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

localization on quantum graphs with random vertex couplings we consider schrodinger operators on a class of periodic quantum graphs with randomly distributed kirchhoff coupling constants at all vertices we obtain necessary conditions for localization on quantum graphs in terms of finite volume criteria for some energy dependent discrete hamiltonians these conditions hold in the strong disorder limit and at the spectral edges

on two open problems of 2 interval patterns the 2 interval pattern problem introduced in stephane vialette on the computational complexity of 2 interval pattern matching problems theoret comput sci 312 2 3 2004 223 249 models general problems with biological structures such as protein contact maps and macroscopic describers of secondary structures of ribonucleic acids given a set of 2 intervals d and a model r the problem is to find a maximum cardinality subset d of d such that any two 2 intervals in d satisfy r where r is a subset of relations on disjoint 2 intervals precedence nest and cross the problem left unanswered at present is that of whether there is a polynomial time solution for the 2 interval pattern problem when r and all the support intervals of d are disjoint in this paper we present a reduction from the clique problem to show that in this case the problem is np hard the disjoint 2 interval pattern matching problem is to decide whether a disjoint 2 interval pattern called the pattern is a substructure of another disjoint 2 interval pattern called the target in general the problem is np hard but when there are restrictions on the form of the pattern the problem can in some cases be solved in polynomial time in particular a polynomial time algorithm has been proposed gramm wabi 2004 and ieee acm tcbb 2004 for the case where the patterns are so called crossing contact maps in this paper we show that the problem is actually np hard and point out an error in the analysis of the above algorithm

nontrivial fucik spectrum of one non selfadjoint operator abstract we provide complete analytical description of the fucik spectrum of the non selfadjoint operator corresponding to four point boundary value problem the non selfadjointness results in a surprising structure of the fucik spectrum with new interesting patterns

hamiltonians on discrete structures jumps of the integrated density of states and uniform convergence we study equivariant families of discrete hamiltonians on amenable geometries and their integrated density of states ids we prove that the eigenspace of a fixed energy is spanned by eigenfunctions with compact support the size of a jump of the ids is consequently given by the equivariant dimension of the subspace spanned by such eigenfunctions from this we deduce uniform convergence w r t the spectral parameter of the finite volume approximants of the ids our framework includes quasiperiodic operators on delone sets periodic and random operators on quasi transitive graphs and operators on percolation graphs

scale free unique continuation estimates and applications to random schrodinger operators we prove a unique continuation principle or uncertainty relation valid for schrodinger operator eigenfunctions or more generally solutions of a schrodinger inequality on cubes of side l in 2 mathbb n 1 it establishes an equi distribution property of the eigenfunction over the box the total l2 mass in the box of side l is estimated from above by a constant times the sum of the l2 masses on small balls of a fixed radius 0 evenly distributed throughout the box the dependence of the constant on the various parameters entering the problem is given explicitly most importantly there is no l dependence

ordered and unordered tree inclusion the following tree matching problem is considered given labeled trees p and t can p be obtained from t by deleting nodes deleting a node u entails removing all edges incident to u and if u has a parent v replacing the edge from v to u by edges from v to the children of u the problem is motivated by the study of query languages for structured text databases simple solutions to this problem require exponential time for ordered trees an algorithm is presented that requires o p t time and space the corresponding problem for unordered trees is also considered and a proof of its np completeness is given an algorithm is presented for the unordered problem this algorithm works in o p t time if the out degrees of the nodes in p are bounded by a constant and in polynomial time if they are o log t

surface states and spectra let d 1 let h 0 be the discrete laplacian on the hilbert space l 2 d 1 with a dirichlet boundary condition and let v be a potential supported on the boundary d 1 we introduce the notions of surface states and surface spectrum of the operator h h 0 v and explore their properties our main result is that if the potential v is random and if the disorder is either large or small enough then in dimension two h has no surface spectrum on h 0 with probability one to prove this result we combine aizenman molchanov theory with techniques of scattering theory

