a counterexample to dispersive estimates for schrodinger operators in higher dimensions in dimension n 3 we show the existence of a compactly supported potential in the differentiability class c alpha alpha frac n 3 2 for which the solutions to the linear schrodinger equation in mathbb r n i partial t u delta u vu quad u 0 f fail to satisfy an evolution estimate of the form u t cdot infty leq c t frac n 2 u 0 cdot 1 this contrasts with known results in dimensions n 3 where a pointwise decay condition on v is generally sufficient to imply dispersive bounds

on lp boundedness of wave operators for 4 dimensional schrodinger operators with threshold singularities abstractlet h v x be a schrodinger operator on l 2 r 4 h 0 assume that v x v x c x for some 8 let w s limt eithe ith0 be the wave operators it is known that w extend to bounded operators in l p r 4 for all 1 p if 0 is neither an eigenvalue nor a resonance of h we show that if 0 is an eigenvalue but not a resonance of h then the w are still bounded in l p r 4 for all p such that 4 3 p 4

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

dispersive estimates for higher dimensional schrodinger operators with threshold eigenvalues i the odd dimensional case we investigate l1 rn l rn dispersive estimates for the schrodinger operator h v when there is an eigenvalue at zero energy and n 5 is odd in particular we show that if there is an eigenvalue at zero energy then there is a time dependent rank one operator ft satisfying ft l1 l t 2 n2 for t 1 such that eithpac ft l1 l t 1 n2 for t 1 with stronger decay conditions on the potential it is possible to generate an operator valued expansion for the evolution taking the form eithpac h t 2 n2a 2 t 1 n2a 1 t n2a0 with a 2 and a 1 finite rank operators mapping l1 rn to l rn while a0 maps weighted l1 spaces to weighted l spaces the leading order terms a 2 and a 1 vanish when certain orthogonality conditions between the potential v and the zero energy eigenfunctions are satisfied we show that under the same orthogonality conditions the remaining t n2a0 term also exists as a map from l1 rn to l rn hence eithpac h satisfies the same dispersive bounds as the free evolution despite the eigenvalue at zero

the local structure of length spaces with curvature bounded above we show that a number of different notions of dimension coincide for length spaces with curvature bounded above we then apply this result showing that if x is a locally compact cat 0 space with cocompact isometry group then the dimension of the tits boundary and the asymptotic cone s of x are determined by the maximal dimension of a flat in x

on the fourth order schrodinger equation in four dimensions dispersive estimates and zero energy resonances abstract we study the fourth order schrodinger operator h 2 v for a decaying potential v in four dimensions in particular we show that the t 1 decay rate holds in the l 1 l setting if zero energy is regular furthermore if the threshold energies are regular then a faster decay rate of t 1 log t 2 is attained for large t at the cost of logarithmic spatial weights zero is not regular for the free equation hence the free evolution does not satisfy this bound due to the presence of a resonance at the zero energy we provide a full classification of the different types of zero energy resonances and study the effect of each type on the time decay in the dispersive bounds

the lp boundedness of wave operators for schrodinger operators with threshold singularities abstract let h v be a schrodinger operator on l 2 r n with real valued potential v for n 4 and let h 0 if v has sufficiently rapid decay the wave operators defined as w s lim t e i t h e i t h 0 are known to be bounded on l p r n for all 1 p if zero is not an eigenvalue and on 1 p n 2 if zero is an eigenvalue we show that these wave operators are also bounded on l 1 r n by direct examination of the integral kernel of the leading term furthermore if r n v x x d x 0 for all eigenfunctions then the wave operators are l p bounded for 1 p n if in addition r n x v x x d x 0 then the wave operators are bounded for 1 p

dispersive estimates for schrodinger equations with threshold resonance and eigenvalue let h v x be a three dimensional schrodinger operator we study the time decay in lp spaces of scattering solutions e ithpcu where pc is the orthogonal projection onto the continuous spectral subspace of l2 r3 for h under suitable decay assumptions on v x it is shown that they satisfy the so called lp lq estimates e ithpcu p 4 t 3 1 2 1 p u q for all 1 q 2 p with 1 p 1 q 1 if h has no threshold resonance and eigenvalue and for all 3 2 q 2 p 3 if otherwise

