gradient estimate for solutions to poisson equations in metric measure spaces abstract let x d be a complete pathwise connected metric measure space with a locally ahlfors q regular measure where q 1 suppose that x d supports a local 1 2 poincare inequality and a suitable curvature lower bound for the poisson equation u f on x d moser trudinger and sobolev inequalities are established for the gradient of u the local holder continuity with optimal exponent of solutions is obtained

neumann problem for p laplace equation in metric spaces using a variational approach existence boundedness and boundary regularity we employ a variational approach to study the neumann boundary value problem for the p laplacian on bounded smooth enough domains in the metric setting and show that solutions exist and are bounded the boundary data considered are borel measurable bounded functions we also study boundary continuity properties of the solutions one of the key tools utilized is the trace theorem for newton sobolev functions and another is an analog of the de giorgi inequality adapted to the neumann problem

conditional stability theorem for the one dimensional klein gordon equation the paper addresses the conditional non linear stability of the steady state solutions of the one dimensional klein gordon equation for large time we explicitly construct the center stable manifold for the steady state solutions using the modulation method of soffer and weinstein and strichartz type estimates the main difficulty in the one dimensional case is that the required decay of the klein gordon semigroup does not follow from strichartz estimates alone we resolve this issue by proving an additional weighted decay estimate and further refinement of the function spaces which allows us to close the argument in spaces with very little time decay

decay for the wave and schrodinger evolutions on manifolds with conical ends part ii let c n be a compact imbedded riemannian manifold of dimension d 1 and define the d 1 dimensional riemannian manifold m x r x w x with r 0 and smooth and the natural metric ds 2 1 r x 2 dx 2 r 2 x ds 2 we require that m has conical ends r x x o x 1 as x the hamiltonian flow on such manifolds always exhibits trapping dispersive estimates for the schrodinger evolution e it m and the wave evolution e it m are obtained for data of the form f x y n u x where y n are eigenfunctions of with eigenvalues u 2 n in this paper we discuss all cases d n 1 if n 0 there is the following accelerated local decay estimate with 0 1 e it m y n f l m c n m t d 1 2 w 1 f l 1 m where w x x and similarly for the wave evolution our method combines two main ingredients a a detailed scattering analysis of schrodinger operators of the form 2 v 2 1 4 2 u on the line where u is real valued and smooth with u l 0 3 l for all l 0 as and v 0 in particular we introduce the notion of a zero energy resonance for this class and derive an asymptotic expansion of the wronskian between the outgoing jost solutions as the energy tends to zero in particular the division into part i and part ii can be explained by the former being resonant at zero energy where the present paper deals with the nonresonant case b estimation of oscillatory integrals by non stationary phase

weak curvature conditions and functional inequalities we give sufficient conditions for a measured length space x d to admit local and global poincare inequalities along with a sobolev inequality we first introduce a condition dm on x d d ef ined in terms of transport of measures we show that dm together with a doubling condition on implies a scaleinvariant local poincare inequality we show that if x d has nonnegative n ricci curvature and has unique minimizing geodesics between almost all pairs of points then it satisfies dm with constant 2 n t he condition dm is preserved by measured gromov hausdorff limits we then prove a sobolev inequality for measured length spaces with n ricci curvature bounded below by k 0 finally we derive a sharp global poincare inequality

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

asymptotic stability of small gap solitons in nonlinear dirac equations we prove dispersive decay estimates for the one dimensional dirac operator and use them to prove asymptotic stability of small gap solitons in the nonlinear dirac equations with quintic and higher order nonlinear terms

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

geometry and analysis of dirichlet forms let e be a regular strongly local dirichlet form on l2 x m and d the associated intrinsic distance assume that the topology induced by d coincides with the original topology on x and that x is compact satisfies a doubling property and supports a weak 1 2 poincare inequality we first discuss the non coincidence of the intrinsic length structure and the gradient structure under the further assumption that the ricci curvature of x is bounded from below in the sense of lott sturm villani the following are shown to be equivalent i the heat flow of e gives the unique gradient flow of u ii e satisfies the newtonian property iii the intrinsic length structure coincides with the gradient structure moreover for the standard resistance dirichlet form on the sierpinski gasket equipped with the kusuoka measure we identify the intrinsic length structure with the measurable riemannian and the gradient structures we also apply the above results to the coarse ricci curvatures and asymptotics of the gradient of the heat kernel

