We consider the Schroedinger operator in R^3 with N point interactions placed at Y=(y_1, ... ,y_N), y_j in R^3, of strength a=(a_1, ... ,a_N). Exploiting the spectral theorem and the rather explicit expression for the resolvent we prove a (weighted) dispersive estimate for the corresponding Schroedinger flow.
In the special case N=1 the proof is directly obtained from the unitary group which is known in closed form.
We prove L^1 --> L^\infty estimates for the linear Schroedinger equation in three dimensions. The potential is assumed to belong to certain L^p spaces, but no pointwise decay estimates and no additional regularity is required.
The possibility of finite-time, dispersive blow up for nonlinear equations of Schroedinger type is revisited. This mathematical phenomena is one of the possible explanations for oceanic and optical rogue waves. In dimension one, the possibility of dispersive blow up for nonlinear Schroedinger equations already appears in [9]. In the present work, the existing results are extended in several ways. In one direction, the theory is broadened to include the Davey-Stewartson and Gross-Pitaevskii equations. In another, dispersive blow up is shown to obtain for nonlinear Schroedinger equations in spat…
We prove a general dispersive estimate for a Schroedinger type equation on a product manifold, under the assumption that the equation restricted to each factor satisfies suitable dispersive estimates. Among the applications are the two-particle Schroedinger equations on Euclidean spaces, and the nonlinear Schroedinger equation on the product of two real hyperbolic spaces.
We prove a dispersive estimate for the evolution of Schroedinger operators $H = -Δ+ V(x)$ in ${\mathbb R}^3$. The potential is allowed to be a complex-valued function belonging to $L^p(\R^3)\cap L^q(\R^3)$, $p < \frac32 < q$, so that $H$ need not be self-adjoint or even symmetric. Some additional spectral conditions are imposed, namely that no resonances of $H$ exist anywhere within the interval $[0,\infty)$ and that eigenfunctions at zero (including generalized eigenfunctions) decay rapidly enough to be integrable.
We prove dispersive estimates for linear Schroedinger equations in two space dimensions. The potential is assumed to be real-valued with some polynomial decay (faster than a negative third power), and zero energy is assumed to be a regular point for the perturbed resolvent.
We prove Strichartz-type estimates for Schroedinger's equation with time-dependent potentials. The time derivative of the potentials need not be integrable, so the total variation of the potentials may be infinite.
We consider non-selfadjoint operators of the kind arising in linearized NLS and prove dispersive bounds for the time-evolution without assuming that the edges of the essential spectrum are regular. Our approach does not depend on any specific properties of NLS. Rather, it is axiomatic on the linear level, and our results are obtained from four assumptions (which are of course motivated by NLS). This work is in three dimensions.
We prove L^1 --> L^\infty estimates for linear Schroedinger equations in dimensions one and three. The potentials are only required to satisfy some mild decay assumptions. No regularity on the potentials is assumed.
We prove a dispersive estimate for the one-dimensional Schroedinger equation, mapping between weighted $L^p$ spaces with stronger time-decay ($t^{-3/2}$ versus $t^{-1/2}$) than is possible on unweighted spaces. To satisfy this bound, the long-term behavior of solutions must include transport away from the origin. Our primary requirements are that $(1+|x|)^3 V$ be integrable and $-Δ+ V$ not have a resonance at zero energy. If a resonance is present (for example in the free case), similar estimates are valid after projecting away from a rank-one subspace corresponding to the resonance.
In this article we prove time-global smoothing effects of dispersive pseudodifferential equations with constant coefficient radially symmetric symbols on real rank one symmetric spaces of noncompact type. We also discuss gain of regularities according to decay rates of initial values for the Schroedinger evolution equation. We introduce some isometric operators and reduce the arguments to the well-known Euclidean case. In our proof, Helgason's Fourier transform and the Radon transform as an elliptic Fourier integral operator play crucial roles.
We prove a dispersive estimate for the evolution of Schroedinger operators H = -Δ+ V(x) in three dimensions. The potential should belong to the closure of bounded compactly-supported functions with respect to the golbal Kato norm. Some additional spectral conditions are imposed, namely that no resonances or eigenfunctions of H exist anywhere on the positive half-line. The proof is an application of a new version of Wiener's L^1 inversion theorem.
We establish a dispersive estimate (with a decay of 1/t), valid for all times, for the Schroedinger evolution on a non-compact 2-dimensional manifold with a trapped geodesic.
