a spectral mapping theorem for perturbed ornstein uhlenbeck operators on l2 rd abstract we consider ornstein uhlenbeck operators on l 2 r d perturbed by a radial potential v under weak assumptions on v we prove a spectral mapping theorem for the generated semigroup the proof relies on a perturbative construction of the resolvent based on angular separation and the gearhart pruss theorem

on the pseudospectrum of elliptic quadratic differential operators we study the pseudospectrum of a class of non selfadjoint differential operators our work consists in a detailed study of the microlocal properties which rule the spectral stability or instability phenomena appearing under small perturbations for elliptic quadratic differential operators the class of elliptic quadratic differential operators stands for the class of operators defined in the weyl quantization by complex valued elliptic quadratic symbols we establish in this paper a simple necessary and sufficient condition on the weyl symbol of these operators which ensures the stability of their spectra when this condition is violated we prove that it occurs some strong spectral instabilities for the high energies of these operators in some regions which can be far away from their spectra we give a precise geometrical description of them which explains the results obtained for these operators in some numerical simulations giving the computation of false eigenvalues far from their spectra by algorithms for eigenvalues computing

exponential decay of eigenfunctions in many body type scattering with second order perturbations abstract we show the exponential decay of eigenfunctions of second order geometric many body type hamiltonians at non threshold energies moreover in the case of first order and small second order perturbations we show that there are no eigenfunctions with positive energy

wave packet pseudomodes of variable coefficient differential operators the pseudospectra of non selfadjoint linear ordinary differential operators with variable coefficients are considered it is shown that when a certain winding number or twist condition is satisfied

the radiation condition at infinity for the high frequency helmholtz equation with source term a wave packet approach abstract we consider the high frequency helmholtz equation with a given source term and a small absorption parameter 0 the high frequency or semi classical parameter is 0 we let and go to zero simultaneously we assume that the zero energy is non trapping for the underlying classical flow we also assume that the classical trajectories starting from the origin satisfy a transversality condition a generic assumption under these assumptions we prove that the solution u radiates in the outgoing direction uniformly in in particular the function u when conveniently rescaled at the scale close to the origin is shown to converge towards the outgoing solution of the helmholtz equation with coefficients frozen at the origin this provides a uniform version in of the limiting absorption principle writing the resolvent of the helmholtz equation as the integral in time of the associated semi classical schrodinger propagator our analysis relies on the following tools i for very large times we prove and use a uniform version of the egorov theorem to estimate the time integral ii for moderate times we prove a uniform dispersive estimate that relies on a wave packet approach together with the above mentioned transversality condition iii for small times we prove that the semi classical schrodinger operator with variable coefficients has the same dispersive properties as in the constant coefficients case uniformly in

cores for parabolic operators with unbounded coefficients abstract let a i j 1 n q i j s x d i j i 1 n b i s x d i be a family of elliptic differential operators with unbounded coefficients defined in r n 1 in m kunze l lorenzi a lunardi nonautonomous kolmogorov parabolic equations with unbounded coefficients trans amer math soc in press under suitable assumptions it has been proved that the operator g a d s generates a semigroup of positive contractions t p t in l p r n 1 for every 1 p where is an infinitesimally invariant measure of t p t here under some additional conditions on the growth of the coefficients of a which cover also some growths with an exponential rate at we provide two different cores for the infinitesimal generator g p of t p t in l p r n 1 for p 1 and we also give a partial characterization of d g p finally we extend the results so far obtained to the case when the coefficients of the operator a are t periodic with respect to the variable s for some t 0

isotropic hypoellipticity and trend to equilibrium for the fokker planck equation with a high degree potential we consider the fokker planck equation with a confining or anti confining potential which behaves at infinity like a possibly high degree homogeneous function hypoellipticity techniques provide the well posedness of the weak cauchy problem in both cases as well as instantaneous smoothing and exponential trend to equilibrium lower and upper bounds for the rate of convergence to equilibrium are obtained in terms of the lowest positive eigenvalue of the corresponding witten laplacian with detailed applications

spectrum of ornstein uhlenbeck operators in lp spaces with respect to invariant measures abstract let a i j 1 n q ij d ij i j 1 n b ij x j d i be a possibly degenerate ornstein uhlenbeck operator in r n and assume that the associated markov semigroup has an invariant measure we compute the spectrum of a in l p for 1 p

