elliptic problems with nonlinearities indefinite in sign abstract let r n n 3 be a bounded open set with smooth boundary and r we study the dirichlet problem formula with 1 q p and h k l nonnegative functions we prove existence nonexistence and multiplicity results depending on and according to the integrability properties of the ratio k p 1 h q 1

the kirkwood buckingham variational method and the boundary value problems for the molecular schrodinger equation the approach based on the multiplicative form of a trial wave function within the framework of the variational method initially proposed by kirkwood and buckingham is shown to be an effective analytical tool in the quantum mechanical study of atoms and molecules as an example the elementary proof is given to the fact that the ground state energy of a molecular system placed into the box with walls of finite height goes to the corresponding eigenvalue of the dirichlet boundary value problem when the height of the walls is growing up to infinity

the discrete spectrum of the perturbed periodic schrodinger operator in the large coupling constant limit let a be a periodic schrodinger operator and let v 0 0 be a decaying potential we study the number of the eigenvalues of the operator a a v 0 inside a fixed interval 1 2 we obtain an asymptotic formula for as

solving the tensorial 3d acoustic wave equation a mimetic finite difference time domain approach abstractgenerating accurate numerical solutions of the acoustic wave equation awe is a key computational kernel for many seismic imaging and inversion problems although finite difference time domain fdtd approaches for generating full wavefield solutions are well developed for cartesian computational domains several challenges remain when applying fdtd approaches to scenarios arguably best described by more generalized geometry in particular how best to generate accurate and stable fdtd solutions for scenarios involving grids conforming to complex topography or internal surfaces we address these issues by developing a mimetic fdtd mfdtd approach that combines four key components a tensorial 3d awe mimetic finite difference mfd operators fully staggered grids fsgs and mfd robin boundary conditions rbc the tensorial formulation of the 3d awe permits wave propagation to be specified on semi analytically defined coordinate meshes designed to conform to complex domain boundaries mfd o

trapping and cascading of eigenvalues in the large coupling limit we consider eigenvaluese of the hamiltonianh v w w compactly supported in the limit forw 0 we find monotonic convergence ofe to the eigenvalues of a limiting operatorh associated with an exterior dirichlet problem and we estimate the rate of convergence for 1 dimensional systems in 1 dimensional systems withw 0 or withw changing sign we do not find convergence instead we find a cascade phenomenon in which as each eigenvaluee stays near a dirichlet eigenvalue for a long interval of lengtho sqrt lambda of the scaling range quickly drops to the next lower dirichlet eigenvalue stays there for a long interval drops again and so on as a result for most large values of the discrete spectrum ofh is close to that ofe but when reaches a transition region the entire spectrum quickly shifts down by one we also explore the behavior of several explicit models as

on the essential spectrum of schrodinger operators with spherically symmetric potentials on determine le spectre essentiel des operateurs de schrodinger avec un potentiel a symetrie spherique v x q x

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

on the integrated density of states for crystals with randomly distributed impurities in the present paper we discuss spectral properties of a periodic schrodinger operator which is perturbed by randomly distributed impurities such operators occur as simple models for crystals or semi conductors with impurities while the spectrum itself is independent of the concentrationp of impurities for 0 p 1 we focus our attention on the limiting behavior of the integrated density of states p of the random schrodinger operator inside a spectral gap of the periodic operator asp 0 denoting byu0 the set of eigenvalues in the gap of the reference problem having precisely one impurity located at the origin say we show that the integrated density of states concentrates around the points ofu0 in the sense that p ue is of orderp for any fixed e neighborhoodue ofu0 while p k c p2 for any compact subsetk of the gap which does not intersectue

global bifurcation for quasilinear elliptic equations on r n keywords degree for fredholm maps of index zero connected sets of solutions nonexistence of nonzero solutions reference ana article 2001 001doi 10 1007 pl00004863view record in web of science record created on 2008 12 10 modified on 2016 08 08

