weighted estimates for the helmholtz equation and some applications we characterize the radial functionsvin rnfor which the a priori inequality u1 2 2 k uv1 2 2 c v 1 2 k2 u 2holds with constant independent ofk the condition is forvto have the x rays transform everywhere bounded we apply these estimates to the well posedness of evolution schrodinger equations with time dependent drift terms and to the restriction of the fourier transform to euclidean spheres

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

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

lp continuity of wave operators in z abstract we recover for discrete schrodinger operators on the lattice z stronger analogues of the results by weder r weder the w k p continuity of the schrodinger wave operators on the line comm math phys 208 1999 507 520 and by d ancona and fanelli p d ancona l fanelli l p boundedness of the wave operator for the one dimensional schrodinger operator comm math phys 268 2006 415 438 on r

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

instability of the boundary in the billiard ball problem we consider the billiard ball problem in the interior of a plane closed convexc 1 curve which is piecewisec 2 if the curvature has a discontinuity then the boundary is unstable i e no caustics exist near the boundary however in the interior there can exist caustics as we show by an example

how high dimensional stadia look like we give the affirmative answer to the long standing question whether or not the mechanism of defocusing can produce a chaotic behavior in high dimensional hamiltonian systems to do this we prove that billiards in a class of regions in r n n 2 with focusing and flat boundary components have nonvanishing lyapunov exponents

explicit formulae for the wave operators of perturbed self adjoint operators abstract we give explicit formulae for the wave operators of the position operator h 0 and the operator h obtained from h 0 by a rank one perturbation then by the spectral theorem we deduce a general formula for wave operators of unbounded self adjoint operators with absolute continuous spectrum

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

dispersive and strichartz estimates for the three dimensional wave equation with a scaling critical class of potentials we consider the three dimensional linear wave equation perturbed by a potential v belonging to a scaling critical class which is the closure of bounded compactly supported functions with respect to the global kato norm without any further smallness assumptions we first prove the dispersive estimate for the solution of the wave equation as a byproduct we prove strichartz estimates under a slightly stronger condition

geometry thermodynamics and finite size corrections in the critical potts model we establish an intriguing connection between geometry and thermodynamics in the critical q state potts model on two dimensional lattices using the q state bond correlated percolation model qbcpm representation we find that the number of clusters nc of the qbcpm has an energylike singularity for qfi1 which is reached and supported by exact results numerical simulation and scaling arguments we also establish that the finite size correction to the number of bonds nb has no constant term and explains the divergence of related quantities as q 4 the multicritical point similar analyses are applicable to a variety of other systems s1063 651x 99 09612 9

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 lp lp estimate for the schrodinger equation on the half line abstract in this paper we prove the lp lp estimate for the schrodinger equation on the half line and with homogeneous dirichlet boundary condition at the origin

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

billiards with pesin region of measure one we give a description of a large class of plane billiards with pesin region of measure one open conditions including properly those founded by wojtkowski w1 forc4 focusing boundaries are obtained lyapunov s forms introduced by lewowicz are used

a transversal fundamental theorem for semi dispersing billiards for billiards with a hyperbolic behavior fundamental theorems ensure an abundance of geometrically nicely situated and sufficiently large stable and unstable invariant manifolds a transversal fundamental theorem has recently been suggested by the present authors to proveglobal ergodicity and then as an easy consequence the k property of semidispersing billiards in particular the global ergodicity of systems ofn 3 elastic hard balls conjectured by the celebratedboltzmann sinai ergodic hypothesis in fact the suggested transversal fundamental theorem has been successfully applied by the authors in the casesn 3 and 4 the theorem generalizes the fundamental theorem of chernov and sinai that was really the fundamental tool to obtainlocal ergodicity of semi dispersing billiards our theorem however is stronger even in their case too since its conditions are simpler and weaker moreover a complete set of conditions is formulated under which the fundamental theorem and its consequences like the zig zag theorem are valid for general semi dispersing billiards beyond the utmost interesting case of systems of elastic hard balls as an application we also give conditions for the ergodicity and consequently the k property of dispersing billiards transversality means the following instead of the stable and unstable foliations occurring in the chernov sinai formulation of the stable version of the fundamental theorem we use the stable foliation and an arbitrary nice one transversal to the stable one