Global in time dispersive estimates for the Schroedinger and wave evolutions are obtained on manifolds with conical ends whose Hamiltonian flow exhibits trapping. This paper deals with the case of initial data with fixed "nonzero angular momentum".
We prove dispersive estimates for Schroedinger operators in dimension three without any assumptions on zero energy. Ie, we allows resonances and/or eigenvalues at zero energy.
We study a non-linear Schroedinger equation with a Hartree-type nonlinearity and a localized random time-dependent external potential. Sharp dispersive estimates for the linear Schroedinger equation with a random time-dependent potential enable us to also treat the case of small semi-linear perturbations. In both the linear and the nonlinear instances, we prove that, on average, energy remains bounded and solutions scatter.
In this paper we develop the classical multiplier technique to prove a virial identity and smoothing estimates (in a perturbative setting) for the electromagnetic variable coefficients Schroedinger equation.
We consider a class of nonlinear Schroedinger 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 $L^2$) 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 d…
In this paper, we study some modified linear restriction estimates of the dynamics generated by Schroedinger operator on metric cone $M$, where the metric cone $M$ is of the form $M=(0,\infty)_r\timesΣ$ with the cross section $Σ$ being a compact $(n-1)$-dimensional Riemannian manifold $(Σ,h)$ and the equipped metric is $g=\mathrm{d}r^2+r^2h$. Assuming the initial data possesses additional regularity in angular variable $θ\inΣ$, we show some linear restriction estimates for the solutions. As applications, we obtain global-in-time Strichartz estimates for radial initial data and show small initi…
Global in time dispersive estimates for the Schroedinger and wave evolutions are obtained on manifolds with conical ends whose Hamiltonian flow exhibits trapping. This paper deals with the case of initial data with "zero angular momentum".
In this paper, we consider the maximal estimates for the solution to an initial value problem of the linear Schroedinger equation with a singular potential. We show a result about the pointwise convergence of solutions to this special variable coefficient Schroedinger equation with initial data $u_0\in H^{s}(\R^n)$ for $s>1/2$ or radial initial data $u_0\in H^{s}(\R^n)$ for $s\geq1/4$ and the solution does not converge when $s<1/4$.
This paper establishes new estimates for linear Schroedinger equations in R^3 with time-dependent potentials. Some of the results are new even in the time-independent case and all are shown to hold for potentials in scaling-critical, translation-invariant spaces. The proof of the time-independent results uses a novel method based on an abstract version of Wiener's Theorem.
In space dimension $n\geq3$, we consider the magnetic Schrödinger Hamiltonian $H=-(\nabla-iA(x))^2$ and the corresponding Schrödinger equation i\partial_tu+Hu=0. We show some explicit examples of potentials $A$, with less than Coulomb decay, for which any solution of this equation cannot satisfy Strichartz estimates, in the whole range of Schrödinger admissibility.
We prove dispersive estimates in R^3 for the Schroedinger evolution generated by the Hamiltonian H = -Δ+V, under optimal decay conditions on V, in the presence of zero energy eigenfunctions and resonances.
We show that the time evolution of the operator $H = -Δ+ i(A \cdot \nabla + \nabla \cdot A) + V$ in R^3 satisfies Strichartz and smoothing estimates under suitable smoothness and decay assumptions on A and V but without any smallness assumptions. We require that zero energy is neither an eigenvalue nor a resonance.
This paper is concerned with derivation of the global or local in time Strichartz estimates for radially symmetric solutions of the free wave equation from some Morawetz-type estimates via weighted Hardy-Littlewood-Sobolev (HLS) inequalities. In the same way we also derive the weighted end-point Strichartz estimates with gain of derivatives for radially symmetric solutions of the free Schroedinger equation. The proof of the weighted HLS inequality for radially symmetric functions involves an application of the weighted inequality due to Stein and Weiss and the Hardy-Littlewood maximal inequali…
For $α>1$ we consider the initial value problem for the dispersive equation $i\partial_t u +(-Δ)^{α/2} u= 0$. We prove an endpoint $L^p$ inequality for the maximal function $\sup_{t\in[0,1]}|u(\cdot,t)|$ with initial values in $L^p$-Sobolev spaces, for $p\in(2+4/(d+1),\infty)$. This strengthens the fixed time estimates due to Fefferman and Stein, and Miyachi. As an essential tool we establish sharp $L^p$ space-time estimates (local in time) for the same range of $p$.
We obtain certain time decay and regularity estimates for 3D Schroedinger equation with a potential in the Kato class by using Besov spaces associated with Schroedinger operators.