high order mimetic finite difference operators satisfying the extended gauss divergence theorem abstract we present high order mimetic finite difference operators that satisfy the extended gauss theorem these operators have the same order of accuracy in the interior and at the boundary no free parameters and optimal bandwidth they are defined over staggered grids using weighted inner products with a diagonal norm we present several examples to demonstrate that mimetic finite difference schemes using these operators produce excellent results

the discrete spectrum in the gaps of the continuous one for non signdefinite perturbations with a large coupling constant given two selfadjoint operators a and v v v we study the motion of the eigenvalues of the operator a t a tv as t increases let 0 and let be a regular point for a we consider the quantities n n n 0 defined as the number of the eigenvalues of the operator a t that pass point from the right to the left from the left to the right or change the direction of their motion exactly at point respectively as t increases from 0 to 0 an abstract theorem on the asymptotics for these quantities is presented applications to schrodinger operators and its generalizations are given

on a theorem of deift and hempel we provide an alternative proof of the main result of deift and hempel 1 on the existence of eigenvalues ofv dimensional schrodinger operatorsh h0 w in spectral gaps ofh0

critical lieb thirring bounds in gaps and the generalized nevai conjecture for finite gap jacobi matrices we prove bounds of the form e i d h dist e e h 1 2 l 1 norm of a perturbation where i is a gap included are gaps in continuum one dimensional periodic schrodinger operators and finite gap jacobi matrices where we get a generalized nevai conjecture about an l 1 condition implying a szeg condition one key is a general new form of the birman schwinger bound in gaps

perturbation theory for shape resonances and large barrier potentials we develop a systematic perturbation and resonance theory for the one dimensional schrodinger equation of the form d 2 dx 2 u x lambda v x e psi x 0 0 leqq x infty where the barrier potentialv x is supported only wherex 1 and is non negative there and is a real parameter tending to infinity we prove that every eigenvalue turns into a resonance or an eigenvalue for finite

a new randomized algorithm to approximate the star discrepancy based on threshold accepting we present a new algorithm for estimating the star discrepancy of arbitrary point sets similar to the algorithm for discrepancy approximation of winker and fang siam j numer anal 34 1997 pp 2028 2042 it is based on the optimization algorithm threshold accepting our improvements include amongst others a nonuniform sampling strategy which is more suited for higher dimensional inputs and additionally takes into account the topological characteristics of given point sets and rounding steps which transform axis parallel boxes on which the discrepancy is to be tested into critical test boxes these critical test boxes provably yield higher discrepancy values and contain the box that exhibits the maximum value of the local discrepancy we provide comprehensive experiments to test the new algorithm our randomized algorithm computes the exact discrepancy frequently in all cases where this can be checked i e where the exact discrepancy of the point set can be computed in feasible time most importantly in higher dimensions the new method behaves clearly better than all previously known methods

eigenvalue branches of the schrodinger operatorh λw in a gap of σ h the authors study the eigenvalue branches of the schrodinger operatorh w in a gap of h in particular they consider questions of asymptotic distribution of eigenvalues and bounds on the number of branches they also address the completeness problem

a new mimetic scheme for the acoustic wave equation a new mimetic finite difference scheme for solving the acoustic wave equation is presented it combines a novel second order tensor mimetic discretizations in space and a leapfrog approximation in time to produce an explicit multidimensional scheme convergence analysis of the new scheme on a staggered grid shows that it can take larger time steps than standard finite difference schemes based on ghost points formulation a set of numerical test problems gives evidence of the versatility of the new mimetic scheme for handling general boundary conditions

the discrete spectrum in the spectral gaps of semibounded operators with non sign definite perturbations abstract given two self adjoint operators a and v v v we study the motion of the eigenvalues of the operator a t a tv as t increases let 0 and let be a regular point for a we consider the quantities n v n v and n 0 v defined as the number of eigenvalues of the operator a t that pass point from the right to the left from the left to the right or change the direction of their motion exactly at point respectively as t increases from 0 to 0 we study asymptotic characteristics of these quantities as in the present paper the results obtained previously o l safronov comm math phys 193 1998 233 243 are extended and given new applications to differential operators