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

equivariant e theory for groupoids acting on c algebras we define equivariant asymptotic morphisms between c0 x algebras and construct the category of homotopy classes of such morphisms based on this category we construct the category of homotopy classes of asymptotic morphisms which are equivariant with respect to an action of a groupoid we then define and study the corresponding equivariant e theory

conormal distributions in the shubin calculus of pseudodifferential operators we characterize the schwartz kernels of pseudodifferential operators of shubin type by means of a fourier bros iagolnitzer transform based on this we introduce as a generalization a new class of tempered distributions called shubin conormal distributions we study their transformation behavior normal forms and microlocal properties

existence uniqueness and invariant measures for stochastic semilinear equations on hilbert spaces a class of stochastic evolution equations with additive noise and weakly continuous drift is considered first regularity properties of the corresponding ornstein uhlenbeck transition semigrouprt are obtained we show thatrt is a compactc0 semigroup in all sobolev spaceswn p which are built on its invariant measure then we show the existence uniqueness compactness and smoothing properties of the transition semigroup for semilinear equations inlp spaces and spacesw1 p as a consequence we prove the uniquencess of martingale solutions to the stochastic equation and the existence of a unique invariant measure equivalent to it is shown also that the density of this measure with respect to is inlp for allp 1

wave packet pseudomodes of twisted toeplitz matrices the pseudospectra of banded nonsymmetric toeplitz or circulant matrices with varying coefficients are considered such matrices are characterized by a symbol that depends on both position x and wave number k it is shown that when a certain winding number or twist condition is satisfied related to hormander s commutator condition for partial differential equations e pseudoeigenvectors of such matrices for exponentially small values of e exist in the form of localized wave packets the symbol need not be smooth with respect to x just differentiable at a point or less c 2004 wiley periodicals inc

elliptic problems on ℝ n with unbounded coefficients in classical sobolev spaces we discuss the existence and uniqueness in h1 n and the h2 n regularity of the solutions of au f when f l2 n and a is a second order linear elliptic operator whose first and zeroth order coefficients may be unbounded at infinity we also investigate whether a generates a c0 or analytic semigroup on l2 the approach in this nonweighted setting is based on a new and general method the idea consists in embedding a into a suitable one parameter family of operators as s with a0 a the properties of as when s 0 make it possible to prove that the boundary integrals arising from simple integration by parts over balls with increasing radius tend to 0 at infinity this provides the needed estimates for uniqueness and regularity

metaplectic representation on wiener amalgam spaces and applications to the schrodinger equation we study the action of metaplectic operators on wiener amalgam spaces giving upper bounds for their norms as an application we obtain new fixed time estimates in these spaces for schrodinger equations with general quadratic hamiltonians and strichartz estimates for the schrodinger equation with potentials v x x 2

the navier stokes equations in ℝn with linearly growing initial data consider the equations of navier stokes on n with initial data u0 of the form u0 x u0 x mx where m is an n n matrix with constant real entries and u0 lp n it is shown that under these assumptions the equations of navier stokes admit a unique local solution in lp n moreover if etm 1 for all t 0 then this mild solution is even analytic in x this is surprising since the underlying semigroup of ornstein uhlenbeck type is not analytic in contrast to the stokes semigroup

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

on the ornstein uhlenbeck operator in spaces of continuous functions we study the realization of the ornstein uhlenbeck operator a in the space of the uniformly continuous and bounded functions in rn we prove that it generates a semigroup which is neither strongly continuous nor analytic but enjoys nice smoothing properties such properties are used to prove a schauder type theorem for the elliptic equation u au f

acoustic field induced in spheres with inhomogeneous density by external sources acoustic or electromagnetic fields induced in the interior of inhomogeneous penetrable bodies by external sources can be evaluated via well known volume integral equations for bodies of arbitrary shape and or composition for which separation of variables fails a direct attack for the solution of these integral equations is the only available approach in a previous paper by the same authors the scalar acoustic field in inhomogeneous spheres of arbitrary compressibility but with constant density was considered in the present one the direct hybrid analytical numerical method applied to the much simpler integral equation for spheres with constant density is generalized to densities that vary with r or even this extension is by no means trivial owing to the appearance of the derivatives of both the density and the unknown function in the volume integral a fact necessitating a more subtle and accuracy sensitive approach again the spherical shape allows use of the orthogonal spherical harmonics and of dini s expansions of a general type for the radial functions the convergence of the latter shown to be superior to other possible sets of orthogonal expansions can be further optimized by the proper selection of a crucial parameter in their eigenvalue equation

on the essential spectrum of certain non commutative oscillators we show here that the spectrum of the family of non commutative harmonic oscillators q w x d for r in the range 1 is 0 and is entirely essential spectrum the previous existing results concern the case 1 case in which q w x d is globally elliptic with a discrete spectrum whose qualitative properties are being extensively studied and ours therefore extend the picture to the range of parameters 1

