structures of s matrices for three body schrodinger operators structures of the s matrix associated with the collision process from 2 clusters to 3 clusters are studied this s matrix is shown to have a continuous kernel except for 2 dimensional spheres on which 2 body subsystems have zero velocity on these spheres the s matrix has in general singularities whose existence arises from the zero eigenvalues and the zero resonances of the 2 body subsystems

the schrodinger propagator for scattering metrics let g be a scattering metric on a compact manifold x with boundary i e a smooth metric giving the interior x the structure of a complete riemannian manifold with asymptotically conic ends an example is any compactly supported perturbation of the standard metric on rn consider the operator h 1 2 v where is the positive laplacian with respect to g and v is a smooth real valued function on x vanishing to second order at iox assuming that g is nontrapping we construct a global parametrix u z w t for the kernel of the schri odinger propagator u t e ith where z w io x and t 0 the parametrix is such that the difference between u and u is smooth and rapidly decreasing both as t iae 0 and as z iae iox uniformly for w on compact subsets of x let r x 1 where x is a boundary defining function for x be an asymptotic radial variable and let w t be the kernel e ir2 2tu t using the parametrix we show that w t belongs to a class of i legendre distributionsi on x i x i r 0 previously considered by hassell vasy when the metric is trapping then the parametrix construction goes through microlocally in the nontrapping part of the phase space we apply this result to determine the singularities of u t f for any tempered distribution f and for any fixed t 0 in terms of the oscillation of f near iox if the metric is nontrapping then we precisely determine the wavefront set of u t f and hence also precisely determine its singular support more generally we are able to determine the wavefront set of u t f for t 0 resp t 0 on the non backward trapped resp non forward trapped subset of the phase space this generalizes results of craig kappeler strauss and wunsch

weyl asymptotics for the laplacian on asymptotically euclidean spaces we prove a trace type formula for the laplacian on an asymptotically euclidean space and use this to obtain weyl asymptotics of the scattering phase other results include an explicit calculation of the leading order singularity of the scattering matrix and results on the behavior of the resolvent and scattering matrix near 0

notes on the wave equation on asymptotically euclidean manifolds abstract the primary object of this paper is to discuss the asymptotics of solutions of the wave equation on an asymptotically euclidean manifold when the initial data have compact support by going over to an appropriate conformal metric it is shown that just as for the ordinary wave equation such a solution has a forward future and a backward past radiation field the same method is then used to define end points of bicharacteristics that begin and end above the boundary the analogue of the sphere at infinity in the euclidean case and to derive a relation between the wave front sets of the two radiation fields the extension of these results to solutions with finite energy is also briefly discussed

scattering metrics and geodesic flow at infinity any compact manifold with boundary admits a riemann metric on its interior taking the form x 4 dx 2 x 2 h near the boundary where x is a boundary defining function and h is a smooth symmetric 2 cotensor restricting to be positive definite and hence a metric h on the boundary the scattering theory associated to the laplacian for such a scattering metric was discussed by the first author and here it is shown as conjectured that the scattering matrix is a fourier integral operator which quantizes the geodesic flow on the boundary for the metric h at time to prove this the poisson operator of the associated generalized boundary problem is constructed as a fourier integral operator associated to a singular legendre manifold

scattering matrices in many body scattering in this paper we show that generalized eigenfunctions of many body hamiltonians h with short range two body interactions have distributional asymptotics at non threshold channels the leading terms of the asymptotics can be used to define a scattering matrix and we show that this is the same up to normalization as that arising from the standard wave operator approach we also prove the existence of local distributional asymptotics for locally approximate generalized eigenfunctions in the more general setting of short range perturbations of a scattering metric defined by melrose in 13

haro strait geometry sloping bottom this paper is a followup to a previous jasa paper which assumed a flat bottom new results include i estimations assuming a sloping bottom where as before the geometric parameters of various examined haro st scenarios are estimated based on the time domain signals alone additionally this new paper considers ii four parts of the time domain signal rather than the previous 2 the time differences for each phone for 1 surface reflected minus direct arrivals 2 bottom reflected minus direct arrivals plus 3 surface then bottom reflected minus direct arrivals plus 4 bottom then surface reflected minus direct arrivals finally iii 12 to 16 phones rather than the previous eight may now be considered as well up to 16 phones are available for some source array paths the determined geometric parameters remain source range source depth individual phone ranges individual phone depths and now include two water depths at the source and array per scenario we have retained constraints

