strichartz estimates for the wave and schrodinger equations with the inverse square potential abstract we prove spacetime weighted l 2 estimates for the schrodinger and wave equation with an inverse square potential we then deduce strichartz estimates for these equations

dispersive estimates for the dirac equation in an aharonov bohm field abstract we prove local smoothing and weighted strichartz estimates for the dirac equation with a aharonov bohm potential the proof inspired by 12 relies on an explicit representation of the solution built in terms of spectral projections

self adjointness for dirac operators via hardy dirac inequalities distinguished self adjoint extensions of dirac operators are constructed for a class of singular potentials including coulombic ones up to and including the critical case x 1 the method uses hardy dirac inequalities and quadratic form techniques

the hardy inequality and the asymptotic behaviour of the heat equation with an inverse square potential we study the well posedness and describe the asymptotic behavior of solutions of the heat equation with inverse square potentials for the cauchy dirichlet problem in a bounded domain and also for the cauchy problem in rn in the case of the bounded domain we use an improved form of the so called hardy poincare inequality and prove the exponential stabilization towards a solution in separated variables in rn we first establish a new weighted version of the hardy poincare inequality and then show the stabilization towards a radially symmetric solution in self similar variables with a polynomial decay rate this work complements and explains well known work by baras and goldstein on the existence of global solutions and blow up for these equations in the present article the sign restriction on the data and solutions is removed the functional framework for well posedness is described and the asymptotic rates calculated examples of non uniqueness are also given

lp lp estimates for the schrodinger equation on the line and inverse scattering for the nonlinear schrodinger equation with a potential abstract in this paper i prove a l p l p estimate for the solutions to the one dimensional schrodinger equation with a potential in l 1 where in the generic case 3 2 and in the exceptional case i e when there is a half bound state of zero energy 5 2 i use this estimate to construct the scattering operator for the nonlinear schrodinger equation with a potential i prove moreover that the low energy limit of the scattering operator uniquely determines the potential and the coupling constant of the nonlinearity using a method that allows as well for the reconstruction of the potential and of the nonlinearity

a new proof of the hardy rellich inequality in any dimension the hardy rellich inequality in the whole space with the best constant was firstly proved by tertikas and zographopoulos in adv math 2007 in higher dimensions n 5 then it was extended to lower dimensions n 3 4 by beckner in forum math 2008 and ghoussoub moradifam in math ann 2011 by applying totally different techniques in this note we refine the method implemented by tertikas and zographopoulos based on spherical harmonics decomposition to give an easy and compact proof of the optimal hardy rellich inequality in any dimension n 3 in addition we provide minimizing sequences which were not explicitly mentioned in the quoted papers in lower dimensions n 3 4 emphasizing their symmetry breaking we also show that the best constant is not attained in the proper functional space

lp boundedness of the wave operator for the one dimensional schrodinger operator given a one dimensional perturbed schrodinger operator h d2 dx2 v x we consider the associated wave operators w defined as the strong l2 limits lim s to pm infty e ish e ish 0 we prove that w are bounded operators on lp for all 1 p provided 1 x 2 v x in l 1 or else 1 x v x in l 1 and 0 is not a resonance for p we obtain an estimate in terms of the hilbert transform some applications to dispersive estimates for equations with variable rough coefficients are given

l p boundedness of wave operators for two dimensional schrodinger operators let be the two dimensional schrodinger operator with the real valued potential v which satisfies the decay condition at infinity for we show that the wave operators are bounded in for any 1 p under the condition that h has no zero bound states or zero resonance extending the corresponding results for higher dimensions as w intertwine h 0 and the absolutely continuous part h p ac of h f h p ac w f h 0 w for any borel function f on 1 this reduces the various l p mapping properties of f h p ac to those of f h 0 the convolution operator by the fourier transform of the function f 2

