polynomial bound on the number of scattering poles on montre que dans la diffusion par un potentiel infiniment differentiable a support compact les poles de l operateur de diffusion sont a distribution polynomiale

resonances for periodic jacobi operators with finitely supported perturbations we describe the resonances and the eigenvalues of a periodic jacobi operator with finitely supported perturbations in the case of small diagonal perturbations we determine their asymptotics

algebraic and scattering aspects of a symmetric solvable potential we study a particular solvable potential and analyse the effect of symmetry on its bound state as well as scattering solutions we determine the transmission and reflection coefficients for the symmetric case and also formulate the problem in terms of an su 1 1 potential group which allows unified treatment of the discrete and the continuous spectra in a natural way we find that bound and scattering states of the symmetric problem supply a basis for the unitary irreducible representations of the su 1 1 potential group and this gives a straightforward group theoretical interpretation of the fact that the complex invariant potential has a real energy spectrum

a bivariate rational interpolation and the properties abstract in this paper a bivariate rational interpolation is constructed using both function values and partial derivatives of the function being interpolated as the interpolation data the interpolation function has a simple and explicit rational mathematical representation with parameters and it can be expressed by the symmetric bases it is proved that the interpolation is stable the concept of integral weights coefficients of the interpolation is given which describes the weight of the interpolation points and the quantity as the interpolation data in the local interpolating region

scattering resonances of convex obstacles for general boundary conditions we study the distribution of resonances for smooth strictly convex obstacles under general boundary conditions we show that under a pinched curvature condition for the boundary of the obstacle the resonances are separated into cubic bands and the distribution in each band satisfies weyl s law

stability for the multifrequency inverse medium problem abstract the solution of a multi frequency 1d inverse medium problem consists of recovering the refractive index of a medium from measurements of the scattered waves for multiple frequencies in this paper rigorous stability estimates are derived when the frequency takes value in a bounded interval it is showed that the ill posedness of the inverse medium problem decreases as the width of the frequency interval becomes larger more precisely under certain regularity assumptions on the refractive index the estimates indicate that the power in holder stability is an increasing function of the largest value in the frequency band finally a lipschitz stability estimate is obtained for the observable part of the medium function defined through a truncated trace formula

distribution of energy levels of quantum free particle on the liouville surface and trace formulae we consider the weyl asymptotic formula left e n leqq r 2 right frac area q 4 pi r 2 n r for eigenvalues of the laplace beltrami operator on a two dimensional torusq with a liouville metric which is in a sense the most general case of an integrable metric we prove that if the surfaceq is non degenerate then the remainder termn r has the formn r r1 2 r where r is an almost periodic function of the besicovitch classb1 and the fourier amplitudes and the fourier frequencies of r can be expressed via lengths of closed geodesics onq and other simple geometric characteristics of these geodesics we prove then that if the surfaceq is generic then the limit distribution of r has a densityp t which is an entire function oft possessing an asymptotics on a real line logp t c t4 ast an explicit expression for the fourier transform ofp t via fourier amplitudes of r is also given we obtain the analogue of the guillemin duistermaat trace formula for the liouville surfaces and discuss its accuracy

complex scaling and the distribution of scattering poles the purpose of this paper is to establish sharp polynomial bounds on the number of scattering poles for a general class of compactly supported self adjoint perturbations of the laplacian in r n odd we also consider more general types of bounds that give sharper estimates in certain situations the general conclusion can be stated as follows the order of growth of the poles is the same as the order of growth of eigenvalues for corresponding compact problems from the few known cases however the exact asymptotics are expected to be different the scattering poles for compactly supported perturbations were rigorously defined by lax and phillips 13 in a more general setting they correspond to resonances the study of which has a long tradition in mathematical physics in the lax phillips theory they appear as the poles of the meromorphic continuation of the scattering matrix and coincide with the poles of the meromorphic continuation of the resolvent of the perturbed operator because of the latter characterization they can be considered as the analogue of the discrete spectral data for problems on noncompact domains the problem of estimating the counting function

weighted rational cubic spline interpolation and its application abstract in qi duan et al korean j comput appl math 6 1 1999 203 215 the authors have discussed constrained interpolation problems by means of rational cubic spline interpolation with linear denominators but there are still some cases in which the constrained interpolation cannot be solved in this paper the weighted rational cubic spline interpolation has been constructed using the rational cubic spline with linear denominator and the rational cubic spline based on function values by these the problems to constrain the weighted rational interpolation curves to lie strictly above or below a given piecewise linear curve and between two given piecewise linear curves can be solved completely also the approximation properties of these weighted rational cubic splines are studied