comments on comparative analysis of acceleration techniques for 2 d and 3 d green s functions in periodic structures along one and two directions the paper by valerio et al 2007 contains a lot of useful information on the problem of efficient calculation of the periodic green s function however making their review of the existing methods the authors have missed one approach which in our opinion deserves attention as well that approach was originally proposed by ivanishin 2002 for calculating the green s function of a rectangular waveguide later it was generalized and successfully applied in skobelev and mukhamedov 1993 and skobelev 2002 for solving 2d and 3d problems associated with analysis of 1d and 2d periodic arrays respectively

perfectly matched layer truncation for parabolic wave equation models solutions of the parabolic wave equation are extended to complex values of the height variable the fast decay of the solutions in certain directions of the complex plane is used to improve parabolic wave equation solvers the solution is marched on a suitably chosen contour in the complex plane rather than on the real line this interpretation of berenger s perfectly matched layer allows extremely efficient truncation of the computational domain with a split step fourier algorithm the theory is developed for both the wide angle and narrow angle parabolic wave equations and numerical results are presented

distribution of resonances for asymptotically euclidean manifolds in this paper we discuss meromorphic continuation of the resolvent and bounds on the number of resonances for scattering manifolds a class of manifolds generalizing euclidian n space subject to the basic assumption of analyticity near infinity we show that resolvent of the laplacian has a meromorphic continuation to a conic neighborhood of the continuous spectrum this involves a geometric interpretation of the complex scaling method in terms of deformations in the grauert tube of the manifold we then show that the number of resonances poles of the meromorphic continuation of the resolvent in a conic neighborhood of mathbb r of absolute value less than r 2 is mathcal o r n under the stronger assumption of global analyticity and hyperbolicity of the geodesic flow we prove a finer weyl type upper bound for the counting function for resonances in small neighborhoods of the real axis this estimate has an exponent which involves the dimension of the trapped set of the geodesic flow

symbolic functional calculus and n body resolvent estimates abstract we construct a functional calculus for symbolic functions of scattering pseudodifferential operators on manifolds with boundary this is applied to obtain mapping results on the scattering wavefront set of the b operator used by gerard et al 1994 commutator algebra and resolvent estimates advanced studies in pure mathematics vol 23 math soc japan tokyo to obtain n body resolvent estimates as an application we construct the plane waves for an n body short range schrodinger operator and obtain a scattering wavefront set estimate

matched field inversion of broadband data using the freeze bath method this paper presents a statistical freeze bath method for geoacoustic inversion that emphasizes the search for a distribution of models that fit the data well contrary to simulated annealing optimization the freeze bath samples the multidimensional model parameter space at constant freeze probability corresponding to fixed temperatures for each parameter the sampling process uses a heat bath algorithm as a boltzmann sampling tool to carry out a global search over the model parameter space the conventional heat bath algorithm is modified to sample on a fuzzy grid in order to access the entire range of parameter values the inversion provides a set of good models that indicates how well the model parameters are constrained by the data and reveals the degree of correlation between parameters the efficiency of the search process is improved by reparameterizing the original model parameters to a new set based on the eigenvectors of the model covariance matrix the inversion performance of the freeze bath

reply to comments on comparative analysis of acceleration techniques for 2 d and 3 d green s functions in periodic structures along one and two directions the authors wish to thank dr skobelev for his interesting remark about the acceleration technique for the three dimensional 3 d green s function of a two dimensional 2 d array first proposed by ivanishin it basically consists of a kummer s decomposition of higher order with respect to those presented the possibility of obtaining similar decompositions is mentioned where details are shown only about the lowest order extraction and it has also been recalled in our paper in connection with the first presented geometry 2 d green s function of a one dimensional array