an optimal wegner estimate and its application to the global continuity of the integrated density of states for random schrodinger operators we prove that the integrated density of states ids of random schrodinger operators with anderson type potentials on l 2 r d for d geq1 is locally holder continuous at all energies with the same holder exponent 0 alpha leq1 as the conditional probability measure for the single site random variable as a special case we prove that if the probability distribution is absolutely continuous with respect to lebesgue measure with a bounded density then the ids is lipschitz continuous at all energies the single site potential u in l 0 infty r d must be nonnegative and compactly supported the unperturbed hamiltonian must be periodic and satisfy a unique continuation principle we also prove analogous continuity results for the ids of random anderson type perturbations of the landau hamiltonian in two dimensions all of these results follow from a new wegner estimate for local random hamiltonians with rather general probability measures

on the lipschitz continuity of the integrated density of states for sign indefinite potentials the present paper is devoted to the study of spectral properties of random schrodinger operators using a finite section method for toeplitz matrices we prove a wegner estimate for some alloy type models where the single site potential is allowed to change sign the results apply to the corresponding discrete model too in certain disorder regimes we are able to prove the lipschitz continuity of the integrated density of states and or localization near spectral edges

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

a remark on the dirichlet neumann decoupling and the integrated density of states abstract we prove an estimate on the difference of the number of eigenvalues for schrodinger operators with dirichlet and neumann boundary conditions in large boxes in the proof we use krein s theory of the spectral shift function as an application we show that the integrated density of states is independent of the choice of boundary conditions

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

anderson localization and lifshits tails for random surface potentials we consider schrodinger operators on l2 rd with a random potential concentrated near the surface rd1 0 rd we prove that the integrated density of states of such operators exhibits lifshits tails near the bottom of the spectrum from this and the multiscale analysis by boutet de monvel and stollmann arch math 80 2003 87 97 we infer anderson localization pure point spectrum and dynamical localization for low energies our proof of lifshits tails relies on spectral properties of schrodinger operators with partially periodic potentials in particular we show that the lowest energy band of such operators is parabolic

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

harmonic analysis on sl 2 r and smoothness of the density of states in the one dimensional anderson model we consider infinite jacobi matrices with ones off diagonal and independent identically distributed random variables with distributionf v dv on diagonal iff has compact support and lies in some sobolev spacel 1 then we prove that the integrated density of states k e isc ine

bound states for nano tubes with a dislocation abstract as a model for an interface in solid state physics we consider two real valued potentials v 1 and v 2 on the cylinder or tube s r r z where we assume that there exists an interval a 0 b 0 which is free of spectrum of v k for k 1 2 we are then interested in the spectrum of h t v t for t r where v t x y v 1 x y for x 0 and v t x y v 2 x t y for x 0 while the essential spectrum of h t is independent of t we show that discrete spectrum related to the interface at x 0 is created in the interval a 0 b 0 at suitable values of the parameter t provided v 2 has some essential spectrum in a 0 we do not require v 1 or v 2 to be periodic we furthermore show that the discrete eigenvalues of h t are lipschitz continuous functions of t if the potential v 2 is locally of bounded variation

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

a computational model for rna multiple structural alignment this paper addresses the problem of aligning multiple sequences of noncoding rna ncrna genes we approach this problem with the biologically motivated paradigm that scoring of ncrna alignments should be based primarily on secondary structure rather than nucleotide conservation we introduce a novel graph theoretic model nlg for analyzing algorithms based on this approach prove that the rna multiple alignment problem is np complete in this model and present a polynomial time algorithm that approximates the optimal structure of size s within a factor of o log 2s

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

structure of the fucik spectrum and existence of solutions for equations with asymmetric nonlinearities let l dom l l2 l2 be a self adjoint operator being open and bounded in rn we give a description of the fucik spectrum of l away from the essential spectrum let be a point in the discrete spectrum of l provided that some non degeneracy conditions are satisfied we prove that the fucik spectrum consists locally of a finite number of curves crossing at each of these curves can be associated to a critical point of the function h x xs21a6 x x l2 restricted to the unit sphere in ker l i the corresponding critical values determine the slopes of these curves we also give global results describing the fucik spectrum and existence results for semilinear equations by performing degree computations between the fucik curves

