dominating sets in chordal graphs a set of vertices d is a dominating set for a graph if every vertex is either in d or adjacent to a vertex which is in d we show that the problem of finding a minimum dominating set in a chordal graph is np complete even when restricted to undirected path graphs but exhibit a linear time greedy algorithm for the problem further restricted to directed path graphs streamlined to handle only trees our algorithm becomes the algorithm of cockayne goodman and hedetniemi an interesting parallel with graph isomorphism is pointed out

eigenvalue estimates for schrodinger operators on metric trees we consider schrodinger operators on radial metric trees and prove lieb thirring and cwikel lieb rozenblum inequalities for their negative eigenvalues the validity of these inequalities depends on the volume growth of the tree we show that the bounds are valid in the endpoint case and reflect the correct order in the weak or strong coupling limit

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

transport in simple networks described by an integrable discrete nonlinear schrodinger equation we elucidate the case in which the ablowitz ladik al type discrete nonlinear schrodinger equation nlse on simple networks e g star graphs and tree graphs becomes completely integrable just as in the case of a simple one dimensional 1d discrete chain the strength of cubic nonlinearity is different from bond to bond and networks are assumed to have at least two semi infinite bonds with one of them working as an incoming bond the present work is a nontrivial extension of our preceding one sobirov et al phys rev e 81 066602 2010 on the continuum nlse to the discrete case we find 1 the solution on each bond is a part of the universal bond independent al soliton solution on the 1d discrete chain but it is multiplied by the inverse of the square root of bond dependent nonlinearity 2 nonlinearities at individual bonds around each vertex must satisfy a sum rule and 3 under findings 1 and 2 there exist an infinite number of constants of motion as a practical issue with the use of an al soliton injected through the incoming bond we obtain transmission probabilities inversely proportional to the strength of nonlinearity on the outgoing bonds

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

the berry keating operator on l 2 mathbb r _ gt rm d x and on compact quantum graphs with general self adjoint realizations the berry keating operator berry and keating 1999 siam rev 41 236 governing the schrodinger dynamics is discussed in the hilbert space and on compact quantum graphs it has been proved that the spectrum of hbk defined on is purely continuous and thus this quantization of hbk cannot yield the hypothetical hilbert polya operator possessing as eigenvalues the nontrivial zeros of the riemann zeta function a complete classification of all self adjoint extensions of hbk acting on compact quantum graphs is given together with the corresponding secular equation in form of a determinant whose zeros determine the discrete spectrum of hbk in addition an exact trace formula and the weyl asymptotics of the eigenvalue counting function are derived furthermore we introduce the squared berry keating operator which is a special case of the black scholes operator used in financial theory of option pricing again all self adjoint extensions the corresponding secular equation the trace formula and the weyl asymptotics are derived for h2bk on compact quantum graphs while the spectra of both hbk and h2bk on any compact quantum graph are discrete their weyl asymptotics demonstrate that neither hbk nor h2bk can yield as eigenvalues the nontrivial riemann zeros some simple examples are worked out in detail

the generalized star product and the factorization of scattering matrices on graphs in this article we continue our analysis of schrodinger operators on arbitrary graphs given as certain laplace operators in the present article we give the proof of the composition rule for the scattering matrices this composition rule gives the scattering matrix of a graph as a generalized star product of the scattering matrices corresponding to its subgraphs we perform a detailed analysis of the generalized star product for arbitrary unitary matrices the relation to the theory of transfer matrices is also discussed

spectral asymptotics of the laplacian on platonic solids graphs we investigate the high energy eigenvalue asymptotics of quantum graphs consisting of the vertices and edges of the five platonic solids considering two different types of the vertex coupling one is the standard condition and the other is the preferred orientation one introduced in the work by exner and tater phys lett a 382 283 287 2018 the aim is to provide another illustration of the fact that the asymptotic properties of the latter coupling are determined by the vertex parity by showing that the octahedron graph differs in this respect from the other four for which the edges at high energies effectively disconnect and the spectrum approaches the one of the dirichlet laplacian on an interval