pseudospectra of semiclassical pseudo differential operators the pseudo spectra or spectral instability of non selfadjoint operators is a topic of current interest in applied mathematics for example in computational fluid dynamics it affects the study of the stability of laminar flows in fact even for the most basic flows the computations entirely fails to predict what is observed in the experiments the explanation is that for non normal operators the resolvent could be very large far away from the spectrum which makes computation of the eigenvalues impossible the occurence of false eigenvalues is due to the existence of quasi modes i e approximate local solutions to the eigenvalue problem the quasi modes appear since the nirenberg treves condition psi is not satisfied for topological reasons less

scattering asymptotics for riemann surfaces in this article we prove the optimal polynomial lower bound for the number of resonances of a surface with hyperbolic ends we also give weyl asymptotics for the relative scattering phase of such a surface the proofs are based on trace formulae analogous to those of the euclidean odd dimensional scattering the main technical ingredient is a new proof of the poisson formula theorem 5 7 which is applicable in the euclidean case as well our lower bound seems to be the first example of an optimal polynomial lower bound for the number of resonances holding for a general class of higher dimensional elliptic operators with no symmetries the previous general lower bounds or asymptotics were either nonoptimal 25 58 9 one dimensional or radial 65 67 and 54 41 1 or they required some degeneracy of the

sharp polynomial bounds on the number of scattering poles of radial potentials abstract it is shown that for scattering by a radially symmetric potential in r n n odd the number of poles in a disk of radius r satisfies an estimate n r c n r 1 n this bound is sharp as shown by the special case of potentials nonvanishing at the boundary where n r k n a n r n 1 o 1 a being the diameter of the support

several complex variables and the distribution of resonances in potential scattering we study resonances associated to schrodinger operators with compactly supported potentials on d d 3 odd we consider potentials depending holomorphically on a parameter open image in new window for certain such families for all z except those in a pluripolar set the associated resonance counting function has order of growth d

error estimation of a kind of rational spline this paper deals with the approximation properties of a kind of rational spline with linear denominator when the function being interpolated is c 3 in an interpolating interval error estimate expressions of interpolating functions are derived convergence is established the optimal error coefficient c i is proved to be symmetric about the parameters of the rational interpolation and it is bounded finally the precise jump measurements of the second derivatives of the interpolating function at the knots are given

resonances for non trapping time periodic perturbations for time periodic perturbations of the wave equation in given by a potential q t x we obtain an upper bound of the number of the resonances we establish for m n large enough a trace formula relating the iterations of the monodromy operator u mt 0 t 0 and the sum jzmj of all resonances counted with their multiplicities

isophasal isopolar and isospectral schrodinger operators and elementary complex analysis we explicitly construct super exponentially decaying potentials on d so that the associated schrodinger operators are isophasal and isopolar we use similar techniques to construct isospectral schrodinger operators with periodic potentials on and isospectral schrodinger operators on

lower bounds for resonance counting functions for obstacle scattering in even dimensions in even dimensional euclidean scattering the resonances lie on the logarithmic cover of the complex plane this paper studies resonances for obstacle scattering in bbb r d with dirichlet neumann or admissible robin boundary conditions when d is even set n m r to be the number of resonances with norm at most r and argument between m pi and m 1 pi then lim sup r rightarrow infty log n m r over log r d if m in bbb z setminus 0

lower bounds on the number of scattering poles ii abstract we obtain lower bounds on the number of scattering poles for a class of abstract compactly supported perturbations of the laplacian in r n n odd they are applied to estimate the number of resonances for obstacle scattering and for hypoelliptic compactly supported perturbations of the laplacian the proof involves a development of lax phillips theory in a generalized setting a poisson formula for scattering poles and some simple tauberian arguments

resonances of a potential well with a thick barrier this work is motivated by the desire to develop a method that allows for easy and accurate calculation of complex resonances of a one dimensional schrodinger s equation whose potential is a low energy well surrounded by a thick barrier the resonance is calculated as a perturbation of the bound state associated with a barrier of infinite thickness we show that the corrector to the bound state energy is exponentially small in the barrier thickness a simple computational strategy that exploits this smallness is devised and numerically verified to be very accurate we also provide a study of high frequency resonances and show how they can be approximated numerical examples are given to illustrate the main ideas in this work