time domain green s function for an infinite sequentially excited periodic planar array of dipoles the present paper is a continuation of previous explorations by the authors aimed at gaining a basic understanding of the time domain td behavior of large periodic phased i e sequentially turned on array antennas and related configurations our systematic investigation of the relevant canonical td dipole excited green s functions has so far included those for infinite and truncated sequentially pulsed line periodic arrays parameterized in terms of radiating propagating and nonradiating evanescent conical td floquet waves fw and truncation induced td fw modulated tip diffractions the present contribution extends these investigations to an infinite periodic sequentially pulsed planar array which generates pulsed plane propagating and evanescent fw starting from the familiar frequency domain fd transformation of the linearly phased element by element summation synthesis into summations of propagating and evanescent fws we access the time domain by fourier inversion the inversion integrals are manipulated in a unified fashion into exact closed forms which are parameterized by the single nondimensional quantity spl eta c v sup p sub u1 where v sup p sub u1 and c are the excitation phase speed along a preferred phasing direction u sub 1 in the array plane and the ambient wave speed respectively the present study deals with the practically relevant rapidly phased propagating case spl eta 1 regime for a future manuscript numerical reference data generated via element by element summation over the fields radiated by the individual dipoles with ultrawide band limited excitation are compared with results obtained much more efficiently by inclusion of a few td fws physical interpretation of the formal td fw solutions is obtained by recourse to asymptotics instantaneous frequencies and wavenumbers and related constructs of special interest is the demonstration that the td fws emerge along equal delay ellipses from the array plane this furnishes a novel and physically appealing interpretation of the planar array td fw phenomenology

subelliptic estimates for quadratic differential operators we prove global subelliptic estimates for quadratic differential operators quadratic differential operators are operators defined in the weyl quantization by complex valued quadratic symbols in a previous joint work with m hitrik we pointed out the existence of a particular linear subvector space in the phase space intrinsically associated to their weyl symbols called singular space which rules spectral properties of non elliptic quadratic operators the purpose of the present paper is to prove that quadratic operators whose singular spaces are reduced to zero are subelliptic with a loss of derivatives depending directly on particular algebraic properties of the hamilton maps of their weyl symbols more generally when singular spaces are symplectic spaces we prove that quadratic operators are subelliptic in any direction of the symplectic orthogonal complements of their singular spaces

an acoustic inverse scattering problem for spheres with radially inhomogeneous compressibility an inverse acoustic scattering problem the main aim of which is to reconstruct the one dimensional variation of the acoustical parameters of a spherical object is investigated the problem is first formulated conventionally through a coupled system of integral equations and then this system is reduced to one dimensional form by using the orthogonality properties of spherical harmonics the inverse problem is solved in an iterative fashion via classical newton algorithm some numerical simulations are carried out to test the feasibility of the method as well as to see the effects of some parameters on the solution it is shown that the method is very effective for the profiles having smooth variations provided that an appropriate initial guess is chosen however some of the classical disadvantages of the newton type algorithms are also observed in numerical experiments which may limit the applicability of the method to a certain extent

a recursive green s function technique for acoustic scattering from heterogeneous objects a fast efficient algorithm for computing acoustic fields scattered by inhomogeneous objects in an otherwise homogeneous space is presented the algorithm called the recursive green s function method rgfm constructs the domain green s function by recursively combining known green s functions from smaller subdomains the fields on the scatterer surface are then computed using a boundary integral formulation proper implementation of the rgfm results in a storage requirement of o n and computational costs of o n3 2 and o n2 for two and three dimensional problems respectively where n is the total number of discrete points in the inhomogeneous region results are compared with those obtained from exact solutions to show the accuracy of the method

numerical linear algebra and solvability of partial differential equations it was observed long ago that the obstruction to the accurate computation of eigenvalues of large non self adjoint matrices is inherent in the problem the basic idea is that the resolvent of a highly non normal operator can be very large far away from the spectrum this leads to an easily observable fact that algorithms for locating eigenvalues will typically find some false eigenvalues these false eigenvalues also explain one of the most surprising phenomena in linear pdes namely the fact discovered by hans lewy in 1957 in berkeley that one cannot always locally solve the pde p u f almost immediately after that discovery hormander provided an explanation of lewy s example showing that almost all operators with non constanct complex valued coefficients are not locally solvable in modern language that was done by considering the essentially dual problem of existence of non propagating singularities the purpose of this article is to review this work in the context of almost eigenvalues and from the point of view of semi classical analysis

a general result about the pseudo spectrum of schrodinger operators for schrdinger operators with complexvalued potentials we give a sufficient geometrical condition on potentials for the existence of pseudospectra we construct some approximate semiclassical mode

optimal non reversible linear drift for the convergence to equilibrium of a diffusion we consider non reversible perturbations of reversible diffusions that do not alter the invariant distribution and we ask whether there exists an optimal perturbation such that the rate of convergence to equilibrium is maximized we solve this problem for the case of linear drift by proving the existence of such optimal perturbations and by providing an easily implementable algorithm for constructing them we discuss in particular the role of the prefactor in the exponential convergence estimate our rigorous results are illustrated by numerical experiments