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

inverse spectral problem for quantum graphs the inverse spectral problem for the laplace operator on a finite metric graph is investigated it is shown that this problem has a unique solution for graphs with rationally independent edges and without vertices having valence 2 to prove the result a trace formula connecting the spectrum of the laplace operator with the set of periodic orbits for the metric graph is established

faster algorithms for vertex partitioning problems parameterized by clique width abstract many np hard problems such as dominating set are fpt parameterized by clique width for graphs of clique width k given with a k expression dominating set can be solved in 4 k n o 1 time however no fpt algorithm is known for computing an optimal k expression for a graph of clique width k if we rely on known algorithms to compute a 2 3 k 1 expression via rank width and then solving dominating set using the 2 3 k 1 expression the above algorithm will only give a runtime of 4 2 3 k n o 1 there have been results which overcome this exponential jump the best known algorithm can solve dominating set in time 2 o k 2 n o 1 by avoiding constructing a k expression bui xuan et al 2013 7 we improve this to 2 o k log k n o 1 indeed we show that for a graph of clique width k a large class of domination and partitioning problems lc vsp including dominating set can be solved in 2 o k log k n o 1 our main tool is a variant of rank width using the rank of a 0 1 matrix over the rational field instead of the binary field

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

characteristic functions of quantum graphs in this paper we give a reduction formula for the characteristic functions of the sturm liouville boundary value problems defined on a tree we also discuss the multiplicity of the eigenvalues and interlacing properties between two spectral sets associated with different boundary conditions

quantum motion on a half line connected to a plane in this paper the free motion of a particle on a manifold that consists of a one dimensional and a two dimensional part connected in one point is discussed the class of admissible hamiltonians is found using the theory of self adjoint extensions particular attention is paid to those hamiltonians that allow the particle to pass through the point singularity the reflection coefficient and other quantities characterizing scattering on the connection point are calculated a possible application is also discussed

imaging geometric graphs using internal measurements abstract this article presents two topologically motivated algorithms for reconstructing a metric graph from solutions to a wave equation on it our algorithms rely on narrowband and visibility measurements and are therefore of considerable value to urban sensing applications this is an advancement over the wideband methods traditionally used to study metric graphs our methodology exposes and separates the impact of the graph topology and geometry and is driven by a complete cohomological characterization of the space of wave equation solutions

picard vessiot theory and ziglin s theorem abstract given a hamiltonian system ziglin s theorem is one of the criteria used to study the integrability of the system for this we make use of the variational equations along a known solution in this paper we obtain some consequences of the application of picard vessiot differential galois theory to ziglin s theorem first we present the general results for the case of a finite arbitrary number of degrees of freedom then we pass to the case of two degrees of freedom where the results can be given with more detail under the hypothesis of ziglin s theorem and some additional technical assumptions the main results theorems 1 and 2 relate the integrability of the hamiltonian with the properties of the differential galois group of the picard vessiot extension associated to the normal reduced variational equations for two degrees of freedom it is possible to study also the resonant case theorems 4 and 5 give the full classification of the galois groups and the related extensions in the integrable case a couple of applications are made to recover ito s theorem and to study the lame equation

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

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

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

on agmon metrics and exponential localization for quantum graphs we investigate the rate of decrease at infinity of eigenfunctions of quantum graphs by using agmon s method to prove l2 and l infty bounds on the product of an eigenfunction with the exponential of a certain metric a generic result applicable to all graphs is that the exponential rate of decay is controlled by an adaptation of the standard estimates for a line which are of classical liouville green wkb form examples reveal that this estimate can be the best possible but that a more rapid rate of decay is typical when the graph has additional structure in order to understand this fact we present two alternative estimates under more restrictive assumptions on the graph structure that pertain to a more rapid decay one of these depends on how the eigenfunction is distributed along a particular chosen path while the other applies to an average of the eigenfunction over edges at a given distance from the root point

quantum graphs with singular two particle interactions we construct quantum models of two particles on a compact metric graph with singular two particle interactions the hamiltonians are self adjoint realizations of laplacians acting on functions defined on pairs of edges in such a way that the interaction is provided by boundary conditions in order to find such hamiltonians closed and semi bounded quadratic forms are constructed from which the associated self adjoint operators are extracted we provide a general characterization of such operators and furthermore produce certain classes of examples we then consider identical particles and project to the bosonic and fermionic subspaces finally we show that the operators possess purely discrete spectra and that the eigenvalues are distributed following an appropriate weyl asymptotic law

dispersion for the schrodinger equation on networks in this paper we consider the schrodinger equation on a network formed by a tree with the last generation of edges formed by infinite strips we give an explicit description of the solution of the linear schrodinger equation with constant coefficients this allows us to prove dispersive estimates which in turn are useful for solving the nonlinear schrodinger equation the proof extends also to the laminar case of positive step function coefficients having a finite number of discontinuities