distribution of poles for scattering on the real line abstract the density of scattering poles is shown to be proportional to the length of the convex hull of the support of the potential in the case of a potential with finite singularities at the endpoints of the support asymptotic formulae for the poles are given while in the c 0 case an example of a potential with infinitely many scattering poles on i r is constructed the scattering amplitude of a compactly supported potential is also characterized

fractal weyl law for open quantum chaotic maps we study a semiclassical quantization of poincar e maps arising in scattering problems with fractal hyperbolic trapped sets the main application is the proof of a fractal weyl upper bound for the number of resonances scattering poles in small domains near the real axis this result includes the case of several convex hard obstacles satisfying a no eclipse condition

resonances for 1d massless dirac operators abstract we consider the 1d massless dirac operator on the real line with compactly supported potentials we study resonances as the poles of scattering matrix or equivalently as the zeros of modified fredholm determinant we obtain the following properties of the resonances 1 asymptotics of counting function 2 estimates on the resonances and the forbidden domain 3 the trace formula in terms of resonances

a new weighted rational cubic interpolation and its approximation a weighted rational cubic spline interpolation has been constructed using two kinds of rational cubic spline with quadratic denominator the degree of smoothness of this spline is c 2 in the interpolating interval when the parameters satisfy a continuous system the sufficient and necessary conditions that constrain the interpolant curves to be convex in the interpolating interval or subinterval are derived also the error estimate formulas of this interpolation are obtained

schrodinger operators with complex valued potentials and no resonances in dimension d 3 we give examples of nontrivial compactly supported complex valued potentials such that the associated schrodinger operators have neither resonances nor eigenvalues if d 2 we show that there are potentials with no resonances or eigenvalues away from the origin these schrodinger operators are isophasal and have the same scattering phase as the laplacian on rd in odd dimensions d 3 we study the fundamental solution of the wave equation perturbed by such a potential if the space variables are held fixed it is superexponentially decaying in time

multiple cue saliency measurement and optimized image composition for image retargeting we present a novel image retargeting approach for high definition displaying based on multiple cue salient region extraction and optimized image composition which allows resizing an image into user specified aspect ratios while preserving prominent visual salient image information firstly we sum up six properties for measuring the salient objects and apply them to region extraction then according to our optimized rules of image composition we evaluate the aesthetic scores with different extracted sub images finally we sort out the target image with the highest aesthetic score and retarget it to fit the final scale of displaying the experimental results show the performance and effectiveness of our approach

existence of resonances in three dimensions ifp is an elliptic self adjoint perturbation of the laplacian on 3 and the coefficients ofp decay super exponentially then we show thatp has infinitely many resonances the resonances are defined here as the poles of the meromorphic continuation of p 2 1

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

sum rules for jacobi matrices and their applications to spectral theory we discuss the proof of and systematic application of case s sum rules for jacobi matrices of special interest is a linear combination of two of his sum rules which has strictly positive terms among our results are a complete classification of the spectral measures of all jacobi matrices j for which j j 0 is hilbert schmidt and a proof of nevai s conjecture that the szego condition holds if j j 0 is trace class

breit wigner approximation and the distribution of resonances for operators with a discrete spectrum j 2 the counting function of j s n trivially satisfies n n j j in scattering situations the natural analogue of the discrete spectrum is given by resonances j and of n by the scattering phase s the relation between the two is now non trivial and we prove that where is the harmonic measure of the upper of half plane and can be taken dependent on this provides a precise high energy version of the breit wigner approximation and relates the properties of s to the distribution of resonances close to the real axis

spreading of lagrangian regularity on rational invariant tori let p h be a self adjoint semiclassical pseudodifferential operator on a manifold m such that the bicharacteristic flow of the principal symbol on t m is completely integrable and the subprincipal symbol of p h vanishes consider a semiclassical family of eigenfunctions or more generally quasimodes u h of p h we show that on a nondegenerate rational invariant torus lagrangian regularity of u h regularity under test operators characteristic on the torus propagates both along bicharacteristics and also in an additional diffractive manner in particular in addition to propagating along null bicharacteristics regularity fills in the interiors of small annular tubes of bicharacteristics