electromagnetic scattering by an inhomogeneous gyroelectric sphere using volume integral equation and orthogonal dini type basis functions the electromagnetic scattering by an inhomogeneous gyroelectric sphere is studied in this work the solution is constructed using the well known electric field volume integral equation efvie the kernel that incorporates the green s function is expanded in spherical vector wave functions and the unknown electric field and electric displacement is expanded in spherical vector wave functions using fully orthogonal dini type basis functions the permittivity tensor of the spherical object supports a continuously varying radial inhomogeneity while its permeability is that of free space this approach allows a fully analytical computation of the unknown matrix coefficients in special cases of homogeneous permittivity tensors the validation of our method is performed by comparisons with published results in the literature new numerical results are given for various continuously varying permittivity tensors

a unified approach to acoustic fields induced in inhomogeneous spheres by external sources abstract the evaluation of acoustic fields induced in inhomogeneous spheres by external sources based on volume integral equations is simplified considerably using ivakin s integral equation and the use of the well known expansion for the green dyadic the method presented here overcomes the difficulties of calculation arising from the possible step discontinuities in the density allowing a quick and accurate evaluation of the fields so the simple hybrid analytical numerical scalar method developed previously for spheres with variable even discontinuous compressibility is easily extended to the general case of inhomogeneous density in a manner requiring the least possible analytical and computational effort

semigroup expansions for non selfadjoint schrodinger operators abstract a large time expansion for the propagator associated to a semiclassical non selfadjoint magnetic schrodinger operator is established in terms of the low lying eigenvalues of the operator

the computation of resonances in open systems using a perfectly matched layer in this paper we consider the problem of computing resonances in open systems we first characterize resonances in terms of improper eigen functions of the helmholtz operator on an unbounded domain the perfectly matched layer pml technique has been successfully applied to the computation of scattering problems we shall see that the application of pml converts the resonance problem to a standard eigenvalue problem still on an infinite domain this new eigenvalue problem involves an operator which resembles the original helmholtz equation transformed by a complex shift in the coordinate system our goal will be to approximate the shifted operator first by replacing the infinite domain by a finite computational domain with a convenient boundary condition and second by applying finite elements on the computational domain we shall prove that the first of these steps leads to eigenvalue convergence to the desired resonance values which is free from spurious computational eigenvalues provided that the size of computational domain is sufficiently large the analysis of the second step is classical finally we illustrate the behavior of the method applied to numerical experiments in one and two spatial dimensions

pseudospectra of the orr sommerfeld operator this paper investigates the pseudospectra and the numerical range of the orr sommerfeld operator for plane poiseuille flow a number z in bf c is in the epsilon pseudospectrum of a matrix or operator a if zi a 1 geq epsilon 1 or equivalently if z is in the spectrum of a e for some perturbation e satisfying e leq epsilon the numerical range of a is the set of numbers of the form au u where cdot cdot is the inner product and u is a vector or function with huh u 1 the spectrum of the orr sommerfeld operator consists of three branches it is shown that the eigenvalues at the intersection of the branches are highly sensitive to perturbations and that the sensitivity increases dramatically with the reynolds number the associated eigenfunctions are nearly linearly dependent even though they form a complete set to understand the high sensitivity of the eigenvalues a model operator is considered related to the airy equation tha

attaining the optimal gaussian diffusion acceleration sampling from probability distributions in high dimensional spaces is generally impractical diffusion processes with invariant equilibrium distributions can be used as a means to generate approximations an important task in such an endeavor is to design an equilibrium preserving drift to accelerate the convergence starting from a reversible diffusion it is desirable to depart for non reversible dynamics via a perturbed drift so that the convergence rate is maximized with the common equilibrium in the gaussian diffusion acceleration this problem can be cast as perturbing the inverse of a given covariance matrix by skew symmetric matrices so that all resulting eigenvalues have identical real part this paper describes two approaches to obtain the optimal rate of gaussian diffusion the asymptotical approach works universally for arbitrary ornstein uhlenbeck processes whereas the direct approach can be implemented as a fast divide and conquer algorithm a comparison with recently proposed lelievre nier pavliotis algorithm is made

pseudospectra of linear operators if a matrix or linear operator a is far from normal its eigenvalues or more generally its spectrum may have little to do with its behavior as measured by quantities such as an or exp ta more may be learned by examining the sets in the complex plane known as the pseudospectra of a defined by level curves of the norm of the resolvent zi a 1 five years ago the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications since that time analogous computations have been carried out for differential and integral operators this paper a companion to the earlier one presents ten examples each chosen to illustrate one or more mathematical or physical principles

