eigenvalues of the laplacian in two dimensions the eigenvalue problem for the laplace operator in two dimensions is classical in mathematics and physics nevertheless computational methods for estimating the eigenvalues are still of much current interest particularly in applications to acoustic and electromagnetic waveguides although our primary interest is with the computational methods there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and in addition are of interest in themselves these results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds some of the more powerful techniques available are illustrated by applying them to a model problem

shape recognition using eigenvalues of the dirichlet laplacian the eigenvalues of the dirichlet laplacian are used to generate three different sets of features for shape recognition and classification in binary images the generated features are rotation translation and size invariant the features are also shown to be tolerant of noise and boundary deformation these features are used to classify hand drawn synthetic and natural shapes with correct classification rates ranging from 88 9 to 99 2 the classification was done using few features only two features in some cases and simple feedforward neural networks or minimum euclidian distance

wake free waves in one and three dimensions a recent paper by gottlieb j math phys 29 2434 1988 provides examples of acoustic wave equations in various dimensions that have nontrivial families of solutions that are progressing waves of order 1 and relates this to whether or not these equations satisfy huygens principle a statement made in that paper related to huygens principle in one space dimension is clarified and it is shown in this connection that in general the relationship between the possession of progressing wave solutions and the satisfaction of huygens principle is more complex than the situation described by gottlieb in addition the attractive properties of progressing waves of order 1 are retained by progressing waves of any finite order and we use this to generalize in several ways gottlieb s results on wake free solutions of the acoustic equation in three dimensions

can one hear the shape of a drum revisited in a landmark paper mark kac in 1966 amer math monthly 73 pp 1 23 showed that geometric properties of regions in r 2 can be obtained by studying the asymptotic properties of the spectrum of the laplacian he also conjectured that the shape of a region might be completely determined by the spectrum we describe recent developments in this field and discuss the status of the kac conjecture as well as similar conjectures for operators other than the laplacian

wave equations with the characteristic propagation property within the class of second order linear self adjoint wave equations in 1 1 dimensions an explicit construction is given of probably all those with the characteristic propagation property that is those whose solutions are without tails

isoperimetric bounds for higher eigenvalue ratios for the n dimensional fixed membrane problem we give several results which extend our recent proof of the payne polya weinberger conjecture to ratios of higher eigenvalues in particular we show that for a bounded domain n the eigenvalues of its dirichlet laplacian obey where m denotes the m th eigenvalue and j p k denotes the k th positive zero of the bessel function j p x certain extensions of this result are given the most general being the bound where k 2 and l m denotes the number of nodal domains of an m th eigenfunction our results imply certain further conjectures of payne polya and weinberger concerning 3 2 and 4 3 in addition we find a resonably good bound on 4 1 we also briefly discuss extensions to schrodinger operators and other elliptic eigenvalue problems

application of complex canonical point transformations to linear second order differential equations we present a canonical method to solve one dimensional linear differential equations making use of pseudodifferential calculus we apply two successive canonical point transformations on the cartesian momentum and position spaces to obtain a nonlinear complex valued canonical transformation which maps a very simple linear differential equation into the desired differential equation this method yields a closed contour integral representation for the exact solution in terms of arbitrary functions which may be determined from the mapping equations in a similar way to that followed in classical mechanics this method does not require the completeness condition on the intermediary states and avoids calculation of the kernel of the generator we explicitly develop the case of second order differential equations and give some standard examples to show how this method works

theory of group representations and integration of nonlinear systems xa zz exp kx a abstract a constructive proof of complete integrability of the system xa zf z exp kx a and its explicit solution are given for k coinciding with the cartan matrix of simple lie algebras the technique for dealing with this system of essentially nonlinear two dimensional partial differential equations involves lax type representation and representation theory of the corresponding group

a note on the gap between the first two eigenvalues for the schrodinger operator by means of a commutation formula the author gives a simple proof of the upper bound of wong et al on the gap between the first two eigenvalues in the schrodinger operator unfortunately this proof does not seem to generalise into higher dimensions

sequences of point transformations and linear canonical transformations in classical and quantum mechanics sequences of point transformations and canonical linear transformations are considered in classical and quantum mechanics it is shown that the unitary representations of such transformations can be obtained in general in the sense that in the limit classical behavior is retrieved in the particular case of one point transformation combined with two linear transformations the results found in this way are exact a new class of differential equations is thereby solved by quadratures