motion estimation using a complex valued wavelet transform this paper describes a new motion estimation algorithm that is potentially useful for both computer vision and video compression applications it is hierarchical in structure using a separable two dimensional 2 d discrete wavelet transform dwt on each frame to efficiently construct a multiresolution pyramid of subimages the dwt is based on a complex valued pair of four tap fir filters with gabor like characteristics the resulting complex dwt cdwt effectively implements an analysis by an ensemble of gabor like filters with a variety of orientations and scales the phase difference between the subband coefficients of each frame at a given subpel bears a predictable relation to a local translation in the region of the reference frame subtended by that subpel that relation is used to estimate the displacement field at the coarsest scale of the multiresolution pyramid each estimate is accompanied by a directional confidence measure in the form of the parameters of a quadratic matching surface the initial estimate field is progressively refined by a coarse to fine strategy in which finer scale information is appropriately incorporated at each stage the accuracy efficiency and robustness of the new algorithm are demonstrated in comparison testing against hierarchical implementations of intensity gradient based and fractional precision block matching motion estimators

the forward problem for the electromagnetic helmholtz equation with critical singularities abstract we study the forward problem of the magnetic schrodinger operator with potentials that have a strong singularity at the origin we obtain new resolvent estimates and give some applications on the spectral measure and on the solutions of the associated evolution problem

a dirac field interacting with point nuclear dynamics the system describing a single dirac electron field coupled with classically moving point nuclei is presented and studied the model is a semi relativistic extension of corresponding time dependent one body hartree fock equation coupled with classical nuclear dynamics already known and studied both in quantum chemistry and in rigorous mathematical literature we prove local existence of solutions for data in h sigma with sigma in 1 frac 3 2 in the course of the analysis a second new result of independent interest is discussed and proved namely the construction of the propagator for the dirac operator with several moving coulomb singularities

variational methods in relativistic quantum mechanics this review is devoted to the study of stationary solutions of lin ear and nonlinear equations from relativistic quantum mechanics involving the dirac operator the solutions are found as critical points of an energy func tional contrary to the laplacian appearing in the equations of nonrelativistic quantum mechanics the dirac operator has a negative continuous spectrum which is not bounded from below this has two main consequences first the energy functional is strongly indefinite second the euler lagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap between the negative and positive continuous spectra moreover since we work in the space domain r 3 the palais smale condition is not satisfied for these reasons the problems discussed in this review pose a challenge in the calculus of variations the existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems in the first part we consider the fixed eigenvalue problem for models of a free self interacting relativistic particle they allow to describe the localized state of a spin 1 2 particle a fermion which propagates without changing its shape this includes the soler models and the maxwell dirac or klein gordon dirac equations the second part is devoted to the presentation of min max principles al lowing to characterize and compute the eigenvalues of linear dirac operators with an external potential in the gap of their essential spectrum many con sequences of these min max characterizations are presented among them a new kind of hardy like inequalities and a stable algorithm to compute the eigenvalues in the third part we look for normalized solutions of nonlinear eigenvalue problems the eigenvalues are lagrange multipliers lying in a spectral gap we review the results that have been obtained on the dirac fock model which is a nonlinear theory describing the behavior of n interacting electrons in an external electrostatic field in particular we focus on the problematic definition of the ground state and its nonrelativistic limit in the last part we present a more involved relativistic model from quan tum electrodynamics in which the behavior of the vacuum is taken into ac count it being coupled to the real particles the main interesting feature of this model is that the energy functional is now bounded from below providing us with a good definition of a ground state

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

energy critical nonlinear schrodinger equation with inverse square potential and subcritical perturbation abstract in this study we consider the global well posedness and scattering theory for the energy critical nonlinear schrodinger equation with inverse square potential and subcritical perturbation in three spatial dimensions we establish the global well posedness theory and scattering theory for a 1 4 1 25 and we improve the results obtained by zhang and zheng 19

remarks on endpoint strichartz estimates for schrodinger equations with the critical inverse square potential abstract the purpose of this paper is to study the validity of global in time strichartz estimates for the schrodinger equation on r n n 3 with the negative inverse square potential x 2 in the critical case n 2 2 4 it turns out that the situation is different from the subcritical case n 2 2 4 in which the full range of strichartz estimates is known to be hold more precisely splitting the solution into the radial and non radial parts we show that i the radial part satisfies a weak type endpoint estimate which can be regarded as an extension to higher dimensions of the endpoint strichartz estimate with radial data for the two dimensional free schrodinger equation ii other endpoint estimates in lorentz spaces for the radial part fail in general iii the non radial part satisfies the full range of strichartz estimates