on the rates of decay to equilibrium in degenerate and defective fokker planck equations abstract we establish sharp long time asymptotic behaviour for a family of entropies to defective fokker planck equations and show that much like defective finite dimensional odes their decay rate is an exponential multiplied by a polynomial in time the novelty of our study lies in the amalgamation of spectral theory and a quantitative non symmetric hypercontractivity result as opposed to the usual approach of the entropy method

wkb analysis of bohmian dynamics we consider a semiclassically scaled schrodinger equation with wkb initial data we prove that in the classical limit the corresponding bohmian trajectories converge locally in measure to the classical trajectories before the appearance of the first caustic in a second step we show that after caustic onset this convergence in general no longer holds in addition we provide numerical simulations of the bohmian trajectories in the semiclassical regime that illustrate the above results 2014 wiley periodicals inc

spectra and semigroup smoothing for non elliptic quadratic operators we study non elliptic quadratic differential operators quadratic differential operators are non selfadjoint operators defined in the weyl quantization by complex valued quadratic symbols when the real part of their weyl symbols is a non positive quadratic form we point out the existence of a particular linear subspace in the phase space intrinsically associated to their weyl symbols called a singular space such that when the singular space has a symplectic structure the associated heat semigroup is smoothing in every direction of its symplectic orthogonal space when the weyl symbol of such an operator is elliptic on the singular space this space is always symplectic and we prove that the spectrum of the operator is discrete and can be described as in the case of global ellipticity we also describe the large time behavior of contraction semigroups generated by these operators

on the pointwise convergence of the integral kernels in the feynman trotter formula we study path integrals in the trotter type form for the schrodinger equation where the hamiltonian is the weyl quantization of a real valued quadratic form perturbed by a potential v in a class encompassing that considered by albeverio and ito in celebrated papers of fourier transforms of complex measures essentially v is bounded and has the regularity of a function whose fourier transform is in l 1 whereas the strong convergence in l 2 in the trotter formula as well as several related issues at the operator norm level are well understood the original feynman s idea concerned the subtler and widely open problem of the pointwise convergence of the corresponding probability amplitudes that are the integral kernels of the approximation operators we prove that for the above class of potentials such a convergence at the level of the integral kernels in fact occurs uniformly on compact subsets and for every fixed time except for certain exceptional time values for which the kernels are in general just distributions actually theorems are stated for potentials in several function spaces arising in harmonic analysis with corresponding convergence results proofs rely on banach algebras techniques for pseudo differential operators acting on such function spaces

network oriented dyadic time domain green s function for a sequentially excited infinite planar array of dipoles in free space during the past few years we have performed an effective wideband analysis for characterizing the electrodynamic behavior of phased array antennas infinite periodic structures frequency selective surfaces and related applications with emphasis on gaining physical insight into the phenomenology of short pulse radiation the present contribution shows the current status of our network oriented dyadic time domain green s function td gf for a planar array of sequentially excited dipoles that constitutes a prototype study of sequentially short pulsed radiation by infinite periodic arrays the dispersive effects of the td floquet waves fw and the consequences of a te and tm field decomposition are discussed in details in this network formulation the equivalent td transmission line tl voltages and currents excited by equivalent td current generators are determined an expression for the tm voltage tl gf is provided for the first time in terms of incomplete lipschitz hankel integrals and bessel functions together with some remarks for its evaluation as shown in previous publications for the scalar potentials real td physical observables are determined by pairing td floquet waves with positive and negative indices it is shown that the vectorial td radiated field is reconstructed at any time with the superposition of a few td fws

from semigroups to subelliptic estimates for quadratic operators using an approach based on the techniques of fbi transforms we give a new simple proof of the global subelliptic estimates for non selfadjoint non elliptic quadratic differential operators under a natural averaging condition on the weyl symbols of the operators established by the second author the loss of the derivatives in the subelliptic estimates depends directly on algebraic properties of the hamilton maps of the quadratic symbols using the fbi point of view we also give accurate smoothing estimates of gelfand shilov type for the associated heat semigroup in the limit of small times

l1 properties of intrinsic schrodinger semigroups abstract we investigate the l 1 properties of the intrinsic markov semigroup associated with a schrodinger operator on r n which possesses a positive ground state we discover cases for which this semigroup is norm analytic for positive times and others for which the semigroup is norm discontinuous in the strongest possible sense in the case of the harmonic oscillator we show that the generator of the intrinsic semigroup has totally different spectrum depending on whether one works in l 1 r e x 2 dx or l 2 r e x 2 dx in more general cases we show that the equality of the l 1 and l 2 spectrum is closely related to whether the schrodinger semigroup is intrinsically ultracontractive