a deterministic non stochastic low frequency method for geoacoustic inversion it is well known that multiple frequency sources are necessary for accurate geoacoustic inversion this paper presents an inversion method which uses the low frequency lf spectrum only to estimate bottom properties even in the presence of expected errors in source location phone depths and ocean sound speed profiles matched field processing mfp along a vertical array is used the lf method first conducts an exhaustive search of the five parameter search space sediment thickness sound speed at the top of the sediment layer the sediment layer sound speed gradient the half space sound speed and water depth at 25 hz and continues by retaining only the high mfp value parameter combinations next frequency is slowly increased while again retaining only the high value combinations at each stage of the process only those parameter combinations which give high mfp values at all previous lf predictions are considered an ever shrinking set it is important to note that a complete search of each relevant parameter space seems to be necessary not only at multiple sequential frequencies but also at multiple ranges in order to eliminate sidelobes i e false solutions even so there are no mathematical guarantees that one final unique solution will be found

spectral and scattering theory for symbolic potentials of order zero the spectral and scattering theory is investigated for a generalization to scattering metrics on two dimensional compact manifolds with boundary of the class of smooth potentials on r2 which are homogeneous of degree zero near infinity the most complete results require the additional assumption that the restriction of the potential to the circle s at infinity be morse generalized eigenfunctions associated to the essential spectrum at non critical energies are shown to originate both at minima and maxima although the latter are not germane to the l2 spectral theory asymptotic completeness is shown both in the traditional l2 sense and in the sense of tempered distributions this leads to a definition of the scattering matrix the structure of which will be described in a future publication

generalization of the split step pade solution the split step pade solution j acoust soc am 93 1736 1742 1993 is generalized to simultaneously permit arbitrarily large range steps in range independent regions and dense sampling in range although the range step size is limited by the rate of range dependence range steps of many wavelengths are possible for many range dependent ocean acoustics problems at each range point in a dense grid the acoustic pressure is defined over depth in terms of a linear combination of solutions of tridiagonal systems on a multiprocessor computer each of the tridiagonal systems is solved on a different processor and the acoustic field can be generated throughout a large range independent region in the time required by standard finite difference solutions to take one range step the generalized split step pade solution is also applicable to elastic media

asymptotic properties of solutions to 3 particle schrodinger equations we construct a generalized fourier transformation f associated with the 3 body schrodinger operator h a v a x a and characterize all solutions of h u 0 in the agmon hormander space b as the image of f these stationary solutions admit asymptotic expansions in b in terms of spherical waves associated with scattering channels

dispersive waves and pml performance for large stencil differencing in the parabolic equation the discrete reflection absorption performance of perfectly matched layer pml boundary conditions is a function of the field discretization scheme throughout the computational domain and the form of the complex coordinate extension that characterizes the pml region most previous work analyzing pml reflection performance has assumed use of low order second or fourth small stencil three or five point spatial discretizations whose dispersion is poor outside a narrow low wavenumber band large stencil differencing schemes regardless of order are easily created with physically realistic wide band dispersive characteristics but the effect their adoption has on pml performance has not been analyzed previously in the context of a semi discrete approximation to the parabolic paraxial equation this work derives the analytic form of the discrete pml reflection coefficient for an arbitrary order arbitrary stencil interior differencing scheme coupled to an arbitrary order arbitrary stencil pml region this result is then used to show that the absorption performance of the pml regions is largely insensitive to changes in interior differencing as a consequence the benefits of superior dispersion characteristics provided by use of large stencil differencing in the interior physical domain can be had without incurring discrete reflection penalties at the pml

a modification of the kummer s method for efficient computation of the 2 d and 3 d green s functions for 1 d periodic structures an auxiliary function in the form of standing spherical waves with attenuation is proposed in the kummer s method for accelerating the convergence of the spectral series representing the green s function of doubly periodic structures in free space expressions for the amplitude and phase constants of the auxiliary waves versus their attenuation constant are derived at which the spectral difference series formed as a result of the kummer s transformation converges in the worst case as the floquet mode propagation constant in the power of minus 5 9 and 13 for the cases of using one two and three auxiliary waves respectively while the spatial series formed by the auxiliary waves converges exponentially some examples allowing determination of optimum values for the attenuation constant are considered and some comparative results characterizing the effectiveness of the technique proposed are presented and discussed