We prove global smoothing and Strichartz estimates for the Schroedinger, wave, Klein-Gordon equations and for the massless and massive Dirac systems, perturbed with singular electromagnetic potentials. We impose a smallness condition on the magnetic part, while the electric part can be large. The decay and regularity assumptions on the coefficients are close to critical.
We consider the cubic nonlinear Schrödinger equation in two space dimensions with an attractive potential. 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, localized in space initial data, converge to the set of bound states. Therefore, the center manifold in this problem is a global attractor. The proof hinges on dispersive estimates that we obtain for the non-autonomous, non-Hamiltonian, linearized dynamics around the bound states.
We consider the Schrödinger type operator ${\mathcal A}=(1+|x|^α)Δ-|x|^β$, for $α\in [0,2]$ and $β\ge 0$. We prove that, for any $p\in (1,\infty)$, the minimal realization of operator ${\mathcal A}$ in $L^p(\R^N)$ generates a strongly continuous analytic semigroup $(T_p(t))_{t\ge 0}$. For $α\in [0,2)$ and $β\ge 2$, we then prove some upper estimates for the heat kernel $k$ associated to the semigroup $(T_p(t))_{t\ge 0}$. As a consequence we obtain an estimate for large $|x|$ of the eigenfunctions of ${\mathcal A}$. Finally, we extend such estimates to a class of divergence type elliptic operat…
Given a one dimensional perturbed Schroedinger operator H=-(d/dx)^2+V(x) we consider the associated wave operators W_+, W_- defined as the strong L^2 limits as s-> \pm\infty of the operators e^{isH} e^{-isH_0} We prove that the wave operators are bounded operators on L^p for all 1<p<\infty, provided (1+|x|)^2 V(x) is integrable, or else (1+|x|)V(x) is integrable and 0 is not a resonance. For p=\infty we obtain an estimate in terms of the Hilbert transform. Some applications to dispersive estimates for equations with variable rough coefficients are given.
The Fourier transforms of the products of two respectively three solutions of the free Schroedinger equation in one space dimension are estimated in mixed and, in the first case weighted, L^p - norms. Inserted into an appropriate variant of the Fourier restriction norm method, these estimates serve to prove local well-posedness of the Cauchy problem for the cubic nonlinear Schroedinger (NLS) equation with data u_0 in the function space ^L^r:=^H^r_0, where for s \in R the spaces ^H^r_s are defined by the norms ||u_0||_{^H^r_s}:=||^u_0||_{L^r'_ξ}, 1/r + 1/r'=1. Similar arguments, combined with a…
We consider in the whole plane the Hamiltonian coupling of semilinear Schroedinger equations which have critical growth in the sense of Moser. We prove that the (nonempty) set S of ground state solutions is compact up to translations. Moreover, ground states are uniformly bounded and uniformly decaying at infinity. Then we prove that actually the ground state is positive and radially symmetric. We apply those results to prove the existence of semiclassical ground states solutions to the singularly perturbed system when the external potential stays bounded away from zero. Namely, as the adimens…
We prove a family of identities that involve the solutions to the free Schreodinger equation. As a consequence of these identities we shall deduce a lower bound for the local smoothing estimate and a uniqueness criterion.
We prove Strichartz estimates for the Schroedinger operator $H = -Δ+ V(t,x)$ with time-periodic complex potentials $V$ belonging to the scaling-critical space $L^{n/2}_x L^\infty_t$ in dimensions $n \ge 3$. This is done directly from estimates on the resolvent rather than using dispersive bounds, as the latter generally require a stronger regularity condition than what is stated above. In typical fashion, we project onto the continuous spectrum of the operator and must assume an absence of resonances. Eigenvalues are permissible at any location in the spectrum, including at threshold energies,…
The Cauchy problem for a coupled system of the Schroedinger and the KdV equation is shown to be globally well-posed for data with infinite energy. The proof uses refined bilinear Strichartz estimates and the I-method introduced by Colliander, Keel, Staffilani, Takaoka, and Tao.
We prove dispersive estimates at high frequency in dimension two for both the wave and the Schrodinger groupes for a very large class of real-valued potentials.
We obtain a dispersive long-time decay in weighted energy norms for solutions of 3D Klein-Gordon equation with magnetic and scalar potentials. The decay extends the results obtained by Jensen and Kato for the Schroedinger equation with scalar potential. For the proof we develop the spectral theory of Agmon, Jensen and Kato and minimal escape velocities estimates of Hunziker, Sigal and Soffer.