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

can one hear the shape of a graph we show that the spectrum of the schroperator on a finite metric graph determines uniquely the connectivity matrix and the bond lengths provided that the lengths are non commensurate and the connectivity is simple no parallel bonds between vertices and no loops connecting a vertex to itself that is one can hear the shape of the graph we also consider a related inversion problem a compact graph can be converted into a scattering system by attaching to its vertices leads to infinity we show that the scattering phase determines uniquely the compact part of the graph under similar conditions as above

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

inverse eigenvalue problems on directed graphs the differential operators id and d2 p are constructed on certain finite directed weighted graphs two types of inverse spectral problems are considered first information about the graph weights and boundary conditions is extracted from the spectrum of d2 second the compactness of isospectral sets for d2 p is established by computation of the residues of the zeta function

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

spectral properties of matrices associated with some directed graphs we study the spectral properties of certain non self adjoint matrices associated with large directed graphs asymptotically the eigenvalues converge to certain curves apart from a finite number that have limits not on these curves

the isospectral fruits of representation theory quantum graphs and drums we present a method which enables one to construct isospectral objects such as quantum graphs and drums one aspect of the method is based on representation theory arguments which are shown and proved the complementary part concerns techniques of assembly which are both stated generally and demonstrated for this purpose quantum graphs are grist to the mill we develop the intuition that stands behind the construction as well as the practical skills of producing isospectral objects we discuss the theoretical implications which include sunada s theorem of isospectrality sunada 1985 ann math 121 169 arising as a particular case of this method a gallery of new isospectral examples is presented and some known examples are shown to result from our theory

on the algorithmic complexity of edge total domination let g v e be a graph with vertex set v and edge set e a subset f e is an edge total dominating set if every edge e e is adjacent to at least one edge in f the edge total domination problem is to find a minimum edge total dominating set of g in the present paper we prove that the edge total domination problem is np complete for planar graphs with maximum degree three and for undirected path graphs a subclass of chordal graphs and a superclass of trees we also design a linear time algorithm for solving this problem in a tree

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

regular and irregular geodesics on spherical harmonic surfaces abstract the behavior of geodesic curves on even seemingly simple surfaces can be surprisingly complex in this paper we use the hamiltonian formulation of the geodesic equations to analyze their integrability properties in particular we examine the behavior of geodesics on surfaces defined by the spherical harmonics using the morales ramis theorem and kovacic algorithm we are able to prove that the geodesic equations on all surfaces defined by the sectoral harmonics are not integrable and we use poincare sections to demonstrate the breakdown of regular motion

inverse spectral problems for the sturm liouville operator on a d star graph abstract in this work we consider inverse spectral problems for the sturm liouville differential operator on a d star type graph with standard matching conditions in the internal vertex where the integer d 2 by using the yurko s method yurko 2008 27 yurko 2009 28 we show that 1 if the potential function q j x on a fixed edge e j is prescribed on the interval 2 then the reciprocal of d of the spectrum suffices to determine q j x on the whole interval 0 2 the 2 over d of the spectrum suffices to determine q j x on a fixed edge e j

a new proof of the voronoi summation formula we present a short alternative proof of the voronoi summation formula which plays an important role in dirichlet s divisor problem and has recently found an application in physics as a trace formula for a schrodinger operator on a noncompact quantum graph

non integrability of the generalized spring pendulum problem we investigate a generalization of the three dimensional spring pendulum system the problem depends on two real parameters k a where k is the young modulus of the spring and a describes the nonlinearity of elastic forces we show that this system is not integrable when k a we carefully investigated the case k a when the necessary condition for integrability given by the morales ruiz ramis theory is satisfied we discuss an application of the higher order variational equations for proving the non integrability in this case