krein s formula and one dimensional multiple well abstract it is shown that all the discrete eigenvalues of a one dimensional schrodinger operator with a multiple well potential possess an asymptotic expansions in power of h 1 2 when h 0 a formula for all the coefficients of these expansions is given the method uses two main tools perturbation by boundary conditions and exponential decay of eigenfunctions which are developed in this article as a by product of this work the exponential localization of eigenvectors when h goes to zero can be proved

differential inequalities for riesz means and weyl type bounds for eigenvalues we derive differential inequalities and difference inequalities for riesz means of eigenvalues of the dirichlet laplacian r z x k z k

wave splitting and lattice dynamics a wave splitting approach used elsewhere to solve a black hole scattering problem is adapted to the formal solution of arbitrary self adjoint wave equations in 1 1 dimensions and yields potentially useful results in this more general case it is then shown that the self adjoint wave equation being solved and the non self adjoint linear wave equations satisfied by the one way component waves are naturally related to a pair of motions of the 1 1 dimensional toda lattice that together comprise a motion of the kac van moerbeke lattice this provides a partial explanation for the peculiar fact that every motion of the kac van moerbeke lattice can be viewed as two interpolated motions of the toda lattice

solvable systems of wave equations and non abelian toda lattices relates equivalence classes of coupled systems of n linear wave equations to motions of an n n matrix dynamical systems the two dimensional non abelian toda lattice in particular the correspondence is shown to relate those coupled systems of wave equations with progressing wave general solutions to motions of the finite non abelian toda lattice with free ends generalizing a known result for the n 1 case some non trivial motions of such toda lattices are found and the corresponding coupled wave equations and their progressing wave general solutions are given other consequences of the correspondence and possible application of the progressing waves are discussed

a diffusion mechanism for obstacle detection from size change information a mechanism fur the visual detection of obstacles is presented a new immediacy measure representing the imminence of collision between an object and a moving observer is defined a diffusion process on the image domain whose initial condition is determined by the motion field normal to the object s boundary is shown to converge asymptotically to the immediacy measure a network of locally connected cells derived from a finite difference approximation of the diffusion equation estimates the immediacy measure from normal velocity and boundary information provided by a motion measurement and segmentation stage the algorithm s performance on real image sequences is demonstrated

xi and eta function resummation of finite series general case given a spectrum of positive numbers lambda m from which a xi function z s sigma m lambda sm can be constructed the reorganization of series of the type f s t identical to m lambda smf lambda mt into power series in t is examined in detail using the method of xi function resummation for summand functions f lambda mt having power series expansions in lambda mt with infinite radius of convergence and which satisfy other conditions of a rather general nature the author finds that f s t can be reorganized to f s t n antbn k cktdk in t r s t where r s t vanishes exponentially as t to 0 the numbers an bn cn dn can all be computed in terms of the xi function z s r s t is difficult to evaluate but important general features of this function can be determined the power series expansion of f s t can be regarded as a generalization of the heat kernel expansion for which f lambda mt exp lambda mt and s 0 to nonzero complex variable s which is useful and to many other summand functions f lambda mt remarkably the xi function resummation method can be applied as easily to divergent series f s t as it can to convergent ones the method is therefore both a rearrangement procedure for convergent series and a summation prescription for divergent series

bounds on eigenvalues of dirichlet laplacian in this paper we investigate an eigenvalue problem of dirichlet laplacian on a bounded domain in an n dimensional euclidean space rn if k 1 is the k 1 th eigenvalue of dirichlet laplacian on then we prove that for n 41 and k geq 41 lambda k 1 leq k frac2n lambda 1 and for any n and k lambda k 1 leq c 0 n k k frac2n lambda 1 with c 0 n k leq j 2 n 2 1 j 2 n 2 1 1 where jp k denotes the k th positive zero of the standard bessel function jp x of the first kind of order p from the asymptotic formula of weyl and the partial solution of the conjecture of polya we know that our estimates are optimal in the sense of order of k