scattering theory for riemannian laplacians we introduce a notion of scattering theory for the laplace beltrami operator on non compact connected and complete riemannian manifolds a principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity another condition is certain bounds of derivatives up to order one of the trace of this quantity these conditions are shown to be optimal for existence and completeness of a wave operator our theory does not involve prescribed asymptotic behavior of the metric at infinity like asymptotic euclidean or hyperbolic metrics studied previously in the literature a consequence of the theory is spectral theory for the laplace beltrami operator including identification of the continuous spectrum and absence of singular continuous spectrum

approximation and equidistribution of phase shifts spherical symmetry consider a semiclassical hamiltonian h v h h 2 delta v e where h 0 is a semiclassical parameter is the positive laplacian on mathbb r d v is a smooth compactly supported central potential function and e 0 is an energy level in this setting the scattering matrix sh e is a unitary operator on l 2 mathbb s d 1 hence with spectrum lying on the unit circle moreover the spectrum is discrete except at 1

semiclassical estimates in asymptotically euclidean scattering we consider long range semiclassical perturbations of the laplacian on asymptotically euclidean manifolds we obtain precise resolvent estimates under non trapping assumptions the novelty lies in a systematic use of geometric microlocal methods

a variational data assimilation system for the range dependent acoustic model using the representer method theoretical derivations this study presents the theoretical framework for variational data assimilation of acoustic pressure observations into an acoustic propagation model namely the range dependent acoustic model ram ram uses the split step pade algorithm to solve the parabolic equation the assimilation consists of minimizing a weighted least squares cost function that includes discrepancies between the model solution and the observations the minimization process which uses the principle of variations requires the derivation of the tangent linear and adjoint models of the ram the mathematical derivations are presented here and for the sake of brevity a companion study presents the numerical implementation and results from the assimilation simulated acoustic pressure observations

propagation estimates for n body schroedinger operators we prove propagation estimates of strong type for long rangen body hamiltonians emphasis is on phase space analysis in the free channel region

greene approximation wide angle parabolic equation for radio propagation this paper presents the greene second order parabolic equation pe solution to model radio propagation over irregular terrain this solution is implemented by finite difference and shift map technique the second order finite difference approach performs well for slopes up to about 15 and discontinuous slope changes up to about 30 which is better than can be achieved using the split step fourier approach the results prove that the greene pe solution has greater propagation angles than does the claerbout pe solution thus the greene second order solution can certainly give better results for large angle propagation over complicated terrain boundaries besides we treat the important problem of accuracy of the different approximations of the pe based propagation models and derive the general pe solutions with respect to the terrain slope for the different approximations it is useful to have collected in one paper the most important pe approximations for radio propagation

the wave equation on asymptotically de sitter like spaces in this paper we obtain the asymptotic behavior of solutions of the klein gordon equation on lorentzian manifolds x g which are de sitter like at infinity such manifolds are lorentzian analogues of the so called riemannian conformally compact or asymptotically hyperbolic spaces under global assumptions on the null bicharacteristic flow namely that the boundary of the compactification x is a union of two disjoint manifolds y and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to and to the other manifold as the parameter goes to we also define the scattering operator and show that it is a fourier integral operator associated to the bicharacteristic flow from y to y

analysis of waveguide antenna arrays with protruding dielectric elements the two dimensional problem of radiation of te and tm waves from a waveguide array with protruding smooth dielectric elements of arbitrary shape is considered and solution algorithms are suggested the algorithms are based on applying the method of auxiliary sources for the representation of electromagnetic fields outside and inside the protrusions in combination with the method of integral equations for the electric field at the waveguide aperture the point matching of the field tangential components on the protrusion to free space boundary and at the waveguide aperture is used to reduce the problem to a system of linear algebraic equations for the amplitudes of the auxiliary filamentary currents and of the waveguide aperture electric field which is assumed to be piecewise constant the amplitudes obtained from the solution of the system are used for computing the array reflection coefficient and element pattern which are shown in some cases to be significantly dependent on the protrusion shape examples of arrays with flat topped element patterns resulting from array geometry numerical optimization are also presented