algorithm and hardness results on neighborhood total domination in graphs abstract a set d v of a graph g v e is called a neighborhood total dominating set of g if d is a dominating set and the subgraph of g induced by the open neighborhood of d has no isolated vertex given a graph g min ntds is the problem of finding a neighborhood total dominating set of g of minimum cardinality the decision version of min ntds is known to be np complete for bipartite graphs and chordal graphs in this paper we extend this np completeness result to undirected path graphs chordal bipartite graphs and planar graphs we also present a linear time algorithm for computing a minimum neighborhood total dominating set in proper interval graphs we show that for a given graph g v e min ntds cannot be approximated within a factor of 1 e log v unless np dtime v o log log v and can be approximated within a factor of o log where is the maximum degree of the graph g finally we show that min ntds is apx complete for graphs of degree at most 3

non integrability of a class of painleve iv equations as hamiltonian systems in this paper we will prove the rational non integrability of a class of hamiltonian systems associated with painleve iv equation by using morales ramis theory and kovacic s algorithm which to some extent also implies the non integrability of the fourth painleve equation itself

the number of eigenvalues of the matrix schrodinger operator on the half line with general boundary conditions we prove a bound of the bargmann birman schwinger type on the number of eigenvalues of the matrix schrodinger operator on the half line with the most general self adjoint boundary condition at the origin and with selfadjoint matrix potentials that are integrable and have a finite first moment

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

darboux points and integrability of hamiltonian systems with homogeneous polynomial potential in this paper we study the integrability of natural hamiltonian systems with a homogeneous polynomial potential the strongest necessary conditions for their integrability in the liouville sense have been obtained by a study of the differential galois group of variational equations along straight line solutions these particular solutions can be viewed as points of a projective space of dimension smaller by one than the number of degrees of freedom we call them darboux points we analyze in detail the case of two degrees of freedom we show that except for a radial potential the number of darboux points is finite and it is not greater than the degree of the potential moreover we analyze cases when the number of darboux points is smaller than maximal for two degrees of freedom the above mentioned necessary condition for integrability can be expressed in terms of one nontrivial eigenvalue of the hessian of potential calculated at a darboux point we prove that for a given potential these nontrivial eigenvalues calculated for all darboux points cannot be arbitrary because they satisfy a certain relation which we give in an explicit form we use this fact to strengthen maximally the necessary conditions for integrability and we show that in a generic case for a given degree of the potential there is only a finite number of potentials which satisfy these conditions we also describe the nongeneric cases as an example we give a full list of potentials of degree four satisfying these conditions then investigating the differential galois group of higher order variational equations we prove that except for one discrete family among these potentials only those which are already known to be integrable are integrable we check that a finite number of potentials from the exceptional discrete family are not integrable and we conjecture that all of them are not integrable

the algorithmic complexity of mixed domination in graphs let g v e be a simple graph with vertex set v and edge set e a subset email protected email protected e is a mixed dominating set if every element email protected email protected e w is either adjacent or incident to an element of w the mixed domination problem is to find a minimum mixed dominating set of g in this paper we first prove that a connected graph is a tree if and only if its total graph is strongly chordal and thus we obtain a polynomial time algorithm for this problem in trees further we design another linear time labeling algorithm for this problem in trees at the end of the paper we show that the mixed domination problem is np complete even when restricted to split graphs a subclass of chordal graphs

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

recovering a quantum graph spectrum from vertex data we study the question to what extent spectral information on a schrodinger operator on a finite compact metric graph subject to standard or type matching conditions can be recovered from a corresponding titchmarsh weyl function on the boundary of the graph in contrast to the case of ordinary or partial differential operators the knowledge of the titchmarsh weyl function is in general not sufficient for recovering the complete spectrum of the operator or the potentials on the edges however it is shown that those eigenvalues with sufficiently high depending on the cyclomatic number of the graph multiplicities can be recovered moreover we prove that under certain additional conditions the titchmarsh weyl function even contains the full spectral information

on the non integrability of the stark zeeman hamiltonian system in this paper we present a proof of the non integrability in the liouvillian sense of the stark zeeman hamiltonian in particular we generalize the result of kummer and saenz about the non integrability of the pure zeeman hamiltonian the proof we give is an application of the theorem of morales and ramis 1998 about non integrability based on differential galois theory