poincare type inequalities and finding good parameterizations a very important question in geometric measure theory is how geometric features of a set translate into analytic information about it reifenberg bull am math soc 66 312 313 1960 proved that if a set is well approximated by planes at every point and at every scale then the set is a bi holder image of a plane it is known today that carleson type conditions on these approximating planes guarantee a bi lipschitz parameterization of the set in this paper we consider an n ahlfors regular rectifiable set m subset mathbb r n d that satisfies a poincare type inequality involving lipschitz functions and their tangential derivatives then we show that a carleson type condition on the oscillations of the tangent planes of m guarantees that m is contained in a bi lipschitz image of an n plane we also explore the poincare type inequality considered here and show that it is in fact equivalent to other poincare type inequalities considered on general metric measure spaces

derivations and alberti representations abstract we relate generalized lebesgue decompositions of measures in terms of curve fragments alberti representations and weaver derivations this correspondence leads to a geometric characterization of the local norm on the weaver cotangent bundle of a metric measure space x the local norm of a form df sees how fast f grows on curve fragments seen by this implies a new characterization of differentiability spaces in terms of the a e equality of the local norm of df and the local lipschitz constant of f as a consequence the lip lip inequality of keith must be an equality we also provide dimensional bounds for the module of derivations in terms of the assouad dimension of x

time decay estimates for the wave equation with potential in dimension two abstract we study the wave equation with potential u t t u v u 0 in two spatial dimensions with v a real valued decaying potential with h v we study a variety of mapping estimates of the solution operators cos t h and sin t h h under the assumption that zero is a regular point of the spectrum of h we prove a dispersive estimate with a time decay rate of t 1 2 a polynomially weighted dispersive estimate which attains a faster decay rate of t 1 log t 2 for t 2 finally we prove dispersive estimates if zero is not a regular point of the spectrum of h

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

p poincare inequality versus poincare inequality some counterexamples we point out some of the differences between the consequences of p poincare inequality and that of poincare inequality in the setting of doubling metric measure spaces based on the geometric characterization of poincare inequality given in durand cartagena et al mich math j 60 2011 we obtain a geometric property implied by the support of a p poincare inequality and demonstrate by examples that an analogous geometric characterization for finite p is not possible the examples we give are metric measure spaces which are doubling and support an poincare inequality but support no finite p poincare inequality in particular these examples show that one cannot expect a self improving property for poincare inequality in the spirit of keith zhong ann math 167 2 575 599 2008 we also show that the persistence of poincare inequality under measured gromov hausdorff limits fails for poincare inequality

discrete approximations of metric measure spaces of controlled geometry abstract we find a necessary and sufficient condition for a doubling metric space to carry a 1 p poincare inequality the condition involves discretizations of the metric space and poincare inequalities on graphs

relaxation of solitons in nonlinear schrodinger equations with potential abstract in this paper we study dynamics of solitons in the generalized nonlinear schrodinger equation nls with an external potential in all dimensions except for 2 for a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space centered near different critical points of the potential we call those solutions which are centered near the minima of the potential and which minimize energy restricted to l 2 unit sphere trapped solitons or just solitons in this paper we prove under certain conditions on the potentials and initial conditions that trapped solitons are asymptotically stable moreover if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position the dynamical law of motion of the soliton i e effective equations of motion for the soliton s center and momentum is close to newton s equation but with a dissipative term due to radiation of the energy to infinity

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

some remarks on the schrodinger equation with a potential in lrtlsx we study the dispersive properties of the linear schrodinger equation with a time dependent potential v t x we show that an appropriate integrability condition in space and time on v i e the boundedness of a suitable lrtlsx norm is sufficient to prove the full set of strichartz estimates we also construct several counterexamples which show that our assumptions are optimal both for local and for global strichartz estimates in the class of large unsigned potentials v lrtlsx

quasisymmetric parametrizations of two dimensional metric spheres according to the classical uniformization theorem every smooth riemannian surface z homeomorphic to the 2 sphere is conformally diffeomorphic to s2 the unit sphere inr3 equipped with the riemannian metric induced by the ambient euclidean metric the availability of a similar uniformization procedure for spheres with a generalized conformal structure is highly desirable in particular in connection with thurston s hyperbolization conjecture this was addressed by cannon in his combinatorial riemann mapping theorem 7 he considers topological surfaces equipped with a sequence of shinglings a combinatorial structure that leads to a notion of approximate conformal moduli of rings he then finds conditions that imply the existence of coordinate systems on the surface that relate these combinatorial moduli to classical analytic moduli in the plane in this paper we develop a uniformization theory for a different type of generalized conformal structure we start with a metric space z homeomorphic to s2 and ask for conditions under which z can be mapped onto s2 by a quasisymmetric homeomorphism the class of quasisymmetries is an appropriate analog of conformal1 mappings in a metric space context quasisymmetric homeomorphisms also arise in the theory of gromov hyperbolic metric spaces quasi isometries between gromov hyperbolic spaces induce quasisymmetric boundary homeomorphisms our setup has the advantage that we can exploit recent notions and methods from analysis