scattering theory for nonlinear schrodinger equations with inverse square potential abstract we study the long time behavior of solutions to nonlinear schrodinger equations with some critical rough potential of a x 2 type the new ingredients are the interaction morawetz type inequalities and sobolev norm property associated with p a a x 2 we use such properties to obtain the scattering theory for the defocusing energy subcritical nonlinear schrodinger equation with inverse square potential in energy space h 1 r n

on the wave equation with a large rough potential we prove an optimal dispersive l decay estimate for a three dimensional wave equation perturbed with a large nonsmooth potential belonging to a particular kato class the proof is based on a spectral representation of the solution and suitable resolvent estimates for the perturbed operator

strichartz estimates for the magnetic schrodinger equation abstract we prove global scale invariant strichartz estimates for the linear magnetic schrodinger equation with small time dependent magnetic field this is done by constructing an appropriate parametrix as an application we show a global regularity type result for schrodinger maps in dimensions n 6

scattering theory for nls with inverse square potential in 2d abstract in this paper we study the long time behavior of the solution of nonlinear schrodinger equation with a singular potential we prove scattering below the ground state for the radial nls with inverse square potential in dimension two i u t l a u u p u when 2 p a 0 and l a a x 2 this work extends the result in 12 15 16 to dimension 2d the key point is a modified version of arora dodson murphy s approach 2 and that the equivalence of sobolev norms associated to the operator l a and the operator l a with aharonov bohn potential for radial function

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

the w k p continuity of the schrodinger wave operators on the line we prove that the wave operators for the schrodinger equation on the line are continuous on the sobolev spaces w k p 1 p moreover if the potential is exceptional and where f 1 x 0 is a jost solution at zero energy the wave operators are continuous on w k 1 and on w k

endpoint strichartz estimates for the magnetic schrodinger equation abstract we prove strichartz estimates for the schrodinger equation with an electromagnetic potential in dimension n 3 the decay and regularity assumptions on the potentials are almost critical i e close to the coulomb case in addition we require repulsivity and a nontrapping condition which are expressed as smallness of suitable components of the potentials while the potentials themselves can be large the proof is based on smoothing estimates and new sobolev embeddings for spaces associated to magnetic potentials

time decay of scaling critical electromagnetic schrodinger flows we obtain a representation formula for solutions to schrodinger equations with a class of homogeneous scaling critical electromagnetic potentials as a consequence we prove the sharp l 1 to l infty time decay estimate for the 3d inverse square and the 2d aharonov bohm potentials

yosida approximation and nonlinear hyperbolic equation we are concerned with the initial value problem for nonlinear evolution equations of the form u t m a 1 2 u t 2 au t u t f t on 0 u 0 u o u o u 1 here a is a nonnegative selfadjoint operator in a real hilbert space h 0 is a constant and m is a c 1 class function satisfying m r m o 0 for r 0 with m o constant the equation has its motivation in the mathematical description of small amplitude vibrations of an elastic string

nonlinear transform for robust dense block based motion estimation we present a noniterative multiresolution motion estimation strategy involving block based comparisons in each detail band of a laplacian pyramid a novel matching score is developed and analyzed the proposed matching score is based on a class of nonlinear transformations of laplacian detail bands yielding 1 bit or 2 bit representations the matching score is evaluated in a dense full search motion estimation setting with synthetic video frames and an optical flow data set together with a strategy for combining the matching scores across resolutions the proposed method is shown to produce smoother and more robust estimates than mean square error mse in each detail band and combined it tolerates more of nontranslational motion such as rotation validating the analysis while providing much better localization of the motion discontinuities we also provide an efficient implementation of the motion estimation strategy and show that the computational complexity of the approach is closely related to the traditional mse block based full search motion estimation procedure

strichartz estimates and local smoothing estimates for asymptotically flat schrodinger equations abstract in this article we study global in time strichartz estimates for the schrodinger evolution corresponding to long range perturbations of the euclidean laplacian this is a natural continuation of a recent article d tataru parametrices and dispersive estimates for schrodinger operators with variable coefficients amer j math 130 2008 571 634 of the third author where it is proved that local smoothing estimates imply strichartz estimates by d tataru parametrices and dispersive estimates for schrodinger operators with variable coefficients amer j math 130 2008 571 634 the local smoothing estimates are known to hold for small perturbations of the laplacian here we consider the case of large perturbations in three increasingly favorable scenarios i without non trapping assumptions we prove estimates outside a compact set modulo a lower order spatially localized error term ii with non trapping assumptions we prove global estimates modulo a lower order spatially localized error term and iii for time independent operators with no resonance or eigenvalue at the bottom of the spectrum we prove global estimates for the projection onto the continuous spectrum