hardy littlewood sobolev inequalities for a class of non symmetric and non doubling hypoelliptic semigroups in his seminal 1934 paper on brownian motion and the theory of gases kolmogorov introduced a second order evolution equation which displays some challenging features in the opening of his 1967 hypoellipticity paper hormander discussed a general class of degenerate ornstein uhlenbeck operators that includes kolmogorov s as a special case in this note we combine semigroup theory with a nonlocal calculus for these hypoelliptic operators to establish new inequalities of hardy littlewood sobolev type in the situation when the drift matrix has nonnegative trace our work has been influenced by ideas of e stein and varopoulos in the framework of symmetric semigroups one of our objectives is to show that such ideas can be pushed to successfully handle the present degenerate non symmetric setting

em field induced in inhomogeneous dielectric spheres by external sources the electromagnetic field induced in the interior of inhomogeneous dielectric bodies by external sources can be evaluated by solving the well known electric field integrodifferential equation efide for spheres with constant magnetic permeability but variable dielectric constant a direct mainly analytical solution can be used even in case when the inhomogeneity in renders separation of variables inapplicable this approach constitutes a generalization of the hybrid analytical numerical scalar method developed by the authors in two recent papers for the corresponding acoustic scalar field induced in spheres with variable density and or compressibility this extension by no means trivial owing to the vector and integrodifferential nature of the equation is based on field vector expansions using the set of three harmonic surface vectors orthogonal and complete over the surface of the sphere for their angular dependence and dini s expansions of a general type for their radial functions the use of the latter has been shown to be superior to other possible sets of orthogonal expansions and as far as its convergence is concerned it may further be improved by properly choosing a crucial parameter in their eigenvalue equation the restriction to the spherical shape is imposed here to allow use of the well known expansion of green s dyadic in spherical eigenvectors of the vector wave equation

asymptotic behavior in time periodic parabolic problems with unbounded coefficients abstract we study asymptotic behavior in a class of nonautonomous second order parabolic equations with time periodic unbounded coefficients in r r d our results generalize and improve asymptotic behavior results for markov semigroups having an invariant measure we also study spectral properties of the realization of the parabolic operator u a t u u t in suitable l p spaces

generalized mehler semigroups and applications we construct and study generalized mehler semigroups p t t 0 and their associated markov processesm the construction methods for p t t 0 are based on some new purely functional analytic results implying in particular that any strongly continuous semigroup on a hilbert spaceh can be extended to some larger hilbert spacee with the embeddingh e being hilbert schmidt the same analytic extension results are applied to construct strong solutions to stochastic differential equations of typedx t c dw t ax t dt with possibly unbounded linear operatorsa andc onh on a suitably chosen larger spacee for gaussian generalized mehler semigroups p t t 0 with corresponding markov processm the associated non symmetric dirichlet forms ed e are explicitly calculated and a necessary and sufficient condition for path regularity ofm in terms of e d e is proved then using dirichlet form methods it is shown thatm weakly solves the above stochastic differential equation if the state spacee is chosen appropriately finally we discuss the differences between these two methods yielding strong resp weak solutions

generalized ornstein uhlenbeck semigroups littlewood paley stein inequalities and the p a meyer equivalence of norms abstract let be a nondegenerate gaussian measure on a hilbert space h for an arbitrary selfadjoint nonnegative operator a we consider the semigroup e tl e ta on l p where stands for the second quantization operator we provide an explicit characterization of the domains of i l m 2 in l p in terms of gaussian sobolev spaces thus extending the p a meyer result on equivalence of norms the main tools are the littlewood paley stein inequalities which are proved under minimal assumptions by a purely analytic method following e stein 1970 topics in harmonic analysis princeton univ press princeton nj

periodicity induced generalized fourier transform pair the field radiated by an infinite periodic structure can be expressed in terms of floquet waves fws both in the frequency domain fd and time domain td see felsen l b and capolino f ieee trans antennas propagat vol 48 p 921 31 2000 a new periodicity induced generalized fourier transform ft pair is derived relating fd fws to td fws and vice versa based on tabulated transforms and physical conditions at infinity the new fts are directly related to the simple canonical problem of a line array of sequentially excited dipoles that is a basic building block for more general phased periodic structures

semiclassical non concentration near hyperbolic orbits for a large class of semiclassical pseudodifferential operators including schrodinger operators p h h2 g v x on compact riemannian manifolds we give logarithmic lower bounds on the mass of eigenfunctions outside neighbourhoods of generic closed hyperbolic orbits more precisely we show that if a is a pseudodifferential operator which is microlocally equal to the identity near the hyperbolic orbit and microlocally zero away from the orbit then u c log 1 h h p h u clog 1 h i a u this generalizes earlier estimates of colin de verdiere and parisse y colin de verdiere b parisse equilibre instable en regime semi classique i concentration microlocale comm partial differential equations 19 1994 1535 1563 equilibre instable en regime semi classique ii conditions de bohr sommerfeld ann inst h poincare phys theor 61 1994 347 367 obtained for a special case and of burq and zworski n burq m zworski geometric control in the presence of a black box j amer math soc 17 2004 443 471 for real hyperbolic orbits