explicit formula for schrodinger wave operators on the half line for potentials up to optimal decay abstract we give an explicit formula for the wave operators for perturbations of the dirichlet laplacian by a potential on the half line the potential is assumed to decay strictly faster than the polynomial of degree minus two the formula consists of the main term given by the scattering operator and a function of the generator of the dilation group and a hilbert schmidt remainder term our method is based on the elementary construction of the generalized fourier transforms in terms of the solutions of the volterra integral equations the hilbert schmidt property of the remainder term follows from a decay estimate for the jost solution which is established by performing a perturbation expansion of sufficiently high order as a corollary a topological interpretation of levinson s theorem is established via an index theorem approach

semi focusing billiards hyperbolicity in this paper we answer affirmatively the question concerning the existence of hyperbolic billiards in convex domains of 3 we also prove that a related class of semi focusing billiards has mixed dynamics i e their phase space is an union of two invariant sets of positive measure such that the dynamics is integrable on one set and is hyperbolic on the other these billiards are the first rigorous examples of billiards in domains of 3 with divided phase space

principles for the design of billiards with nonvanishing lyapunov exponents we introduce a large class of billiards with convex pieces of the boundary which have nonvanishing lyapunov exponents

smoothing and dispersive estimates for 1d schrodinger equations with bv coefficients and applications we prove smoothing estimates for schrodinger equations i t x a x x 0 with a x bv real and bounded from below we then bootstrap these estimates to obtain optimal strichartz and maximal function estimates all of which turn out to be identical to the constant coefficient case we also provide counterexamples showing a bv to be in a sense a minimal requirement finally we provide an application to sharp well posedness for a generalized benjamin ono equation

scattering for nls with a delta potential abstract we prove h 1 scattering for the mass supercritical defocusing nonlinear schrodinger equation perturbed by a repulsive delta potential

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

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

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

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

on the flows associated to selfadjoint operators on metric measure spaces let x be a metric space with a doubling measure satisfying mu b gtrsim r b n for any ball b with any radius r b 0 let l be a non negative selfadjoint operator on l 2 x we assume that e tl satisfies a gaussian upper bound and that the flow e itl satisfies a typical l 1 l infty dispersive estimate of the form begin aligned vert e itl vert l 1 rightarrow l infty lesssim t frac n 2 end aligned then we prove a similar l 1 l infty dispersive estimate for a general class of flows e it phi l with phi r of power type near 0 and near infty in the case of fractional powers phi l l nu nu in 0 1 we deduce dispersive estimates for e itl nu with data in sobolev besov or hardy spaces h p l with p in 0 1 associated to the operator l

structure of wave operators for a scaling critical class of potentials we prove a structure formula for the wave operators in bbb r 3 w pm mathop hbox s lim limits t to pm infty e it delta v p c e it delta and their adjoints for a scaling invariant class of scalar potentials v in b b left v mid sum k in bbb z 2 k 2 big chi x in 2 k 2 k 1 x v x big l 2 infty right under the assumption that zero is neither an eigenvalue nor a resonance for delta v the formula implies the boundedness of wave operators on l p spaces 1 leq p leq infty on weighted l p spaces and on sobolev spaces as well as multilinear estimates for e ith p c when v decreases rapidly at infinity we obtain an asymptotic expansion of the wave operators the first term of the expansion is of order langle y rangle 4 commutes with the laplacian and exists when v in langle x rangle 3 2 epsilon l 2 we also prove that the scattering operator s w w is an integrable combination of isometries the proof is based on an abstract version of wiener s theorem applied in a new function space

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

dispersive estimate for the 1d schrodinger equation with a steplike potential abstract we prove a sharp dispersive estimate p a c u t x c t 1 2 u 0 l 1 r for the one dimensional schrodinger equation i u t u x x v x u v 0 x u 0 where 1 x 2 v l 1 r and v 0 is a step function real valued and constant on the positive and negative real axes

an exhaustive criterion for the non existence of invariant circles for area preserving twist maps we present a necessary and sufficient condition for the non existence of rotational invariant circles for area preserving twist maps of the cyclinder or annulus based on the cone crossing and killends criteria of mackay and percival 1985 given a number of technical restrictions on the implementation of these criteria this condition leads to a proof of mackay and percival s finite computation conjecture

wave operator bounds for one dimensional schrodinger operators with singular potentials and applications boundedness of wave operators for schrodinger operators in one space dimension for a class of singular potentials admitting finitely many dirac delta distributions is proved applications are presented to for example dispersive estimates and commutator bounds