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

spectral analysis of percolation hamiltonians we study the family of hamiltonians which corresponds to the adjacency operators on a percolation graph we characterise the set of energies which are almost surely eigenvalues with finitely supported eigenfunctions this set of energies is a dense subset of the algebraic integers the integrated density of states has discontinuities precisely at this set of energies we show that the convergence of the integrated densities of states of finite box hamiltonians to the one on the whole space holds even at the points of discontinuity for this we use an equicontinuity from the right argument the same statements hold for the restriction of the hamiltonian to the infinite cluster in this case we prove that the integrated density of states can be constructed using local data only finally we study some mixed anderson quantum percolation models and establish results in the spirit of wegner and delyon and souillard

extracting constrained 2 interval subsets in 2 interval sets 2 interval sets were used in s vialette pattern matching over 2 intervals sets in proc 13th annual symposium combinatorial pattern matching cpm 2002 in lecture notes in computer science vol 2373 springer verlag 2002 pp 53 63 s vialette on the computational complexity of 2 interval pattern matching theoret comput sci 312 2 3 2004 223 249 for establishing a general representation for macroscopic describers of rna secondary structures in this context we have a 2 interval for each legal local fold in a given rna sequence and a constrained pattern made of disjoint 2 intervals represents a putative rna secondary structure we focus here on the problem of extracting a constrained pattern in a set of 2 intervals more precisely given a set of 2 intervals and a model r describing if two disjoint 2 intervals in a solution can be in precedence order be allowed to nest and or be allowed to cross we consider the problem of finding a maximum cardinality subset of disjoint 2 intervals such that any two 2 intervals in agree with r the different combinations of restrictions on model r alter the computational complexity of the problem and need to be examined separately in this paper we improve the time complexity of s vialette on the computational complexity of 2 interval pattern matching theoret comput sci 312 2 3 2004 223 249 for model r by giving an optimal o nlogn time algorithm where n is the cardinality of the 2 interval set we also give a graph like relaxation for model r that is solvable in time finally we prove that the considered problem is np complete for model r even for same length intervals and give a fixed parameter tractability result based on the crossing structure of

spectral properties of the laplacian on bond percolation graphs bond percolation graphs are random subgraphs of the d dimensional integer lattice generated by a standard bond percolation process the associated graph laplacians subject to dirichlet or neumann conditions at cluster boundaries represent bounded self adjoint ergodic random operators with off diagonal disorder they possess almost surely the non random spectrum 0 4d and a self averaging integrated density of states the integrated density of states is shown to exhibit lifshits tails at both spectral edges in the non percolating phase while the characteristic exponent of the lifshits tail for the dirichlet neumann laplacian at the lower upper spectral edge equals d 2 and thus depends on the spatial dimension this is not the case at the upper lower spectral edge where the exponent equals 1 2

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

a note on the relation between the fucik spectrum and pareto eigenvalues abstract in this paper we summarize some known results concerning the so called fucik spectrum arising in problems with jumping nonlinearities and the pareto spectrum or the k spectrum connected with linear complementarity problems we show a surprising relation between these two notions and illustrate it on several examples

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

approximating l2 invariants and the atiyah conjecture let g be a torsion free discrete group and let denote the field of algebraic numbers in we prove that g fulfills the atiyah conjecture if g lies in a certain class of groups d which contains in particular all groups that are residually torsion free elementary amenable or are residually free this result implies that there are no nontrivial zero divisors in g the statement relies on new approximation results for l2 betti numbers over g which are the core of the work done in this paper another set of results in the paper is concerned with certain number theoretic properties of eigenvalues for the combinatorial laplacian on l2 cochains on any normal covering space of a finite cw complex we establish the absence of eigenvalues that are transcendental numbers whenever the covering transformation group is either amenable or in the linnell class we also establish the absence of eigenvalues that are liouville transcendental numbers whenever the covering transformation group is either residually finite or more generally in a certain large bootstrap class 2003 wiley periodicals inc