threshold accepting a general purpose optimization algorithm appearing superior to simulated annealing abstract a new general purpose algorithm for the solution of combinatorial optimization problems is presented the new threshold accepting method is even simpler structured than the wellknown simulated annealing approach the power of the new algorithm is demonstrated by computational results concerning the traveling salesman problem and the problem of the construction of error correcting codes moreover deterministic versions of the new heuristic turn out to perform nearly equally well consuming only a fraction of the computing time of the stochastic versions as an example the deterministic threshold accepting method yields very near to optimum tours for the famous 442 cities traveling salesman problem of grotschel within 1 to 2 s of cpu time

local linear regression for function learning an analysis based on sample discrepancy local linear regression models a kind of nonparametric structures that locally perform a linear estimation of the target function are analyzed in the context of empirical risk minimization erm for function learning the analysis is carried out with emphasis on geometric properties of the available data in particular the discrepancy of the observation points used both to build the local regression models and compute the empirical risk is considered this allows to treat indifferently the case in which the samples come from a random external source and the one in which the input space can be freely explored both consistency of the erm procedure and approximating capabilities of the estimator are analyzed proving conditions to ensure convergence since the theoretical analysis shows that the estimation improves as the discrepancy of the observation points becomes smaller low discrepancy sequences a family of sampling methods commonly employed for efficient numerical integration are also analyzed simulation results involving two different examples of function learning are provided

an isoperimetric inequality for fundamental tones of free plates we establish an isoperimetric inequality for the fundamental tone first nonzero eigenvalue of the free plate of a given area proving the ball is maximal given 0 the free plate eigenvalues and eigenfunctions u are determined by the equation u u u together with certain natural boundary conditions the boundary conditions are complicated but arise naturally from the plate rayleigh quotient which contains a hessian squared term d 2 u 2

uniform point sets and the collision test monte carlo and quasi monte carlo methods are popular numerical tools used in many applications the quality of the pseudorandom sequence used in a monte carlo simulation is essential to the accuracy of its estimates likewise the quality of the low discrepancy sequence determines the accuracy of a quasi monte carlo simulation there is a vast literature on statistical tests that help us assess the quality of a pseudorandom sequence however for low discrepancy sequences assessing quality by estimating discrepancy is a very challenging problem leaving us with no practical options in very high dimensions in this paper we will discuss how a certain interpretation of the well known collision test for pseudorandom sequences can be used to obtain useful information about the quality of low discrepancy sequences numerical examples will be used to illustrate the applications of the collision test

construction algorithms for digital nets with low weighted star discrepancy we introduce a new construction method for digital nets which yield point sets in the s dimensional unit cube with low star discrepancy the digital nets are constructed using polynomials over finite fields it has long been known that there exist polynomials which yield point sets with low unweighted star discrepancy this result was obtained by niederreiter by the means of averaging over all polynomials hence concrete examples of good polynomials were not known in many cases here we show that good polynomials can be found by computer search the search algorithm introduced in this paper is based on minimizing a quantity closely related to the star discrepancy it has been pointed out that many integration problems can be modeled by weighted function spaces and it has been shown that in this case point sets with low weighted discrepancy are required hence it is particularly useful to be able to adjust a point set to some given weights we are able to extend our results from the unweighted case to show that this can be done using our construction algorithms this way we can find point sets with low weighted star discrepancy making such point sets especially useful for many applications