global l p estimates for degenerate ornstein uhlenbeck operators we consider a class of degenerate ornstein uhlenbeck operators in mathbb r n of the kind mathcal a equiv sum i j 1 p 0 a ij partial x i x j 2 sum i j 1 n b ij x i partial x j where aij bij are constant matrices aij is symmetric positive definite on mathbb r p 0 p0 n and bij is such that mathcal a is hypoelliptic for this class of operators we prove global lp estimates 1 p of the kind left vert partial x i x j 2 u right vert l p left mathbb r n right leq c left left vert mathcal a u right vert l p left mathbb r n right left vert u right vert l p left mathbb r n right right rm for i j 1 2 ldots p 0 and corresponding weak type 1 1 estimates this result seems to be the first case of global estimates in lebesgue lp spaces for complete hormander s operators sum x i 2 x 0 proved in absence of a structure of homogeneous group we obtain the previous estimates as a byproduct of the following one which is of interest in its own left vert partial x i x j 2 u right vert l p left s right leq c left vert lu right vert l p left s right for any u in c 0 infty left s right where s is the strip mathbb r n times left 1 1 right and l is the kolmogorov fokker planck operator mathcal a partial t to get this estimate we use in a crucial way the left invariance of l with respect to a lie group structure in mathbb r n 1 and some results on singular integrals on nonhomogeneous spaces recently proved in bramanti revista matematica iberoamericana 2009 in press

probabilistic weyl laws for quantized tori for the toeplitz quantization of complex valued functions on a 2n dimensional torus we prove that the expected number of eigenvalues of small random perturbations of a quantized observable satisfies a natural weyl law 1 3 in numerical experiments the same weyl law also holds for false eigenvalues created by pseudospectral effects

computation of pseudospectra by continuation the concept of pseudospectrum was introduced by trefethen to explain the behavior of nonnormal operators many phenomena for example hydrodynamic instability and convergence of iterative methods for linear systems cannot be accounted for by eigenvalue analysis but are more understandable by examining the pseudospectra the straightforward way to compute pseudospectra involves many applications of the singular value decomposition svd this paper presents several fast continuation methods to calculate pseudospectra of different types of matrices

contraction semigroups of elliptic quadratic differential operators we study the contraction semigroups of elliptic quadratic differential operators elliptic quadratic differential operators are the non selfadjoint operators defined in the weyl quantization by complex valued elliptic quadratic symbols we establish in this paper that under the assumption of ellipticity as soon as the real part of their weyl symbols is a non zero non positive quadratic form the norm of contraction semigroups generated by these operators decays exponentially in time

an invariant class of wave packets for the wigner transform abstract generalised hagedorn wave packets appear as exact solutions of schrodinger equations with quadratic possibly complex potential and are given by a polynomial times a gaussian we show that the wigner transform of generalised hagedorn wave packets is a wave packet of the same type in phase space the proofs build on a parametrisation via lagrangian frames and a detailed analysis of the polynomial prefactors including a novel laguerre connection our findings directly imply the recently found tensor product structure of the wigner transform of hagedorn wave packets

field induced in inhomogeneous spheres by external sources i the scalar case the evaluation of acoustic or electromagnetic fields induced in the interior of inhomogeneous penetrable bodies by external sources is based on well known volume integral equations this is particularly true for bodies of arbitrary shape and or composition for which separation of variables fails in this paper the investigation focuses on acoustic scalar fields in inhomogeneous spheres of arbitrary composition i e with r or even dependent medium parameters the volume integral equation is solved by a hybrid analytical numerical method which takes advantage of the orthogonal properties of spherical harmonics and in particular of the so called dini s expansions of the radial functions whose convergence is optimized the numerical part comes at the end it involves the evaluation of certain definite integrals and the matrix inversion for the expansion coefficients of the solution the scalar case treated here serves as a steppingstone for the solution of the more difficult electromagnetic problem

exponential return to equilibrium for hypoelliptic quadratic systems we study the problem of convergence to equilibrium for evolution equations associated to general quadratic operators quadratic operators are non selfadjoint differential operators with complex valued quadratic symbols under appropriate assumptions a complete description of the spectrum of such operators is given and the exponential return to equilibrium with sharp estimates on the rate of convergence is proven some applications to the study of chains of oscillators and to the generalized langevin equation are given