finding common structured patterns in linear graphs a linear graph is a graph whose vertices are linearly ordered this linear ordering allows pairs of disjoint edges to be either preceding nesting or crossing given a family of linear graphs and a non empty subset email protected we are interested in the maximum common structured pattern mcsp problem find a maximum size edge disjoint graph with edge pairs all comparable by one of the relations in r that occurs as a subgraph in each of the linear graphs of the family the mcsp problem generalizes many structure comparison and structure prediction problems that arise in computational molecular biology we give tight hardness results for the mcsp problem for structured patterns and structured patterns furthermore we prove that the problem is approximable within ratios i email protected k for structured patterns ii k 1 2 for structured patterns and iii o klogk for structured patterns where k is the size of the optimal solution and k email protected i 1 k1 i is the kth harmonic number also we provide combinatorial results concerning different types of structured patterns that are of independent interest in their own right

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

constrained open mapping theorem with applications abstract we prove a generalization of graves open mapping theorem for a class of mappings which can be approximated at a reference point by a set valued one having particular properties the nonlinear mapping is restricted to a closed convex subset of a banach space we apply the results to derive necessary and sufficient conditions ensuring the existence of a differentiable selection for the inverse mapping a slight generalization of a sufficient condition by j klamka on so called constrained exact local controllability of nonlinear and semi linear dynamic systems is also proved

linear wegner estimate for alloy type schrodinger operators on metric graphs we study spectra of alloy type random schrodinger operators on metric graphs for finite edge subsets we prove a wegner estimate which is linear in the volume i e the total length of the edges and the length of the energy interval the single site potential needs to have fixed sign the metric graph does not need to have a periodic structure a further result is the existence of the integrated density of states for ergodic random hamiltonians on metric graphs with a z structure for certain models the two above results together imply the lipschitz continuity of the integrated density of states

half eigenvalues of periodic sturm liouville problems we consider the nonlinear sturm liouville problem 1 pu qu au bu u in 0 2 2 u 0 u 2 pu 0 pu 2 where 1 p q l1 0 2 with p 0 a e on 0 2 a b l1 0 2 is a real parameter and u t max u t 0 for t 0 2 values of for which 1 2 has a non trivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions the set of half eigenvalues will be denoted by h we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having certain nodal properties and we obtain certain spectral and degree theoretic properties associated with h these properties yield results on the existence and non existence of solutions of the problem 3 pu qu f t u h in 0 2 together with 2 where h l1 0 2 f 0 2 r r is a caratheodory function and the limits a t lim f t b t lim f t exist for a e t 0 2 when the functions a and b are constant the set h is closely related to the fucik spectrum of the problem and equivalent solvability results are obtained using the two approaches however when a and b are not constant the half eigenvalue approach yields stronger results

on the fucik spectrum of the wave operator and an asymptotically linear problem abstract we study generalized solutions of the nonlinear wave equation u t t u s s a u b u p s t u with periodic conditions in t and homogeneous dirichlet conditions in s under the assumption that the ratio of the period to the length of the interval is two when p 0 and is a nonzero eigenvalue of the wave operator we give a proof of the existence of two families of curves which may coincide in the fucik spectrum intersecting at this result is known for some classes of self adjoint operators which does not cover the situation we consider here but in a smaller region than ours our approach is based on a dual variational formulation and is also applicable to other operators such as the laplacian in addition we prove an existence result for the non homogeneous situation when the pair a b is not between the fucik curves passing through 0 0 and p is a continuous function sublinear at infinity

log holder continuity of the integrated density of states for stochastic jacobi matrices we consider the integrated density of states k e of a general operator on l2 v of the formh h0 v where h 0 u n sum limits left i right 1 u n i and vu n v n u n wherev is a general bounded ergodic stationary process on v we show that k e k e c log e e 1 when e e 1 2 the key is a thouless formula for the strip