comportement semi classique pour l operateur de schrodinger a potentiel periodique abstract in this paper the schrodinger operator with a periodic potential is considered let v a smooth periodic function we study the semi classical behavior for a continuum spectrum of h2 v h 0 we are interested by localization and width of bands we give the interaction matrix up to an exponentially small error measured by agmon s distance between the wells a detailed investigation of the spectrum is made for the case where v has one nondegenerate minima per unit cell we also investigate the spectral properties of h2 v v where v is a smooth positive perturbation with compact support

eigenvalue branches of the perturbed maxwell operator m λd in a gap of σ m the propagation of guided waves in photonic crystal fibers pcfs is studied a pcf can be regarded as a perfectly two dimensional photonic crystal with a line defect along the axial direction this problem can be treated as an eigenvalue problem for a family of noncompact self adjoint operators under the assumption that the background spectrum has a gap we prove that a line defect can create an eigenvalue of any given fixed value in the gap provided that the defect is strong enough based on a decoupling of regions in r2 by means of dirichlet and neumann boundaries then using the trace ideal estimates we study asymptotic distribution of eigenvalues and bounds on the number of eigenvalue branches in particular we show that if the defect is weak enough no eigenvalues can be created inside the gap

the shape resonance for a class of schrodinger operatorsh 2 2m v onl 2 n with potentials having minima embedded in the continuum of the spectrum and non trapping tails we show the existence of shape resonances exponentially close to the real axis as 0 the resonant energies are given by a convergent perturbation expansion in powers of a parameter exhibiting the expected exponentially small behaviour for tunneling

deficiency indices and singular boundary conditions in quantum mechanics we consider schrodinger operators h in l2 rn n n with countably infinitely many local singularities of the potential which are separated from each other by a positive distance it is proved that due to locality each singularity yields a separate contribution to the deficiency index of h in the special case where the singularities are pointlike and the potential exhibits certain symmetries near these points we give an explicit construction of self adjoint boundary conditions

the discrete spectrum of the periodic maxwell operator perturbed by a decreasing potential the propagation of electromagnetic waves in dielectric medium with decreasing perturbations defects is studied we use the maxwell operator m p x and the perturbed operator m p x q x to describe the background medium and perturbed medium respectively where 0 is the coupling constant under the assumption that m has a spectral gap g we prove that a defect can create an eigenvalue of any given fixed value g for some 0 provided that the support of the defect is large enough we also consider the asymptotic distribution of eigenvalues in the gap and bounds on the number of eigenvalue branches

spectral stability under tunneling we study the spectral properties of multiple well schrodinger operators on n we give in particular upper bounds on energy shifts due to tunnel effect and localization properties of wave packets our methods are based on agmon type estimates for resolvents in classically forbidden regions and geometric perturbation theory our results are valid also for an infinite number of wells arbitrary spectral type and in non semi classical regimes

eigenvalue excluding for perturbed periodic one dimensional schrodinger operators subject of investigation in this paper is a one dimensional schrodinger equation where the potential is a sum of a periodic function and a perturbation decaying at graphic 1 it is well known that the essential spectrum consists of spectral bands and that there may or may not be additional eigenvalues below the lowest band or in the gaps between the bands while enclosures for gap eigenvalues can comparatively easily be obtained from numerical approximations e g by d weinstein s bounds there seems to be no method available so far which is able to exclude eigenvalues in spectral gaps i e which identifies subregions of a gap which contain no eigenvalues here we propose such a method it makes heavy use of computer assistance nevertheless the results are completely rigorous in the strict mathematical sense because all computational errors are taken into account 1 embed inline graphic 1 gif

good lattice rules based on the general weighted star discrepancy we study the problem of constructing rank 1 lattice rules which have good bounds on the weighted star discrepancy here the non negative weights are general weights rather than the product weights considered in most earlier works in order to show the existence of such good lattice rules we use an averaging argument and a similar argument is used later to prove that these lattice rules may be obtained using a component by component cbc construction of the generating vector under appropriate conditions on the weights these lattice rules satisfy strong tractability bounds on the weighted star discrepancy particular classes of weights known as order dependent and finite order weights are then considered and we show that the cost of the construction can be very much reduced for these two classes of weights