gagliardo nirenberg inequalities on manifolds abstract we prove gagliardo nirenberg inequalities on some classes of manifolds lie groups and graphs

on the focusing critical semi linear wave equation the wave equation partial derivative tt psi delta psi psi 5 0 in r 3 is known to exhibit finite time blowup for data of negative energy furthermore it admits the special static solutions phi x a 3a 1 4 1 a x 2 1 2 for all a 0 which are linearly unstable we view these functions as a curve in the energy space h over dot 1 x l 2 we prove the existence of a family of perturbations of this curve that lead to global solutions possessing a well defined long time asymptotic behavior as the sum of a bulk term plus a scattering term moreover this family forms a co dimension one manifold m of small diameter in a suitable topology loosely speaking m acts as a center stable manifold with the curve phi a as an attractor in m

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

quasiconformal and bi lipschitz homeomorphisms uniform domains and the quasihyperbolic metric let d be a proper subdomain of r and kd the quasihyperbolic metric defined by the conformal metric tensor ds2 dist x dd 2ds2 the geodesies for this and related metrics are shown by purely geometric methods to exist and have lipschitz continuous first derivatives this is sharp for kd we also obtain sharp estimates for the euclidean curvature of such geodesies we then use these results to prove a general decomposition theorem for uniform domains in r in terms of embeddings of bi lipschitz balls we also construct a counterexample to the higher dimensional analogue of the decomposition theorem of gehring and osgood the defining density dist x 3d 1 need not be differentiable we primarily are interested in studying the geometry of the quasihyperbolic meric and its geodesies we show that the geodesies of this metric are c11 i e the arclength parametrisation has lipschitz continuous derivatives we show that this is best possible and obtain a sharp result on the euclidean curvature of such geodesies these results are proved for a more general class of metrics in particular metrics defined by locally lipschitz densities we then use these results to prove a gener alisation of the decomposition theorem of gehring and osgood for uniform domains in ri space i wish to express my sincere thanks to f w gehring for suggesting many of these problems for many helpful ideas and simplifications throughout and for allowing me to present his proof of theorem 3 7 i also wish to thank him and d herron for carefully reading the manuscript

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

isoperimetric inequality from the poisson equation via curvature in this paper we establish an isoperimetric inequality in a metric measure space via the poisson equation let x d be a complete pathwise connected metric space with locally ahlfors q regular measure where q 1 that supports a local l2 poincare inequality we show that for the poisson equation u g if the local l norm of the gradient du can be bounded by the lorentz norm lq 1 of g then we obtain an isoperimetric inequality and a sobolev inequality in x d with optimal exponents by assuming a suitable curvature lower bound we establish such optimal bounds on 2011 wiley periodicals inc

weighted sobolev spaces on metric measure spaces abstract we investigate weighted sobolev spaces on metric measure spaces x d m denoting by the weight function we compare the space w x d m which always concides with the closure h x d m of lipschitz functions with the weighted sobolev spaces w 1 p x d m and h 1 p x d m defined as in the euclidean theory of weighted sobolev spaces under mild assumptions on the metric measure structure and on the weight we show that w x d m h x d m we also adapt the results proved by muckenhoupt and the ones proved by zhikov to the metric measure setting considering appropriate conditions on that ensure the equality w 1 p x d m h x d m

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

modulus and the poincare inequality on metric measure spaces the purpose of this paper is to develop the understanding of modulus and the poincare inequality as defined on metric measure spaces various definitions for modulus and capacity are shown to coincide for general collections of metric measure spaces consequently modulus is shown to be upper semi continuous with respect to the limit of a sequence of curve families contained in a converging sequence of metric measure spaces moreover several competing definitions for the poincare inequality are shown to coincide if the underlying measure is doubling one such characterization considers only continuous functions and their continuous upper gradients and extends work of heinonen and koskela applications include showing that the p poincare inequality with a doubling measure for p 1 persists through to the limit of a sequence of converging pointed metric measure spaces this extends results of cheeger a further application is the construction of new doubling measures in euclidean space which admit a 1 poincare inequality

definitions of quasiconformality summary we establish that the infinitesimal h definition for quasiconforreal mappings on carnot groups implies global quasisymmetry and hence the absolute continuity on almost all lines our method is new even in r where we obtain that the limsup condition in the h definition can be replaced by a liminf condition this leads to a new removability result for quasi conformal mappings in euclidean spaces an application to parametrizations of chord arc surfaces is also given