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

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

the integrated density of states of the random graph laplacian we analyse the density of states of the random graph laplacian in the percolating regime a symmetry argument and knowledge of the density of states in the nonpercolating regime allows us to isolate the density of states of the percolating cluster dspc alone thereby eliminating trivially localised states due to finite subgraphs we derive a nonlinear integral equation for the integrated dspc and solve it with a population dynamics algorithm we discuss the possible existence of a mobility edge and give strong evidence for the existence of discrete eigenvalues in the whole range of the spectrum

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

on the computational complexity of 2 interval pattern matching problems the focus of this paper is on the computational complexity of pattern matching problems over set of 2 intervals these problems occur in the context of molecular biology when a structured pattern i e a rna secondary structure given in the form of a 2 interval pattern has to be found in a sequence database we show that finding a 2 interval pattern in a set of 2 intervals is a np complete problem even if no 2 interval of the pattern precedes the other but can be solved in polynomial time for several interesting special cases in particular it is shown that the pseudo knot free rna secondary structure case is polynomial time solvable in our 2 interval formalism also we investigate the computational complexity of finding the longest 2 interval pattern in a set of 2 intervals and prove several np completeness results as well as polynomial time solvable special cases

the lp theory of the spectral shift function the wegner estimate and the integrated density of states for some random operators we develop the l p theory of the spectral shift function for p 1 applicable to pairs of self adjoint operators whose difference is in the trace ideal p for 0 p 1 this result is a key ingredient of a new short proof of the wegner estimate applicable to a wide variety of additive and multiplicative random perturbations of deterministic background operators the proof yields the correct volume dependence of the upper bound this implies the local holder continuity of the integrated density of states at energies in the unperturbed spectral gap under an additional condition of the single site potential local holder continuity is proved at all energies this new wegner estimate together with other standard results establishes exponential localization for a new family of models for additive and multiplicative perturbations

sparse dynamic programming i linear cost functions dynamic programming solutions to a number of different recurrence equations for sequence comparison and for rna secondary structure prediction are considered these recurrences are defined over a number of points that is quadratic in the input size however only a sparse set matters for the result efficient algorithms for these problems are given when the weight functions used in the recurrences are taken to be linear the time complexity of the algorithms depends almost linearly on the number of points that need to be considered when the problems are sparse this results in a substantial speed up over known algorithms

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

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

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

a supersymmetric transfer matrix and differentiability of the density of states in the one dimensional anderson model let h v on 2 z where v x 9 xez are d r v s with common probability distribution v let h t e itvdv v and let k e be the integrated density of states it is proven i if h is n times differentiable with h j t 0 l t for some 0 7 0 l n then k e is a cn function in particular if v has compact support and h t 0 l h 1 with 0 then k e is c this allows v to be singular continuous ii lfh t 0 e a tl for some 0 then k e is analytic in a strip about the real axis the proof uses the supersymmetric replica trick to rewrite the averaged green s function as a two point function of a one dimensional supersymmetric field theory which is studied by the transfer matrix method

random hamiltonians ergodic in all but one direction letv 1 andv 2 be two ergodic random potentials on d we consider the schrodinger operatorh h0 v withh0 and forx x1 x2 v omega x left begin array 20 c v omega 1 x if x 1 over smash scriptstyle vphantom x 0 end array right we prove certain ergodic properties of the spectrum for this model and express the integrated density of states in terms of the density of states of the stationary potentialsv 1 andv 2 finally we prove the existence of the density of surface states forh

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

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

regularity of the surface density of states abstract we prove that the integrated surface density of states of continuous or discrete anderson type random schrodinger operators is a measurable locally integrable function rather than a signed measure or a distribution this generalizes our recent results on the existence of the integrated surface density of states in the continuous case and those of a chahrour in the discrete case the proof uses the new l p bound on the spectral shift function recently obtained by combes hislop and nakamura also we provide a simple proof of their result on the holder continuity of the integrated density of bulk states

nonlinear four point problem non resonance with respect to the fucik spectrum abstract we study the solvability of a nonlinear four point boundary value problem we focus on the non resonance case with respect to the fucik spectrum and we show the difference between regions of type i and regions of type ii qualitatively different patterns of regions of both types can be obtained using various settings of problem parameters

