectral pair correlation function at large and small values of spectral spacings in the limit as the number of bonds approaches infinity and compare these predictions with results of numerical calculations our results represent the first example of intermediate statistics from the symplectic symmetry class

interpolation problems for holomorphic functions abstract in this note we discuss the nevanlinna pick problem corona problem and caratheodory fejer problem for bounded holomorphic functions

schrodinger operators on graphs and geometry iii general vertex conditions and counterexamples schrodinger operators on metric graphs with general vertex conditions are studied explicit spectral asymptotics is derived in terms of the spectrum of reference laplacians a geometric version of

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

matrix representations of fourth order boundary value problems with finite spectrum abstract we show that a class of regular self adjoint fourth order boundary value problems is equivalent to a certain class of matrix problems equivalent here means that they have exactly the same eigenvalues such an equivalence was previously known only in the second order case

de branges spaces on compact riemann surfaces and a beurling lax type theorem abstract using the notion of commutative operator vessels this work investigates de branges rovnyak spaces whose elements are sections of a line bundle of multiplicative half order differentials on a compact real riemann surface as a special case we obtain a beurling lax type theorem in the setting of the corresponding hardy space on a finite bordered riemann surface

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

error estimate of eigenvalues of perturbed second order discrete sturm liouville problems abstract this paper is concerned with eigenvalues of perturbed second order vector discrete sturm liouville problems by some variational properties of eigenvalues of discrete sturm liouville problems error estimates of eigenvalues of perturbed problems sufficiently close to a given sturm liouville problem are given under a certain non singularity condition perturbations of the coefficient functions of the difference equation the weight function and the coefficients of the boundary condition are all considered this together with higher dimension involved results in a certain complexity of the problem and difficulty of study as a direct consequence continuous dependence of eigenvalues on problems is obtained under the non singularity condition in addition an example is presented to illustrate the necessity of the non singularity condition

relationship between scattering matrix and spectrum of quantum graphs abstractwe investigate the equivalence between spectral characteristics of the laplace op erator on a metric graph and the associated unitary scattering operator we provethat the statistics of level spacings and moments of observations in the eigenbasescoincide in the limit that all bond lengths approach a positive constant value 1 introduction quantum graphs have attracted much attention in recent years due both to their ap plicability as physical models and their interesting mathematical properties we referthe reader to 1 and the forthcoming volume 2 for a pot pourri of new results recentreviews dedicated to quantum graphs include 3 4 in this article we focus on one feature of interest which is the use of graph models toprobe the the universality of quantum systems one of the unsolved paradoxes of quantummechanics is the observation that a great many quantum systems are remarkably similarwhen one makes statistical observations in the semi classical r egime this manifests itselfboth in the energy levels and associated energy eigenfunctions despite a great deal ofe ort this universality is poorly understood mathematically generic quantum graphsexhibit this universal behavior and represent the most likely system for which a fullmathematically rigorous proof of this universality will rst be found important steps inthis direction have been taken in 5 a quantum graph can be de ned in two di erent but related ways a complete de scription appears in the following section one may consider a self adjoint realisationof the laplace operator or a scattering matrix approach mathematically the scatteringapproach appears to be more tractable and has formed the basis of most investigations 6 7 8 9 10 moreover the spectra arising from these quantizations are subtly di erent this is not as confusing as it might seem since the statistical properties of both versionsof the spectrum are believed to coincide when averaged over a large interval it is thepurpose of this article to put a concrete mathematical foundation behind this belief the plan of the article is as follows in the next section we give precise de nitions ofthe two ways to describe quantum graphs and in section 3 describe our main results insection 4 we describe the tools used and then present in section 5 the proofs of our results

inverse scattering on matrices with boundary conditions we describe inverse scattering for the matrix schrodinger operator with general self adjoint boundary conditions at the origin using the marchenko equation our approach allows the recovery of the potential as well as the boundary conditions it is easily specialized to inverse scattering on star shaped graphs with boundary conditions at the node

the m matrix inverse problem for the sturm liouville equation on graphs abstractwe consider an inverse spectral problemfor sturm liouville boundaryvalue problemson a graph with formally self adjoint boundary conditions at the nodes where the giveninformation is the m matrix based on the results found in s currie b a watson m matrix asymptotics for sturm liouville problems on graphs j com appl math doi 10 1016 j cam 2007 11 019 using the green s function we prove that the poles ofthe m matrix are at the eigenvalues of the associated boundary value problem and aresimple located on the real axis and that the residue at a pole is a negative semi de nitematrix with rank equal to the multiplicity of the eigenvalue we de ne the so callednorming constants and relate them to the spectral measure and the m matrix thisenables us to recover from the m matrix the boundary conditions and the potential up to a unitary equivalence for co normal boundary conditions keywords inverse problem di erential operators on graphs m function sturm liouville 2000 msc 34a55 34b45 34b20 34l05 34b27

sturm liouville problems with finite spectrum abstract for every positive integer n we construct a class of regular self adjoint and non self adjoint sturm liouville problems with exactly n eigenvalues these n eigenvalues can be located anywhere in the complex plane in the non self adjoint case and anywhere along the real line in the self adjoint case the latter complements the well known general result for right definite sturm liouville problems with an infinite number of eigenvalues which must go to infinity asymptotically like n2 does with an appropriate and natural interpretation of a zero the eigenfunctions have the usual zero properties

dependence of eigenvalues of sturm liouville problems on the boundary abstract the eigenvalues of sturm liouville sl problems depend not only continuously but smoothly on boundary points the derivative of the n th eigenvalue as a function of an endpoint satisfies a first order differential equation this for arbitrary separated or coupled self adjoint regular boundary conditions in addition as the length of the interval shrinks to zero all higher eigenvalues march off to plus infinity this is also true for the first i e lowest dirichlet eigenvalue but not for the lowest neumann eigenvalue the latter has a finite limit

levinson s theorem and higher degree traces for aharonov bohm operators we study levinson type theorems for the family of aharonov bohm models from different perspectives the first one is purely analytical involving the explicit calculation of the wave operators and allowing to determine precisely the various contributions to the left hand side of levinson s theorem namely those due to the scattering operator the terms at 0 energy and at energy the second one is based on non commutative topology revealing the topological nature of levinson s theorem we then include the parameters of the family into the topological description obtaining a new type of levinson s theorem a higher degree levinson s theorem in this context the chern number of a bundle defined by a family of projections on bound states is explicitly computed and related to the result of a 3 trace applied on the scattering part of the model