wesd weighted spectral distance for measuring shape dissimilarity this paper presents a new distance for measuring shape dissimilarity between objects recent publications introduced the use of eigenvalues of the laplace operator as compact shape descriptors here we revisit the eigenvalues to define a proper distance called weighted spectral distance wesd for quantifying shape dissimilarity the definition of wesd is derived through analyzing the heat trace this analysis provides the proposed distance with an intuitive meaning and mathematically links it to the intrinsic geometry of objects we analyze the resulting distance definition present and prove its important theoretical properties some of these properties include 1 wesd is defined over the entire sequence of eigenvalues yet it is guaranteed to converge 2 it is a pseudometric 3 it is accurately approximated with a finite number of eigenvalues and 4 it can be mapped to the 0 1 interval last experiments conducted on synthetic and real objects are presented these experiments highlight the practical benefits of wesd for applications in vision and medical image analysis

variable coefficient wave equations with exact spreading solutions we discuss a family of variable coefficient linear wave equations that are exactly solved by a generalization of progressing waves that spread out as they propagate this can be viewed as a generalization to some variable coefficient equations of the dispersion theory method that is standard for constant coefficient equations the method is illustrated by solving some physically interesting equations

an inverse problem of the wave equation for a general doubly connected region in r2 with a finite number of piecewise smooth robin boundary conditions the spectral distribution m t w 1 exp ite w 1 2 where e w w 1 are the eigenvalues of the negative laplacian d k 1 2 x k 2in the x 1 x 2 plane is studied for a variety of domains where 0

ground state energy bounds for potentials x ν a theory is developed from which both upper and lower analytic bounds on schrodinger eigenvalues can be obtained we propose a recursion algorithm with which ground energies for certain potentials can be rigorously bounded to arbitrary precision these analytic and numerical methods together with existing techniques are applied to the ground state problem for power potentials x 0

optimal bounds for ratios of eigenvalues of one dimensional schrodinger operators with dirichlet boundary conditions and positive potentials consider the schrodinger equation u v x u u on the intervali wherev x 0 forx i and where dirichlet boundary conditions are imposed at the endpoints ofi we prove the optimal bound frac lambda n lambda 1 leqq n 2 for n 2 3 4 on the ratio of thenth eigenvalue to the first eigenvalue for this problem this leads to a complete treatment of bounds on ratios of eigenvalues for such problems extensions of these results to singular problems are also presented a modified prufer transformation and comparison techniques are the key elements of the proof

linear wave equations as motions on a toda lattice a bijection is defined from the set of motions on the infinite toda lattice of strings to a set of sequences of linear wave equations in 1 1 dimensions the sequences being generated by a generalisation of the classical darboux map the bijection is applied to find probably all such wave equations for which characteristic initial data propagate without spreading

discriminative segmentation based evaluation through shape dissimilarity segmentation based scores play an important role in the evaluation of computational tools in medical image analysis these scores evaluate the quality of various tasks such as image registration and segmentation by measuring the similarity between two binary label maps commonly these measurements blend two aspects of the similarity pose misalignments and shape discrepancies not being able to distinguish between these two aspects these scores often yield similar results to a widely varying range of different segmentation pairs consequently the comparisons and analysis achieved by interpreting these scores become questionable in this paper we address this problem by exploring a new segmentation based score called normalized weighted spectral distance nwsd that measures only shape discrepancies using the spectrum of the laplace operator through experiments on synthetic and real data we demonstrate that nwsd provides additional information for evaluating differences between segmentations which is not captured by other commonly used scores our results demonstrate that when jointly used with other scores such as dice s similarity coefficient the additional information provided by nwsd allows richer more discriminative evaluations we show for the task of registration that through this addition we can distinguish different types of registration errors this allows us to identify the source of errors and discriminate registration results which so far had to be treated as being of similar quality in previous evaluation studies

a sharp bound for the ratio of the first two eigenvalues of dirichlet laplacians and extensions the first author s work was partially supported by fondecyt chile project 0132 88 and by a summer research fellowship provided by the research council of the university of missouri columbia he would like to thank the physics department and others at the universidad de chile for their hospitality during his visit in april 1990 when much of this research was completed the second author was supported in part by fondecyt chile projects 0132 88 and 1238 90 both authors also thank fritz gesztesy for general comments and encouragement

upper bounds on the eigenvalue ratio for one dimensional schrodinger operators abstract we apply a commutation formula and the rayleigh ritz inequality to study the ratio of the first two eigenvalues of one dimensional schrodinger operators with dirichlet boundary conditions the potentials we consider here are allowed to change sign we give conditions under which the first eigenvalue is positive and establish some new results on the upper bounds of the eigenvalue ratio