hermite interpolation by piecewise rational surface in this paper a bivariate rational hermite interpolation is developed to create a space surface using both function values and the first order partial derivatives of the function being interpolated as the interpolation data the interpolation function has a simple and explicit rational mathematical representation with parameters there are two schemes to construct the interpolation functions an example shows that both of them approximate the function being interpolated very well the basis of this interpolation is derived and the interpolating function can be c1 in all the interpolation region when the parameters satisfy a simple condition when a patch of the interpolating surface does not satisfy the need of design say it is too high or too low at a point and its neighbourhood the values of the interpolating function in these points need to be decreased or increased under the condition that the interpolating data are not changed for this sufficient conditions are derived and it is shown that this can be done just by selecting suitable parameters in the interpolation

on the resonances and eigenvalues for a 1d half crystal with localised impurity we consider the schrdinger operator h on the half line with a periodic potential p plus a compactly supported potential q for generic p its essential spectrum has an infinite sequence of open gaps we determine the asymptotics of the resonance counting function and show that for sufficiently high energy each non degenerate gap contains exactly one eigenvalue or antibound state giving asymptotics for their positions conversely for any potential q and for any sequences n 0 1 and xn 0 there exists a potential p such that xn is the length of the n th gap n and h has exactly n eigenvalues and 1 n antibound state in each high energy gap moreover we show that between any two eigenvalues in a gap there is an odd number of antibound states and hence deduce an asymptotic lower bound on the number of antibound states in an adiabatic limit

a bivariate rational interpolation with a bi quadratic denominator in this paper a new rational interpolation with a bi quadratic denominator is developed to create a space surface using only values of the function being interpolated the interpolation function has a simple and explicit rational mathematical representation when the knots are equally spaced the interpolating function can be expressed in matrix form and this form has a symmetric property the concept of integral weights coefficients of the interpolation is given which describes the weight of the interpolation points in the local interpolating region

distribution of scattering resonances for generic schrodinger operators abstract let v be the schrodinger operator acting on l 2 r d c with d 3 odd here v is a bounded real or complex valued function vanishing outside the closed ball of center 0 and radius a if v belongs to the class m a of potentials introduced by christiansen we show that when r the resonances of v scaled down by the factor r are asymptotically distributed with respect to an explicit probability distribution on the lower unit half disc of the complex plane the rate of convergence is also considered for subclasses of potentials

a new c2 rational interpolation based on function values and constrained control of the interpolant curves in this paper a new method is developed to create a high order smoothness interpolation using values of the function being interpolated this is a kind of rational cubic interpolation with quadratic denominator this rational spline not only belongs to c 2 in the interpolating interval but could also be used to constrain the shape of the interpolant curve such as to force it to be in the given region all because of the selectable parameters in the rational spline itself the more important achievement mathematically of this method is that the uniqueness of the interpolating function for the given data would be replaced by the uniqueness of the interpolating curve for the given data and the selected parameters

time decay of semigroups generated by dissipative schrodinger operators abstract we establish a representation formula for semigroups of contractions in terms of a global limiting absorption principle as applications we prove long time asymptotics and dispersive estimate of the semigroup generated by i h where h is a dissipative schrodinger operator

painleve formulas of the limiting distributions for nonnull complex sample covariance matrices in a recent study of large non null sample covariance matrices a new sequence of functions gener alizing the gue tracy widom distribution of random matrix theory was obtained this paper derives painleve formulas of these functions and use them to prove that they are indeed distribution functions applications of these new distribution functions to last passage percolation queues in tandem and to tally asymmetric simple exclusion process are also discussed as a part of the proof a representation of orthogonal polynomials on the unit circle in terms of an operator on a discrete set is presented

the lyapunov function for schrodinger operators with a periodic 2 2 matrix potential abstract we consider the schrodinger operator on the real line with a 2 2 matrix valued 1 periodic potential the spectrum of this operator is absolutely continuous and consists of intervals separated by gaps we define a lyapunov function which is analytic on a two sheeted riemann surface on each sheet the lyapunov function has the same properties as in the scalar case but it has branch points which we call resonances we prove the existence of real as well as non real resonances for specific potentials we determine the asymptotics of the periodic and the anti periodic spectrum and of the resonances at high energy we show that there exist two type of gaps 1 stable gaps where the endpoints are the periodic and the anti periodic eigenvalues 2 unstable resonance gaps where the endpoints are resonances i e real branch points of the lyapunov function we also show that periodic and anti periodic spectrum together determine the spectrum of the matrix hill operator