3d quadratic nls equation with electromagnetic perturbations abstract in this paper we study the asymptotic behavior of a quadratic schrodinger equation with electromagnetic potentials we prove that small solutions scatter the proof builds on earlier work of the author for quadratic nls with a non magnetic potential the main novelty is the use of various smoothing estimates for the linear schrodinger flow in place of boundedness of wave operators to deal with the loss of derivative as a byproduct of the proof we obtain boundedness of the wave operator of the linear electromagnetic schrodinger equation on an l 2 weighted space for small potentials as well as a dispersive estimate for the corresponding flow

regular articlegeneralized strichartz inequalities for the wave equation we make a synthetic exposition of the generalized strichartz inequalities for the wave equation obtained in 6 together with the limiting cases recently obtained in 13 with as simple proofs as possible the proofs combine stationary phase estimates dyadic decompositions the hardy littlewood inequality or the holder inequality in time and abstract duality and interpolation arguments

regularity for a schrodinger equation with singular potentials and application to bilinear optimal control abstract we study the schrodinger equation i t u u v 0 u v 1 u 0 on r 3 0 t where v 0 x t x a t 1 with a w 2 1 0 t r 3 is a coulombian potential singular at finite distance and v 1 is an electric potential possibly unbounded the initial condition u 0 h 2 r 3 is such that r 3 1 x 2 2 u 0 x 2 dx the potential v 1 is also real valued and may depend on space and time variables we prove that if v 1 is regular enough and at most quadratic at infinity this problem is well posed and the regularity of the initial data is conserved for the solution we also give an application to the bilinear optimal control of the solution through the electric potential

de rham factors we construct a theory of de rham epsilon factors given a smooth projective variety x over a field of characteristic 0 and a open image in new window module open image in new window we may consider the determinant of de rham cohomology open image in new window as a graded super line let s be the singular support of open image in new window u an open in x y x u and a 1 form on u x such that u s then we show the existence of a canonically defined graded super line open image in new window canonically isomorphic to open image in new window the key property is the local nature of the epsilon factor it only depends on the restriction of open image in new window and to any open neighborhood of y restricting to the case of curves gives a theory of epsilon factors for curves previously defined by beilinson bloch deligne and esnault

maximal functions associated to filtrations abstract let t be a bounded linear or sublinear operator from lp y to lq x a maximal operator t f x supj t f yj x is associated to any sequence of subsets yj of y under the hypotheses that q p and the sets yj are nested we prove that t is also bounded classical theorems of menshov and zygmund are obtained as corollaries multilinear generalizations of this theorem are also established these results are motivated by applications to the spectral analysis of schrodinger operators

global small solutions to the critical radial dirac equation with potential abstract we solve globally a radial cubic dirac equation perturbed with a small potential with data of small critical norm h 1 the main tool is a new endpoint estimate of the perturbed dirac flow for a class of radial type initial data

improved time decay for a class of scaling critical electromagnetic schrodinger flows we consider a schrodinger hamiltonian h a a with scaling critical and time independent external electromagnetic potential and assume that the angular operator l associated to h is positive definite we prove the following if e ith a a l1 l t n 2 then x g n e ith a a x g n l1 l t n 2 g n g n being a positive number explicitly depending on the ground level of l and the space dimension n we prove similar results also for the heat semi group generated by h a a

on parabolic equations with critical electromagnetic potentials abstract we consider a class of parabolic equations with critical electromagnetic potentials for which we obtain a classification of local asymptotics unique continuation results and an integral representation formula for solutions

weighted estimates for powers and smoothing estimates of schrodinger operators with inverse square potentials abstract let l a be a schrodinger operator with inverse square potential a x 2 on r d d 3 the main aim of this paper is to prove weighted estimates for fractional powers of l a the proof is based on weighted hardy inequalities and weighted inequalities for square functions associated to l a as an application we obtain smoothing estimates regarding the propagator e i t l a