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

using integrability to produce chaos billiards with positive entropy a new open class of convex 2 dimensional planar billiards with positive lyapunov exponent almost everywhere is constructed we introduce the notion of a focusing arc and show that such arcs can be used to build billiard systems with positive lyapunov exponents we prove that under smallc6 perturbations focusing arcs remain focusing and thereby show that perturbations of the bunimovich stadium billiard have positive lyapunov exponents

a theorem on ergodicity of two dimensional hyperbolic billiards ergodicity of two dimensional billiards which satisfy some general conditions is proved this theorem is applied to one concrete class of billiards that contains in particular billiards in the stadium

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

scattering in the energy space for the nls with variable coefficients we consider the nls with variable coefficients in dimension n ge 3 begin aligned i partial t u lu f u 0 qquad lv nabla b cdot a x nabla b v c x v qquad nabla b nabla ib x end aligned on mathbb r n or more generally on an exterior domain with dirichlet boundary conditions for a gauge invariant defocusing nonlinearity of power type f u simeq u gamma 1 u we assume that l is a small long range perturbation of delta plus a potential with a large positive part the first main result of the paper is a bilinear smoothing interaction morawetz estimate for the solution as an application under the conditional assumption that strichartz estimates are valid for the linear flow e itl we prove global well posedness in the energy space for subcritical powers gamma 1 frac 4 n when the domain is mathbb r n by extending the strichartz estimates due to tataru am j math 130 3 571 634 2008 we prove that the conditional assumption is satisfied and deduce well posedness and scattering in the energy space

hyperbolic billiards with nearly flat focusing boundaries i abstract the standard wojtkowski markarian donnay bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the focusing boundary this is due to the physical principle that is used in the proofs the so called defocusing mechanism of geometrical optics in this paper we construct examples of hyperbolic billiards with a focusing boundary component of arbitrarily small curvature whose diameter is bounded by a constant independent of that curvature our proof employs a nonstandard cone bundle that does not solely use the familiar dispersing and defocusing mechanisms

many dimensional nowhere dispersing billiards with chaotic behavior abstract we consider some examples of domains in the d dimensional euclidean space d 3 that generate billiards with chaotic behavior time correlations in these billiards tend to zero in the limit t the rate of decay of the correlations is a power law for billiards in discrete time and could be a power law or quasiexponential for billiards with continuous time

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

static and time dependent perturbations of the classical elliptical billiard the elliptical billiard problem defines a two dimensional integrable discrete dynamical system integrability not being a robust property we study some static and time dependent perturbations of this problem for the static case we observe the transition from integrability to chaos on some perturbations of the ellipse then we study time dependent perturbations supposing that the boundary deforms periodically with the time remaining always an ellipse we investigate numerically the now four dimensional phase space looking mainly at the question of whether or not the velocity of a given trajectory may increase indefinitely

endpoint estimates and global existence for the nonlinear dirac equation with potential abstract we prove endpoint estimates with angular regularity for the wave and dirac equations perturbed with a small potential the estimates are applied to prove global existence for the cubic dirac equation perturbed with a small potential for small initial h 1 data with additional angular regularity this implies in particular global existence in the critical energy space h 1 for small radial data

dispersive estimates for the three dimensional schrodinger equation with rough potentials the three dimensional schrodinger propogator e ith h v is a bounded map from l 1 to l with norm controlled by t 3 2 provided the potential satisfies two conditions an integrability condition limiting the singularities and decay of v and a zero energy spectral condition on h this is shown by expressing the spectral measure of h in terms of its resolvents and proving a family of l p mapping estimates for the resolvents previous results in this direction had required v to satisfy explicit pointwise bounds

evolution equations on non flat waveguides we investigate the dispersive properties of evolution equations on waveguides with a non flat shape more precisely we consider an operator h delta x delta y v x y with dirichlet boundary conditions on an unbounded domain and we introduce the notion of a repulsive waveguide along the direction of the first group of variables x if is a repulsive waveguide we prove a sharp estimate for the helmholtz equation hu u f as consequences we prove smoothing estimates for the schrodinger and wave equations associated to h and strichartz estimates for the schrodinger equation additionally we deduce that the operator h does not admit eigenvalues

smoothing estimates for the variable coefficients schrodinger equation with electromagnetic potentials abstract in this paper we develop the classical multiplier technique to build up a virial identity for the electromagnetic variable coefficients schrodinger equation following the strategy of fanelli and vega 2009 13 we shall use such an identity to prove smoothing estimates for the associated flow in a perturbative setting