some remarks on degenerate hypoelliptic ornstein uhlenbeck operators abstract we study degenerate hypoelliptic ornstein uhlenbeck operators in l 2 spaces with respect to invariant measures the purpose of this article is to show how recent results on general quadratic operators apply to the study of degenerate hypoelliptic ornstein uhlenbeck operators we first show that some known results about the spectral and subelliptic properties of ornstein uhlenbeck operators may be directly recovered from the general analysis of quadratic operators with zero singular spaces we also provide new resolvent estimates for hypoelliptic ornstein uhlenbeck operators we show in particular that the spectrum of these non selfadjoint operators may be very unstable under small perturbations and that their resolvents can blow up in norm far away from their spectra furthermore we establish sharp resolvent estimates in specific regions of the resolvent set which enable us to prove exponential return to equilibrium

coupled mode solutions in generalized ocean environments a two way coupled mode model has been developed to solve propagation problems in fluid layered range dependent ocean environments the model incorporates piecewise linear depth dependent range varying sound speed profiles svps a fast and efficient normal mode model which is based on a propagator matrix method is used to obtain the eigenvalues and eigenfunctions the coupling coefficients are expressed in terms of depth integrals of products of airy functions and are solved analytically the coupled equations are solved exactly by an approach referred to as the u k method which incorporates the lanczos method previously used in solving similar coupled channel equations in nuclear theory and successfully applied to nuclear response problems in heavy nuclei knobles solutions of coupled mode equations with a large dimension in underwater acoustics j acoust soc am 96 1741 1747 1994 highlights of the formalism will be presented which expand on the overview given previously stotts et al solutions of coupled mode equations with a large dimension in underwater acoustics j acoust soc am 95 2909 a 1994 several benchmark comparisons will be presented including propagation over a hill upslope wedge propagation and an analytic depth and range dependent svp in a flat waveguide

projective hilbert space structures at exceptional points a non hermitian complex symmetric 2 2 matrix toy model is used to study projective hilbert space structures in the vicinity of exceptional points eps the bi orthogonal eigenvectors of a diagonalizable matrix are puiseuxexpanded in terms of the root vectors at the ep it is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behaviour in the ep limit can be resolved by projectively extending the original hilbert space the complementary normalization conditions correspond then to two different affine charts of this enlarged projective hilbert space geometric phase and phase jump behaviour are analysed and the usefulness of the phase rigidity as measure for the distance to ep configurations is demonstrated finally ep related aspects of pt symmetrically extended quantum mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated

diagonalization of bosonic quadratic hamiltonians by bogoliubov transformations abstract we provide general conditions for which bosonic quadratic hamiltonians on fock spaces can be diagonalized by bogoliubov transformations our results cover the case when quantum systems have infinite degrees of freedom and the associated one body kinetic and paring operators are unbounded our sufficient conditions are optimal in the sense that they become necessary when the relevant one body operators commute

levy ornstein uhlenbeck transition semigroup as second quantized operator abstract let be an invariant measure for the transition semigroup p t of the markov family defined by the ornstein uhlenbeck type equation d x a x d t d l on a hilbert space e driven by a levy process l it is shown that for any t 0 p t considered on l 2 is a second quantized operator on a poisson fock space of e at from this representation it follows that the transition semigroup corresponding to the equation on e r driven by an stable noise l 0 2 is neither compact nor symmetric

pseudospectra in non hermitian quantum mechanics we propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non hermitian operators we relate pseudospectral properties to quasi hermiticity similarity to self adjoint operators and basis properties of eigenfunctions the abstract results are illustrated by unexpected wild properties of operators familiar from pt symmetric quantum mechanics

on global existence and trend to the equilibrium for the vlasov poisson fokker planck system with exterior confining potential we prove a global existence result with initial data of low regularity and prove the trend to the equilibrium for the vlasov poisson fokker planck system with small non linear term but with a possibly large exterior confining potential in dimension d 2 and d 3 the proof relies on a fixed point argument using sharp estimates at short and long time scales of the semi group associated to the fokker planck operator which were obtained by the first author

quadratic mathcal p mathcal t symmetric operators with real spectrum and similarity to self adjoint operators it is established that a symmetric elliptic quadratic differential operator with real spectrum is similar to a self adjoint operator precisely when the associated fundamental matrix has no jordan blocks this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

generalized mehler formula for time dependent non selfadjoint quadratic operators and propagation of singularities we study evolution equations associated to time dependent dissipative non selfadjoint quadratic operators we prove that the solution operators to these non autonomous evolution equations are given by fourier integral operators whose kernels are gaussian tempered distributions associated to non negative complex symplectic linear transformations and we derive a generalized mehler formula for their weyl symbols some applications to the study of the propagation of gabor singularities characterizing the lack of schwartz regularity for the solutions to non autonomous quadratic evolution equations are given