mimetic discretizations for maxwell s equations we have constructed reliable finite difference methods for approximating the solution to maxwell s equations using accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus in discrete form the numerical approximation does not have spurious modes and mimics many fundamental properties of the underlying physical problem including conservation laws symmetries in the solution and the nondivergence of particular vector fields numerical examples demonstrate the high quality of the method when the medium is strongly discontinuous and for nonorthogonal nonsmooth computational grids

on the asymptotic distribution of the eigenvalue branches of the schrodinger operator h λw in a spectral gap of h on considere des operateurs de schrodinger h v dans l espace de hilbert l 2 r avec un potentiel v satisfaisant 1 vel r on suppose que l intervalle a b est contenu dans l ensemble resolvant de h on obtient des estimations asymptotiques sur le nombre des branches de valeurs propes dans a b des familles d operateurs h w 0 quand ou w est un potentiel a courte portee non negatif

coupling constant thresholds of perturbed periodic hamiltonians we consider schrodinger operators of the form h v w on l2 r 1 2 or 3 with v periodic w short range and a real non negative parameter then the continuous spectrum of h has the typical band structure consisting of intervals separated by gaps in the gaps there may be discrete eigenvalues of h that are functions of the parameter let a b be a gap and e a b an eigenvalue of h we study the asymptotic behavior of e as approaches a critical value 0 called a coupling constant threshold at which the eigenvalue either emerges from or is absorbed into the continuous spectrum a typical question is the following assuming e a as 0 is e a c 0 for some 0 and c 0 or is there an expansion in some other quantity as one expects from previous work in the case v 0 the answer strongly depends on

large coupling convergence with negative perturbations abstract let e and p be nonnegative quadratic forms in the hilbert space h suppose that for every 0 the forms e p are densely defined lower semi bounded and closed let h be the self adjoint operator associated with e p and r lim h 1 1 we discuss convergence of h 1 1 towards r strongly uniformly and with respect to the trace and hilbert schmidt norms we also give estimates of the speed of convergence for the indicated norms conditions ensuring trace and hilbert schmidt norm convergence are also given as well as a condition ensuring trace norm convergence with rate proportional to 1 various examples supporting our results are elaborated

the birman schwinger principle on the essential spectrum let h0 and h be self adjoint operators in a hilbert space we consider the spectral projections of h0 and h corresponding to a semi infinite interval of the real line we discuss the index of this pair of spectral projections and prove an identity which extends the birman schwinger principle onto the essential spectrum we also relate this index to the spectrum of the scattering matrix for the pair h0 h

on eigenvalues in gaps for perturbed magnetic schrodinger operators we study schrodinger operators h0 with a gap in the essential spectrum perturbed by either a decreasing electric potential or a decreasing magnetic field in both cases the strength of the perturbation is measured by a coupling constant 0 here we are mainly interested in the asymptotic behavior as of certain counting functions for the eigenvalues that are produced by the perturbation inside the spectral gap the case where we perturb by a potential can be handled using current technology even if h0 contains a fixed magnetic background for perturbations by magnetic fields however we require rather strong assumptions like exponential decay of the perturbations to obtain a lower bound on the counting function to gain some additional intuition we use separation of variables in the closely related model of a schrodinger operator with constant magnetic field in r2 perturbed by a rotationally symmetric magnetic field that decays at infinity

existence of solutions for semilinear elliptic equations with indefinite linear part we study equations on rn lu w x u p 1 ulu f u where l is an indefinite schrodinger operator 0 w x 0 as and subcritical nonlinearities we prove existence of solutions u h1 rn via an inversion to a dual variational problem

eigenvalue asymptotics for the schrodinger operator with perturbed periodic potential we consider the schrodinger operatorh w v acting inl 2 m m 2 with periodic potentialw perturbed by a potentialv which decays slowly at infinity we study the asymptotic behaviour of the discrete spectrum ofh near any given boundary point of the essential spectrum