asymptotic expansions in time for solutions of schrodinger type equations let ia i p d v be a dissipative operator in l2 rn where p d is an elliptic differential operator of order m with real constant coefficients and v is a compact operator from the weighted sobolev space hm s rn to h0 p s rn s r for some m 1 let r z be the resolvent of a then an asymptotic expansion of r z as z approaches a critical value of the polynomial p is given the coefficient operators in the expansion are computed explicitly by using the resolvent expansion and the results of m murata 9 an asymptotic expansion of e ita as t is given

collapsing of riemannian manifolds and eigenvalues of laplace operator on etudie la perturbation des valeurs propres de l operateur de laplace sous des deformations des varietes qui ne conservent pas les dimensions

on the lp boundedness of wave operators for two dimensional schrodinger operators with threshold obstructions abstract let h v be a schrodinger operator on l 2 r 2 with real valued potential v and let h 0 if v has sufficient pointwise decay the wave operators w s lim t e i t h e i t h 0 are known to be bounded on l p r 2 for all 1 p if zero is not an eigenvalue or resonance we show that if there is an s wave resonance or an eigenvalue only at zero then the wave operators are bounded on l p r 2 for 1 p this result stands in contrast to results in higher dimensions where the presence of zero energy obstructions is known to shrink the range of valid exponents p

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

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

convergence of the alternating minimization algorithm for blind deconvolution abstract blind deconvolution refers to the image processing task of restoring the original image from a blurred version without the knowledge of the blurring function one approach that has been proposed recently t chan c wong ieee trans image process 7 1998 370 375 y you m kaveh ieee trans image process 5 1996 416 428 is a joint minimization model in which an objective function is set up consisting of three terms the data fitting term and the regularization terms for the image and the blur this model implicitly defines a one parameter family of blurred images and point spread functions psfs from which the user can decide usually using additional information which is the best restored image to find a local minimum of the objective function we use an alternating minimization am procedure y you m kaveh ieee trans image process 5 1996 416 428 in which we fix either the blur or the image and minimize respect to the other variable each step of which is a standard non blind deconvolution problem while the model is not convex and thus allows multiple solutions we have found that the am procedure always converges globally but with the converged solution depending on the initial guess in this paper we give an analysis of the am procedure which explains the convergence behavior and the observed robustness of the method

dispersive estimates for schrodinger operators in dimension two we prove l1 2 l 2 for the two dimensional schrodinger operator v with the decay rate t 1 we assume that zero energy is neither an eigenvalue nor a resonance this condition is formulated as in the recent paper by jensen and nenciu on threshold expansions for the two dimensional resolvent

a volume convergence theorem for alexandrov spaces with curvature bounded above we obtain a volume convergence theorem for alexandrov spaces with curvature bounded above with respect to the gromov hausdorff distance as one of the main tools proving this we construct an almost isometry between alexandrov spaces with curvature bounded above with weak singularities which are close to each other furthermore as an application of our researches of convergence phenomena for given positive integer n we prove that if a compact geodesically complete n dimensional cat 1 space has the volume sufficiently close to that of the unit n sphere then it is bi lipschitz homeomorphic to the unit n sphere

a recursive soft decision approach to blind image deconvolution this paper presents a new approach to blind image deconvolution based on soft decision blur identification and hierarchical neural networks traditional blind algorithms require a hard decision on whether the blur satisfies a parametric form before their formulations as the blurring function is usually unknown a priori this precondition inhibits the incorporation of parametric blur knowledge domain into the restoration schemes the new technique addresses this difficulty by providing a continual soft decision blur adaptation with respect to the best fit parametric structure throughout deconvolution the approach integrates the knowledge of well known blur models without compromising its flexibility in restoring images degraded by nonstandard blurs an optimization scheme is developed where a new cost function is projected and minimized with respect to the image and blur domains a nested neural network called the hierarchical cluster model is employed to provide an adaptive perception based restoration its sparse synaptic connections are instrumental in reducing the computational cost of restoration conjugate gradient optimization is adopted to identify the blur due to its computational efficiency the approach is shown experimentally to be effective in restoring images degraded by different blurs