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

discontinuities of the integrated density of states for random operators on delone sets despite all the analogies with usual random models tight binding operators for quasicrystals exhibit a feature that clearly distinguishes them from the former the integrated density of states may be discontinuous this phenomenon is identified as a local effect due to the occurrence of eigenfunctions with bounded support

localization in general one dimensional random systems we discuss two ways of extending the recent ideas of localization from discrete schrodinger operators jacobi matrices to the continuum case one case allows us to prove localization in the goldshade molchanov pastur model for a larger class of functions than previously the other method studies the model v wherev is a random constant in each hyper cube we extend wegner s result on the lipschitz nature of the ids to this model

localization for some continuous random hamiltonians in d dimensions abstract we prove the existence with probability one of an interval of pure point spectrum for some families of continuous random schrodinger operators in d dimensions for anderson like models with positive short range single site potentials we also prove that the corresponding eigenfunctions decay exponentially and that the integrated density of states is lipschitz continuous for the other families of random potentials that we study we show that the corresponding eigenfunctions decay faster than an inverse power of x which depends upon the decay rate of the single site potential to obtain these results we develop an extension of the classical aronszajn donoghue theory for a class of relatively compact perturbations and a spectral averaging method which extends kotani s trick to these more general families of operators

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

on the density of states of schrodinger operators with a random potential using very recent results on ergodic theorems for superadditive processes on rd the authors prove the existence of the density of states for a wide class of random schrodinger operators in particular new non asymptotic estimates on the density of states are obtained and examples are discussed

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

simple fast algorithms for the editing distance between trees and related problems ordered labeled trees are trees in which the left to right order among siblings is significant the distance between two ordered trees is considered to be the weighted number of edit operations insert delete and modify to transform one tree to another the problem of approximate tree matching is also considered specifically algorithms are designed to answer the following kinds of questions 1 what is the distance between two trees 2 what is the minimum distance between t 1 and t 2 when zero or more subtrees can be removed from t 2 3 let the pruning of a tree at node n mean removing all the descendants of node n the analogous question for prunings as for subtrees is answered a dynamic programming algorithm is presented to solve the three questions in sequential time o t 1 times t 2 times min textit depth t 1 textit leaves t 1 times min textit depth t 2 textit leaves t 2 and space o t 1 times t 2 compared with o t 1 times t 2 t

a variational approach to dislocation problems for periodic schrodinger operators abstract as a simple model for lattice defects like grain boundaries in solid state physics we consider potentials which are obtained from a periodic potential v v x y on r 2 with period lattice z 2 by setting w t x y v x t y for x 0 and w t x y v x y for x 0 for t 0 1 for lipschitz continuous v it is shown that the schrodinger operators h t w t have spectrum surface states in the spectral gaps of h 0 for suitable t 0 1 we also discuss the density of these surface states as compared to the density of the bulk our approach is variational and it is first applied to the well known dislocation problem korotyaev 2000 2005 15 16 on the real line we then proceed to the dislocation problem for an infinite strip and for the plane in appendix a we discuss regularity properties of the eigenvalue branches in the one dimensional dislocation problem for suitable classes of potentials

on the fucik spectrum in this paper we study the structure of the fucik spectrum of the set of points b a in r 2 for which the equation

bifurcation surfaces stemming from the fucik spectrum abstract let r n be a bounded domain with smooth boundary and g r r be a nonlinear function we prove existence of two dimensional bifurcation surfaces for the elliptic boundary value problem u a u b u g x u in u 0 where u min 0 u u max 0 u and a b r 2 is a pair of parameters we show that these two dimensional bifurcation surfaces stem from the fucik spectrum of the main difficulty in doing that comes from non smoothness of the operators u u in order to overcome this difficulty a variant implicit function theorem and an abstract two dimensional bifurcation theorem are proved these two theorems do not require smoothness of operators and the abstract two dimensional bifurcation theorem can be regarded as an extension of the well known crandall rabinowitz bifurcation theorem and therefore are of interest for their own sake