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

asymptotic stability of ground states in 2d nonlinear schrodinger equation including subcritical cases abstract we consider a class of nonlinear schrodinger equations in two space dimensions with an attractive potential the nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical in l 2 nonlinearities we study the asymptotic stability of the nonlinear bound states i e periodic in time localized in space solutions our result shows that all solutions with small initial data converge to a nonlinear bound state therefore the nonlinear bound states are asymptotically stable the proof hinges on dispersive estimates that we obtain for the time dependent hamiltonian linearized dynamics around a carefully chosen one parameter family of bound states that shadows the nonlinear evolution of the system due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations

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

cheeger harmonic functions in metric measure spaces revisited let x d be a complete metric measure space with a locally doubling measure that supports a local weak l2 poincare inequality by assuming a heat semigroup type curvature condition we prove that cheeger harmonic functions are lipschitz continuous on x d gradient estimates for cheeger harmonic functions and solutions to a class of non linear poisson type equations are presented

from local to global in quasiconformal structures abstract we exhibit a large class of metric spaces whose infinitesimal quasiconformal structure is strong enough to capture the global quasiconformal structure a sufficient condition for this to happen is described in terms of a poincare type inequality

differentiability and poincare type inequalities in metric measure spaces abstract we demonstrate the necessity of a poincare type inequality for those metric measure spaces that satisfy cheeger s generalization of rademacher s theorem for all lipschitz functions taking values in a banach space with the radon nikodym property this is done by showing the existence of a rich structure of curve fragments that connect nearby points similar in nature to semmes s pencil of curves for the standard poincare inequality using techniques similar to cheeger kleiner 12 we show that our conditions are also sufficient we also develop another characterization of rnp lipschitz differentiability spaces by connecting points by curves that form a rich structure of partial derivatives that were first discussed in 5

asymptotic stability of ground states in 3d nonlinear schrodinger equation including subcritical cases we consider a class of nonlinear schrodinger equation in three space dimensions with an attractive potential the nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical in l2 nonlinearities we study the asymptotic stability of the nonlinear bound states i e periodic in time localized in space solutions our result shows that all solutions with small initial data converge to a nonlinear bound state therefore the nonlinear bound states are asymptotically stable the proof hinges on dispersive estimates that we obtain for the time dependent hamiltonian linearized dynamics around a careful chosen one parameter family of bound states that shadows the nonlinear evolution of the system due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations

weighted energy decay for 3d klein gordon equation abstract we obtain a dispersive long time decay in weighted energy norms for solutions of the 3d klein gordon equation with generic potential the decay extends the results obtained by jensen and kato for the 3d schrodinger equation for the proof we modify the spectral approach of jensen and kato to make it applicable to relativistic equations

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

poincare inequalities uniform domains and extension properties for newton sobolev functions in metric spaces in the setting of metric measure spaces equipped with a doubling measure supporting a weak p poincare inequality with 1 p 1 we obtain a near characterization of n1 p extension domains with loc

locally p admissible measures on r abstract in this note we show that locally p admissible measures on r necessarily come from local muckenhoupt a p weights in the proof we employ the corresponding characterization of global p admissible measures on r in terms of global a p weights due to bjorn buckley and keith together with tools from analysis in metric spaces more specifically preservation of the doubling condition and poincare inequalities under flattening due to durand cartagena and li as a consequence the class of locally p admissible weights on r is invariant under addition and satisfies the lattice property we also show that measures that are p admissible on an interval can be partially extended by periodical reflections to global p admissible measures surprisingly the p admissibility has to hold on a larger interval than the reflected one and an example shows that this is necessary

a differentiable structure for metric measure spaces abstract the main result of this paper is the provision of conditions under which a metric measure space admits a differentiable structure this differentiable structure gives rise to a finite dimensional l cotangent bundle over the given metric measure space and then to a sobolev space h 1 p over the given metric measure space the latter which is reflexive for p 1 this extends results of cheeger geom funct anal 9 1999 3 428 to a wider collection of metric measure spaces

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

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

on the spectral theory and dispersive estimates for a discrete schrodinger equation in one dimension based on the recent work komech et al dispersive estimates for 1d discrete schrodinger and klein gordon equations appl anal 85 1487 2006 for compact potentials we develop the spectral theory for the one dimensional discrete schrodinger operator h v n 1 n 1 2 n vn n we show that under appropriate decay conditions on the general potential and a nonresonance condition at the spectral edges the spectrum of h consists of finitely many eigenvalues of finite multiplicities and the essential absolutely continuous spectrum while the resolvent satisfies the limiting absorption principle and the puiseux expansions near the edges these properties imply the dispersive estimates eithpa c h l 2 l 2 t 3 2 for any fixed 52 and any t 0 where pa c h denotes the spectral projection to the absolutely continuous spectrum of h in addition based on the scattering theory for the discrete jost solutions and the previous results by stefanov and kevrekidis asymptotic behaviour of small