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

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

regular and pathological eigenvalue behavior for the equation λu vuon the semiaxis we study the behavior of the eigenvalue distribution functionn for the equation u vuon r u 0 0 typically n o 1 2 and moreover 1 2n 1 v dxas 0 we obtain criteria for each of these relations examples are presented where lim 0 1 2n does exist but is not equal to 1 v dx

an inverse spectral problem for a star graph of krein strings we solve an inverse spectral problem for a star graph of krein strings where the known spectral data comprises the spectrum associated with the whole graph the spectra associated with the individual edges as well as so called coupling matrices in particular we show that these spectral quantities uniquely determine the weight within the class of borel measures on the graph which give rise to trace class resolvents furthermore we obtain a concise characterization of all possible spectral data for this class of weights

on the mixed domination problem in graphs a mixed dominating set of a simple graph g v e is a subset email protected email protected e such that every vertex or edge not in d is adjacent or incident to at least one vertex or edge in d the mixed domination problem is to determine a minimum mixed dominating set of g this paper studies mixed domination in graphs from an algorithmic point of view in particular a linear time labeling algorithm for the mixed domination problem in cacti is presented in addition we fix an incomplete proof of the np completeness of the mixed domination problem in split graphs zhao et al 2011 23 finally we establish a primal dual algorithm for the mixed domination problem in trees

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

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

a topological state sum model for fermions on the circle a simple state sum model for fermions on a 1 manifold is constructed the model is independent of the triangulation and gives exactly the same partition function as the dirac functional integral with zeta function regularization some implications for more realistic physical models are discussed

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

nodal domains on isospectral quantum graphs the resolution of isospectrality we present and discuss isospectral quantum graphs which are not isometric these graphs are the analogues of the isospectral domains in which were introduced recently in gordon et al 1992 bull am math soc 27 134 8 chapman 1995 am math mon 102 124 buser et al 1994 int math res not 9 391 400 okada and shudo 2001 j phys a math gen 34 5911 22 jakobson et al 2006 j comput appl math 194 141 55 and levitin et al 2006 j phys a math gen 39 2073 82 all based on sunada s construction of isospectral domains sunada t 1985 ann math 121 196 86 after presenting some of the properties of these graphs we discuss a few examples which support the conjecture that by counting the nodal domains of the corresponding eigenfunctions one can resolve the isospectral ambiguity

application of the wittrick williams algorithm to the sturm liouville problem on homogeneous trees a structural mechanics analogy the wittrickwilliams ww algorithm was developed over 30 years ago and has been applied with increasing sophistication to problems in structural mechanics ever since much wider applications to a

some inverse scattering problems on star shaped graphs having in mind applications to the fault detection diagnosis of lossless electrical networks here we consider some inverse scattering problems for schrodinger operators over star shaped graphs we restrict ourselves to the case of minimal experimental setup consisting in measuring at most two reflection coefficients when an infinite homogeneous potential less branch is added to the central node first by studying the asymptotic behavior of only one reflection coefficient in the high frequency limit we prove the identiability of the geometry of this star shaped graph the number of edges and their lengths next we study the potential identification problem by inverse scattering noting that the potentials represent the inhomogeneities due to the soft faults in the network wirings potentials with bounded h1 norms the main result states that under some assumptions on the geometry of the graph the measurement of two reflection coefficients associated to two different sets of boundary conditions at the external vertices of the tree determines uniquely the potentials it can be seen as a generalization of the theorem of the two boundary spectra on an interval

schrodinger operators on graphs and geometry i essentially bounded potentials the inverse spectral problem for schrodinger operators on finite compact metric graphs is investigated the relations between the spectral asymptotics and geometric properties of the underlying graph are studied it is proven that the euler characteristic of the graph can be calculated from the spectrum of the schrodinger operator in the case of essentially bounded real potentials and standard boundary conditions at the vertices several generalizations of the presented results are discussed

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

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

classical nonintegrability of a quantum chaotic su 3 hamiltonian system abstract we prove the nonintegrability of a model hamiltonian system defined on the lie algebra s u 3 suitable for investigation of connections between classical and quantum characteristics of chaos

scattering on graphs and one dimensional approximations to n dimensional schrodinger operators in the present article we develop the spectral analysis of schrodinger operators on lattice type graphs for the basic example of a cubic periodic graph the problem is reduced to the spectral analysis of certain regular differential operators on a fundamental star like subgraph with a selfadjoint condition at the central node and quasiperiodic conditions at the boundary vertices using an explicit expression for the resolvent of lattice type operator we develop in the second section appropriate lippmann schwinger techniques for the perturbed periodic operator and construct the corresponding scattering matrix it serves as a base for the approximation of the multi dimensional schrodinger operator by a one dimensional operator on the graph in the third section of the paper for given n dimensional schrodinger operators with rapidly decreasing potential we construct a lattice type operator on a cubic graph embedded into rn and show that the original n dimensional scattering problem can be approximated in a p