remark on the continuity of the density of states of ergodic finite difference operators we give an elementary proof that for a large class ofd dimensional finite difference operators including tight binding models for electron propagation and models for harmonic phonons with random masses or couplings the integrated density of states is a continuous function of the energy

integrated density of states for random metrics on manifolds we study ergodic random schrodinger operators on a covering manifold where the randomness enters both via the potential and the metric we prove measurability of the random operators almost sure constancy of their spectral properties the existence of a self averaging integrated density of states and a pastur ubin type trace formula

localization for some continuous random schrodinger operators we study the spectrum of random schrodinger operators acting onl2 r d of the following type h delta w sum x in mathbb z d t x v x the t x x in mathbb z d are i i d random variables under weak assumptions onv we prove exponential localization forh at the lower edge of its spectrum in order to do this we give a new proof of the wegner estimate that works without sign assumptions onv

crossings and nestings of matchings and partitions we present results on the enumeration of crossings and nestings for matchings and set partitions using a bijection between partitions and vacillating tableaux we show that if we fix the sets of minimal block elements and maximal block elements the crossing number and the nesting number of partitions have a symmetric joint distribution it follows that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of n as well as over all matchings on 2n as a corollary the number of k noncrossing partitions is equal to the number of k nonnesting partitions the same is also true for matchings an application is given to the enumeration of matchings with no k crossing or with no k nesting mathematics subject classification primary 05a18 secondary 05e10 05a15

analytical and numerical results for the fucik spectrum of the laplacian nontrivial solutions of u u vu 0 in rn with zero dirichlet condition on are studied the collection of the pairs v is the so called fucik spectrum f a new variational formulation for parts of f is presented and analyzed based on this formulation a minimization algorithm for the computation of a part of f is developed alternatively an approach using an implicit function argument is discussed analytically and via newton s method also numerically by combining the variational minimization method with newton s method a new bifurcation result is first observed numerically and then proved rigorously by replacing the variational minimization method by the mountain pass algorithm higher curves in f are found numerically several numerical results are discussed including a further example of the previously recorded phenomenon of crossing of fucik curves originating from different eigenvalues

concavity of eigenvalue sums and the spectral shift function abstract it is well known that the sum of negative positive eigenvalues of some finite hermitian matrix v is concave convex with respect to v using the theory of the spectral shift function we generalize this property to self adjoint operators on a separable hilbert space with an arbitrary spectrum more precisely we prove that the spectral shift function integrated with respect to the spectral parameter from to from to is concave convex with respect to trace class perturbations the case of relative trace class perturbations is also considered

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

analyticity of the density of states and replica method for random schrodinger operators on a lattice we analyze the density of states and some aspects of the replica method for anderson s tight binding model on a lattice of arbitrary dimension with diagonal disorder we give heuristic arguments for the conjectures that the classical value of the exponent of the localization length is 1 2 and that the upper critical dimension d c loc is bounded by 4 d c loc 6

global bounds for the lyapunov exponent and the integrated density of states of random schrodinger operators in one dimension in this paper we prove an upper bound for the lyapunov exponent e and a two sided bound for the integrated density of states n e at an arbitrary energy e 0 of random schrodinger operators in one dimension these schrodinger operators are given by potentials of identical shape centred at every lattice site but with non overlapping supports and with randomly varying coupling constants both types of bounds only involve scattering data for the single site potential they show in particular that both e and n e e 1 2 decay at infinity at least like 1 e 1 2 as an example we consider the random kronig penney model

bounds on the spectral shift function and the density of states we study spectra of schrodinger operators on d first we consider a pair of operators which differ by a compactly supported potential as well as the corresponding semigroups we prove almost exponential decay of the singular values n of the difference of the semigroups as n and deduce bounds on the spectral shift function of the pair of operators

on the density of state for the quantum percolation problem for the site dilution model on the hypercubic lattice zd d or 2 the authors examine the density of states for the tight binding hamiltonian projected onto the infinite cluster it is shown that with probability one the corresponding integrated density of states is discontinuous on a set of energies which is dense in the band this result is proved by constructing states supported on finite regions of the infinite cluster analogous to the kirkpatrick and eggarter zero energy molecular state

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