projectivity of the witt vector affine grassmannian we prove that the witt vector affine grassmannian which parametrizes w k lattices in w k frac 1 p n for a perfect field k of characteristic p is representable by an ind perfect scheme over k this improves on previous results of zhu by constructing a natural ample line bundle along the way we establish various foundational results on perfect schemes notably h descent results for vector bundles

linear schrodinger evolution equations with moving coulomb singularities abstract the cauchy problem for schrodinger evolution equations with a finite number of moving coulomb singularities is investigated the case of a single singularity has been studied by several authors see e g baudouin et al 2005 1 and okazawa et al 2010 19 however it seems to be no previous work on plural singularities we shall show that the problem has a unique classical solution by using a time dependent linear transformation of the unknown function which locally freezes the motion of the whole singularities under the simplest collisionless condition in fact a new existence and uniqueness theorem is available for the transformed problem such an abstract framework is established from the viewpoint of linear evolution equations of hyperbolic type in a hilbert space as an innovative modification of those in okazawa 1998 17 and okazawa and yoshii 2011 20

motion estimation based on mutual information and adaptive multi scale thresholding this paper proposes a new method of calculating a matching metric for motion estimation the proposed method splits the information in the source images into multiple scale and orientation subbands reduces the subband values to a binary representation via an adaptive thresholding algorithm and uses mutual information to model the similarity of corresponding square windows in each image a moving window strategy is applied to recover a dense estimated motion field whose properties are explored the proposed matching metric is a sum of mutual information scores across space scale and orientation this facilitates the exploitation of information diversity in the source images experimental comparisons are performed amongst several related approaches revealing that the proposed matching metric is better able to exploit information diversity generating more accurate motion fields

dispersive estimates for schrodinger operators in dimensions one and three we consider l 1 l estimates for the time evolution of hamiltonians h v in dimensions d 1 and d 3 with bound t 2 we require decay of the potentials but no regularity in d 1 the decay assumption is 1 x v x dx whereas in d 3 it is v x c 1 x 3

resolvent expansion and time decay of the wave functions for two dimensional magnetic schrodinger operators we consider two dimensional schrodinger operators h b v given by eq 1 1 below we prove that under certain regularity and decay assumptions on b and v the character of the expansion for the resolvent h b v 1 as 0 is determined by the flux of the magnetic field b through mathbb r 2 subsequently we derive the leading term of the asymptotic expansion of the unitary group e ith b v as t and show how the magnetic field improves its decay in t with respect to the decay of the unitary group e ith 0 v

local smoothing estimates for the massless dirac coulomb equation in 2 and 3 dimensions we prove local smoothing estimates for the massless dirac equation with a coulomb potential in 2 and 3 dimensions our strategy is inspired by 9 and relies on partial wave subspaces decomposition and spectral analysis of the dirac coulomb operator

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

time decay for solutions of schrodinger equations with rough and time dependent potentials in this paper we establish dispersive estimates for solutions to the linear schrodinger equation in three dimensions frac 1 i partial t psi triangle psi v psi 0 qquad psi s f 0 1 where v t x is a time dependent potential that satisfies the conditions sup t v t cdot l frac 3 2 mathbb r 3 sup x in mathbb r 3 int mathbb r 3 int infty infty frac v hat tau x x y d tau dy c 0 here c0 is some small constant and v hat tau x denotes the fourier transform with respect to the first variable we show that under these conditions 0 1 admits solutions l t l2x 3 l2t l6x 3 for any f l2 3 satisfying the dispersive inequality psi t infty le c t s frac32 f 1 text for all times t s 0 2 for the case of time independent potentials v x 0 2 remains true if int mathbb r 6 frac v x v y x y 2 dxdy 4 pi 2 text and v mathcal k sup x in mathbb r 3 int mathbb r 3 frac v y x y dy 4 pi we also establish the dispersive estimate with an e loss for large energies provided v mathcal k v 2 infty

weighted dispersive estimates for two dimensional schrodinger operators with aharonov bohm magnetic field abstract we consider two dimensional schrodinger operators h with an aharonov bohm magnetic field and an additional electric potential we obtain an explicit leading term of the asymptotic expansion of the unitary group e i t h for t in weighted l 2 spaces in particular we show that the magnetic field improves the decay of e i t h with respect to the unitary group of non magnetic schrodinger operators and that the decay rate in time is determined by the magnetic flux