universal lower bounds on eigenvalue splittings for one dimensional schrodinger operators we provide lower bounds on the eigenvalue splitting for d2 dx2 v x depending only on qualitative properties ofv for example ifv is c on a b ande n en 1 are two successive eigenvalues of d2 dx2 v withu a u b 0 boundary conditions and if lambda mathop max limits e in e n 1 e n x in a b e v x 1 2 then e n e n 1 geqq pi lambda 2 exp left lambda b a right the exponential factor in such bounds are saturated precisely in tunneling examples our results arenot restricted tov s of compact support but only require e n mathop lim limits overline x to infty v x

self adjoint acoustic equations with progressing wave solutions the simplest wave equations are those whose general solutions comprise progressing waves the author constructs a comparatively large possibly exhaustive family of self adjoint acoustic equations in 1 1 dimensions that are simple in this sense

large deviations conditioned on large deviations i markov chain and langevin equation we present a systematic analysis of stochastic processes conditioned on an empirical observable q t defined in a time interval 0 t for large t we build our analysis starting with a discrete time markov chain results for a continuous time markov process and langevin dynamics are derived as limiting cases in the large t limit we show how conditioning on a value of q t modifies the dynamics for a langevin dynamics with weak noise and conditioned on q t we introduce large deviation functions and calculate them using either a wkb method or a variational formulation this allows us in particular to calculate the typical trajectory and the fluctuations around this trajectory when conditioned on a certain value of q t for large t

best possible upper and lower bounds for the zeros of the bessel function _ let j k denote the k th positive zero of the bessel function j x in this paper we prove that for 0 and k 1 2 3 ak 21 3 j k ak 21 3 3 20 ak 21 3 1 3 these bounds coincide with the first few terms of the well known asymptotic expansion j k ak 21 3 3 20 ak 21 3 1 3 as k being fixed where ak is the k th negative zero of the airy function ai x and so are best possible

eigenvalue problems of a two dimensional schrodinger operator with nonparabolic effective mass in this paper we study the eigenvalue problem for the schrodinger operator on a two dimensional disk with nonparabolic effective mass approximation here the effective mass depends on the energy states our results mainly concern with the number of energy states lying in a wire and the monotonicity of energy states with respect to the depth of the wire

simple toda lattice motions and their linear wave equations a structure preserving bijection b m to w from the set m of toda lattice motions to a natural partition w of the set of linear wave equations in 1 1 dimensions is applied to obtain some simple and related results about both m and w the toda lattice motions derived emphasise the existence of qualitatively distinct toda lattices depending on the overall sign in the defining dynamical equations the wave equations that arise illustrate a duality on the set of linear wave equations that is induced by the known duality on m generated by interpolating elements of m into motions of kac van moerbeke lattices

brownian cuboid and its geometric characteristics abstract a class of random cuboid model which is defined brownian cuboid is constructed in this paper based on the structure characteristics of brownian cuboid and the properties of brownian motion geometric characteristics of brownian cuboid are discussed by which the probability estimates of the length of the space diagonal surface area and volume are obtained and moreover the small deviation estimates and some relevant results for the geometric characteristics of brownian cuboid are given additionally we directly derive the upper and lower bounds for the smallest positive zero of the bessel functions of the first kind

representation of quantum mechanical wavefunctions by complex valued extensions of classical canonical transformation generators sufficient conditions are given for the possibility to construct quantum mechanical wavefunctions by the sole knowledge of an appropriate sequence of classical canonical transformations which map a given hamiltonian onto the new position variable the transformation matrix element for each individual step of this sequence is given by the semiclassical limit expression of these matrix elements it is a function of the generator of this transformation step only the wavefunction i e the transformation matrix element for the total transformation is obtained as a multiple integral over the transformation matrix elements of the various intermediate steps the practicability of this procedure is demonstrated by several examples the authors consider time independent systems with one degree of freedom

two new weyl type bounds for the dirichlet laplacian in this paper we prove two new weyl type upper estimates for the eigenvalues of the dirichlet laplacian as a consequence we obtain the following lower bounds for its counting function for 1 one has n 2 n 2 1 h n 1 n 2 1 n 2 and n n 2 n 4 n 2 1 h n 1 4 n 1 n 2 1 n 2 where h n 2n j 2 n 2 1 1j 2 n 2 jn 2 1 1 is a constant which depends on n the dimension of the underlying space and bessel functions and their zeros