propagation of singularities in many body scattering in the presence of bound states abstract in this paper we describe the propagation of singularities of tempered distributional solutions u s of h u 0 where h is a many body hamiltonian h v 0 v a v a and is not a threshold of h under the assumption that the inter particle e g two body interactions v a are real valued polyhomogeneous symbols of order 1 e g coulomb type with the singularity at the origin removed here the term singularity refers to a microlocal description of the lack of decay at infinity thus we prove that the set of singularities of u is a union of maximally extended broken bicharacteristics of h these are curves in the characteristic variety of h which can be quite complicated due to the existence of bound states we use this result to describe the wave front relation of the s matrices we also analyze lagrangian properties of this relation which shows that the relation is not too large in terms of its dimension

relative time delay for perturbations of elliptic operators and semiclassical asymptotics the main goal of this paper is to compare two long range perturbations of constant coefficient operators on rn such that their difference is short range typical examples are schrodinger hamiltonians hj h2 vj with vj x o x v2 x v1 x o x with 0 n we can also consider perturbations by magnetic fields and cases where is the laplace beltrami operator for asymptotically flat metrics on rn for the scattering pair h2 h1 the average time delay d h depending on the energy and the parameter h is well defined it is related to the spectral shift function and also to the scattering phase under suitable assumptions we prove asymptotic results on d as high energy regime and as h 0 semi classical regime

asymptotic behavior of generalized eigenfunctions inn body scattering abstract in this paper an asymptotic expansion is proved for locally at infinity outgoing functions on asymptotically euclidian spaces this is applied to n body scattering where the two body interactions are one step polyhomogeneous symbols of order 1 or 2 hence long range and short range respectively the asymptotic behavior of the n body resolvent applied to schwartz functions is thereby deduced away from the singular set where some of the potentials do not decay at infinity

complex pade approximants for wide angle acoustic propagators procedures for constructing wide angle propagators that attenuate evanescent modes in parabolic equation pe models of underwater sound propagation are examined in this context two new split step pade approximations to exponentiated square root operators are proposed the first method involves a propagator derived from a pade approximant of a square root operator that has been rotated in the complex plane while the propagator of the second method is obtained from a complex coefficient rational approximation of the square root operator numerical evidence confirms that these proposed methods yield improved behavior compared to previously suggested techniques in selected areas of the spectral domain

asymptotic properties of generalized eigenfunctions for three body schrodinger operators we study the spatial asymptotics of generalized eigenfunctions of three body schrodinger operators and derive all the s matrices with initial state of 2 clusters

fredholm theory for elliptic operators on quasi asymptotically conical spaces we consider the mapping properties of generalized laplace type operators l r on the class of quasi asymptotically conical qac spaces which provide a riemannian generalization of the qale manifolds considered by joyce joy00 our main result gives conditions under which such operators are fredholm when between

a new rough surface parabolic equation program for computing low frequency acoustic forward scattering from the ocean surface the parabolic equation acoustic propagation program ram has been extended to handle a rough air water interface treated as a series of stair steps because much finer vertical spacing may be needed to resolve the interface than to propagate a low frequency acoustic wave accurately in the rest of the domain an option to use one vertical spacing near the surface and a coarser one in the rest of the ocean has been implemented the necessary galerkin approximations of derivatives on an unequally spaced grid have been worked out using computer algebra the new program rrsfc is efficient enough to compute the frequency spectrum of the field scattered off a moving surface by treating the surface as a sequence of frozen surfaces a comparison of one such spectrum with actual ocean data is provided in order to make a more detailed assessment of the program both the depth dependent pressure for individual surfaces and the frequency spectrum for sequences of surfaces are compared with the results of numerically exact integral equation calculations for a few constant sound speed test cases

a parametrix for the fundamental solution of the klein gordon equation on asymptotically de sitter spaces abstract in this paper we construct a parametrix for the forward fundamental solution of the wave and klein gordon equations on asymptotically de sitter spaces without caustics we use this parametrix to obtain asymptotic expansions for solutions of u f and to obtain a uniform l p estimate for a family of bump functions traveling to infinity