application of threshold accepting to the evaluation of the discrepancy of a set of points efficient routines for multidimensional numerical integration are provided by quasi monte carlo methods these methods are based on evaluating the integrand at a set of representative points of the integration area a set may be called representative if it shows a low discrepancy however in dimensions higher than two and for a large number of points the evaluation of discrepancy becomes infeasible the use of the efficient multiple purpose heuristic threshold accepting offers the possibility to obtain at least good approximations to the discrepancy of a given set of points this paper presents an implementation of the threshold accepting heuristic an assessment of its performance for some small examples and results for larger sets of points with unknown discrepancy

eigenvalues in gaps and decoupling by neumann boundary conditions abstract continuing earlier investigations we show the existence of eigenvalues of schrodinger operators h w r inside a spectral gap of h v under suitable conditions on v and w our method of proof is based on a decoupling of regions in r v by means of dirichlet or neumann boundary conditions where in particular the use of neumann boundary conditions in the decoupling process leads to a new asymptotic results for eigenvalue counting functions

on the existence of eigenvalues of the schrodinger operatorh λw in a gap of σ h the authors prove a number of results asserting the existence of eigenvalues of the schrodinger operatorh w in a gap of h whereh v andv andw are bounded the existence of these eigenvalues is an important element in the theory of the colour of crystals the basic theorems are proved in v stronger results forv 1 are also presented

constructing sobol sequences with better two dimensional projections direction numbers for generating sobol sequences that satisfy the so called property a in up to 1111 dimensions have previously been given in joe and kuo acm trans math software 29 2003 pp 49 57 however these sobol sequences may have poor two dimensional projections here we provide a new set of direction numbers alleviating this problem these are obtained by treating sobol sequences in d dimensions as t d sequences and then optimizing the t values of the two dimensional projections our target dimension is 21201

strong electric field eigenvalue asymptotics for the perturbed magnetic schrodinger operator we consider the schrodinger operator with constant full rank magnetic field perturbed by an electric potential which decays at infinity and has a constant sign we study the asymptotic behaviour for large values of the electric field coupling constant of the eigenvalues situated in the gaps of the essential spectrum of the unperturbed operator

efficient algorithms for computing the l 2 discrepancy the l 2 discrepancy is a quantitative measure of precision for multivariate quadrature rules it can be computed explicitly previously known algorithms needed o m 2 operations where m is the number of nodes in this paper we present algorithms which require o m log m d operations

compactness results for schrodinger equations with asymptotically linear terms abstract we study the nonlinear problem u v x f x u x r n lim x u x 0 where the schrodinger operator v is semibounded and the nonlinearity f is linearly bounded we establish compactness of palais smale sequences and cerami sequences for the associated energy functional under general spectral theoretic assumptions applying these results we obtain existence of three nontrivial solutions if the energy functional has a mountain pass geometry

spectral shift function in the large coupling constant limit given two selfadjoint operators h0 and v v v we study the spectrum of the operator h h0 v 0 we consider the quantity h h0 r which coincides with krein s spectral shift function of the pair h h0 if v is of trace class and study its asymptotic behavior as applications to differential operators are given

eigenvalue bounds in the gaps of schrodinger operators and jacobi matrices abstract we consider c a b where a is selfadjoint with a gap a b in its spectrum and b is relatively compact we prove a general result allowing b of indefinite sign and apply it to obtain a v d 2 bound for perturbations of suitable periodic schrodinger operators and a not quite lieb thirring bound for perturbations of algebro geometric almost periodic jacobi matrices

convergent expansions for tunneling a new method to compute effects of tunneling in one dimensional multiple well is developed a tunneling parameter built with physical quantities is introduced to measure the splitting between eigenvalues due to tunneling these splittings are given by convergent series in term of this tunneling parameter for a wide class of double well