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

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

an exact trace formula and zeta functions for an infinite quantum graph with a non standard weyl asymptotics we study a quantum hamiltonian that is given by the negative laplacian and an infinite chain of like potentials with strength 0 on the half line and which is equivalent to a one parameter family of laplacians on an infinite metric graph this graph consists of an infinite chain of edges with the metric structure defined by assigning an interval in 0 ln to each edge with length we show that the one parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each 0 and prove that the dirichlet laplacian is the limit of the one parameter family in the strong resolvent sense the spectrum of the resulting dirichlet quantum graph is also purely discrete the eigenvalues are given by n n2 with multiplicities d n where d n denotes the divisor function we can thus relate the spectral problem of this infinite quantum graph to dirichlet s famous divisor problem and infer the non standard weyl asymptotics for the eigenvalue counting function based on an exact trace formula the voronoi summation formula we derive explicit formulae for the trace of the wave group the heat kernel the resolvent and for various spectral zeta functions these results enable us to establish a well defined renormalized secular equation and a selberg like zeta function defined in terms of the classical periodic orbits of the graph for which we derive an exact functional equation and prove that the analogue of the riemann hypothesis is true

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

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

localization for anderson models on metric and discrete tree graphs we establish spectral and dynamical localization for several anderson models on metric and discrete radial trees the localization results are obtained on compact intervals contained in the complement of discrete sets of exceptional energies all results are proved under the minimal hypothesis on the type of disorder the random variables generating the trees assume at least two distinct values this level of generality in particular allows us to treat radial trees with disordered geometry as well as schrodinger operators with bernoulli type singular potentials our methods are based on an interplay between graph theoretical properties of radial trees and spectral analysis of the associated random differential and difference operators on the half line

schrodinger operators on regular metric trees with long range potentials weak coupling behavior consider a regular d dimensional metric tree with root o define the schrodinger operator v where v is a non negative symmetric potential on with neumann boundary conditions at o pro

concrete method for recovering the euler characteristic of quantum graphs trace formulas play a central role in the study of spectral geometry and in particular of quantum graphs the basis of our work is the result by kurasov which links the euler characteristic of metric graphs to the spectrum of their standard laplacian these ideas were shown to be applicable even in an experimental context where only a finite number of eigenvalues from a physical realization of quantum graph can be measured in the present work we analyse sufficient hypotheses which guarantee the successful recovery of we also study how to improve the efficiency of the method and in particular how to minimise the number of eigenvalues required finally we compare our findings with numerical examples surprisingly just a few dozens of eigenvalues can be enough

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

a width parameter useful for chordal and co comparability graphs abstract belmonte and vatshelle tcs 2013 used mim width a graph width parameter bounded on interval graphs and permutation graphs to explain existing algorithms for many domination type problems on those graph classes we investigate new graph classes of bounded mim width strictly extending interval graphs and permutation graphs the graphs k t k t and k t s t are graphs obtained from the disjoint union of two cliques of size t and one clique of size t and one independent set of size t respectively by adding a perfect matching we prove that interval graphs are k 3 s 3 free chordal graphs and k t s t free chordal graphs have mim width at most t 1 permutation graphs are k 3 k 3 free co comparability graphs and k t k t free co comparability graphs have mim width at most t 1 chordal graphs and co comparability graphs have unbounded mim width in general we obtain several algorithmic consequences for instance while minimum dominating set is np complete on chordal graphs it can be solved in time n o t on k t s t free chordal graphs the third statement strengthens a result of belmonte and vatshelle stating that either those classes do not have constant mim width or a decomposition with constant mim width cannot be computed in polynomial time unless p n p we generalize these ideas to bigger graph classes we introduce a new width parameter sim width of stronger modeling power than mim width by making a small change in the definition of mim width we prove that chordal graphs and co comparability graphs have sim width at most 1 we investigate a way to bound mim width for graphs of bounded sim width by excluding k t k t and k t s t as induced minors or induced subgraphs and give algorithmic consequences lastly we show that circle graphs have unbounded sim width and thus also unbounded mim width