asymptotic number of scattering resonances for generic schrodinger operators let v be the schrodinger operator acting on l 2 mathbb r d mathbb c with d geq 3 odd here v is a bounded real or complex function vanishing outside the closed ball of center 0 and of radius a let n v r denote the number of resonances of v with modulus r we show that if the potential v is generic in a sense of pluripotential theory then n v r c d a dr d o r d 3 over 16 epsilon quad mbox as r to infty for any e 0 where c d is a dimensional constant

bounded property and point control of a bivariate rational interpolating surface a bivariate rational interpolation method with parameters was created in an earlier work which was based on function values only this paper will deal with the bounded property and the point control method of the interpolating surface it is proved that the values of the interpolating function in the interpolation region are bounded no matter what the parameters might be this is called the bounded property of the interpolation also the approximation expressions of the interpolation are derived they do not depend on the parameters more important is that the value of the interpolating function at any point in the interpolating region can be modified under the condition that the interpolating data are not changed by selecting suitable parameters so the interpolation surface may be modified for the given interpolation data when needed in practical design in the special case the central point mean value control is studied and an example is given to show the control

resonances for dirac operators on the half line abstract we consider the 1d dirac operator on the half line with compactly supported potentials we study resonances as the poles of scattering matrix or equivalently as the zeros of modified fredholm determinant we obtain the following properties of the resonances 1 asymptotics of counting function 2 estimates on the resonances and the forbidden domain

on the resonances of the laplacian on waveguides abstract the resonances for the dirichlet and neumann laplacian are studied on compactly perturbed waveguides in the absence of resonances an upper bound is proven for the localised resolvent this is then used to prove that the existence of a quasimode whose asymptotics is bounded away from the thresholds implies the existence of resonances converging to the real axis the following upper bound to the number of resonances is also proven k j res dist k j physical plane k j 2 k j 3

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

asymptotic distribution of resonances in one dimension abstract we determine the leading asymptotics of the resonance counting function for a class of schrodinger operators in one dimension whose potentials may have non compact support

sharp upper bounds on the number of the scattering poles abstract we study the scattering poles of a compactly supported black box perturbations of the laplacian in r n n odd we prove a sharp upper bound of the counting function n r modulo o r n in terms of the counting function of the reference operator in the smallest ball around the black box in the most interesting cases we prove a bound of the type n r a n r n o r n with an explicit a n we prove that this bound is sharp in a few special spherically symmetric cases where the bound turns into an asymptotic formula

resonances in one dimension and fredholm determinants we discuss resonances for schrodinger operators in whole and half line problems one of our goals is to connect the fredholm determinant approach of froese to the fourier transform approach of zworski another is to prove a result on the number of antibound states namely in a half line problem there are an odd number of antibound states between any two bound states

the hurwitz theorem for bessel functions and antibound states in spectral theory an 1889 result of hurwitz is that the bessel function j v c v 0 c 0 has no zeros for 2 n v n 1 with integer n a new application of this result is given which describes the behaviour of jost functions and antibound states for one dimensional schrodinger equations with exponentially decaying potential a new hurwitz type theorem with similar applications is developed for hypergeometric functions extensions are obtained where the power series are terminated after 2 n 1 terms and larger zero free intervals 2 n v are found

asymptotics for a resonance counting function for potential scattering on cylinders abstract we study certain resonance counting functions for potential scattering on infinite cylinders or half cylinders under certain conditions on the potential we obtain asymptotics of the counting functions with an explicit formula for the constant appearing in the leading term

sharp bounds on the number of scattering poles for perturbations of the laplacian sharp bounds on the numbern r of the scattering poles in the disc z r for a large class of compactly supported perturbations not necessarily selfadjoint of the laplacian in n n 3 odd are obtained in particular in the elliptic case the estimaten r cr n c is proved

a weighted bivariate blending rational interpolation based on function values abstract this paper presents a new weighted bivariate blending rational spline interpolation based on function values this spline interpolation has the following advantages firstly it can modify the shape of the interpolating surface by changing the parameters under the condition that the values of the interpolating nodes are fixed secondly the interpolating function is c 1 continuous for any positive parameters thirdly the interpolating function has a simple and explicit mathematical representation fourthly the interpolating function only depends on the values of the function being interpolated so the computation is simple in addition this paper discusses some properties of the interpolating function such as the bases of the interpolating function the matrix representation the bounded property the error between the interpolating function and the function being interpolated