We prove global existence and scattering for the defocusing, cubic nonlinear Schrödinger equation in $H^s(\rr^3)$ for $s > {4/5}$. The main new estimate in the argument is a Morawetz-type inequality for the solution $ϕ$. This estimate bounds $\|ϕ(x,t)\|_{L^4_{x,t}(\rr^3 \times \rr)}$, whereas the well-known Morawetz-type estimate of Lin-Strauss controls $\int_0^{\infty}\int_{\rr^3}\frac{(ϕ(x,t))^4}{|x|} dx dt
We prove mixed norm space-time estimates for solutions of the Schroedinger equation, with initial data in $L^p$ Sobolev or Besov spaces, and clarify the relation with adjoint restriction.
We prove an optimal 4G Theorem for the Gaussian kernel. We also propose a new general method of estimating Schroedinger perturbations of transition densities, and give applications to the Gaussian kernel.
We prove dispersive estimates for the linear Schrödinger evolution associated to an operator -Δ+ V, where the potential is a signed measure of fractal dimension at least 3/2.
This paper is devoted to the study of time-dependent hyperbolic systems and the derivation of dispersive estimates for their solutions. It is based on a diagonalisation of the full symbol within adapted symbol classes in order to extract the essential information about representations of solutions. This is combined with a multi-dimensional van der Corput lemma to derive dispersive estimates.
In this note we provide dispersive estimates for Fourier integrals with parameter-dependent phase functions in terms of geometric quantities of associated families of Fresnel surfaces. The results are based on a multi-dimensional van der Corput lemma due to the first author. Applications to dispersive estimates for hyperbolic systems and scalar higher order hyperbolic equations are also discussed.
We consider radial solutions to the Cauchy problem for the linear wave equation with a small short-range electromagnetic potential (the "square version" of the massless Dirac equation with a potential) and zero initial data. We prove two a priori estimates that imply, in particular, a dispersive estimate.
The aim of these notes is to describe some recent results concerning dispersive estimates for principally normal pseudodifferential operators. The main motivation for this comes from unique continuation problems. Such estimates can be used to prove Carleman inequalities, which in turn yield unique continuation results for various partial differential operators with rough potentials.
We prove a dispersive estimate for the solutions of the linearized Water-Waves equations in dimension 1 in presence of a flat bottom. We prove a decay with respect to time t of order 1/3 for solutions with initial data in weighted Sobolev spaces. We also give variants to this result with different decays for a more convenient use of the dispersive estimate. We then give an existence result for the full Water-Waves equations in weighted spaces for practical uses of the proven dispersive estimates.
We prove optimal (that is, without loss of derivatives) dispersive estimates for the Schrodinger group $e^{it(-Δ+V)}$ for a class of real-valued potentials $V\in C^k(R^n)$ with $k>(n-3)/2$, where $n=4,5$.
We prove dispersive estimates at low frequency in dimensions n greater or equal to 4 for the wave equation for a very large class of real-valued potentials, provided the zero is neither an eigenvalue nor a resonance.
We consider wave models with lower order terms and recollect some recent results on energy and dispersive estimates for their solution based on symbolic type estimates for coefficients and partly stabilisation conditions. The exposition is complemented by a collection of open problems.
In this paper, we consider the Cauchy problem for the fractional Schrödinger equation $i D_t^αu + (-Δ)^{\fracβ{2}} u =0$ with $0<α<1$, $β>0$. We establish the dispersive estimates for the solutions. In particular, we prove that the decay rates are sharp.
Consider a bilinear interaction between two linear dispersive waves with a generic resonant structure (roughly speaking, space and time resonant sets intersect transversally). We derive an asymptotic equivalent of the solution for data in the Schwartz class, and bilinear dispersive estimates for data in weighted Lebesgue spaces. An application to water waves with infinite depth, gravity and surface tension is also presented.
We study the $L^1-L^\infty$ dispersive estimate of the inhomogeneous fourth-order Schrödinger operator $H=Δ^{2}-Δ+V(x)$ with zero energy obstructions in $\mathbf{R}^{3}$. For the related propagator $e^{-itH}$, we prove that for $0<t\leq 1$, then $e^{-itH}P_{ac}(H)$ satisfies the $|t|^{-3/4}$-estimate. For $t>1$, we prove that:\,\, 1) if zero is a regular point of $H$, then $e^{-itH}P_{ac}(H)$ satisfies the $|t|^{-3/2}$- dispersive estimate.\,\, 2) if zero is a resonance of $H$, there exists a time dependent operator $F_{t}$ such that $e^{-itH}P_{ac}(H)-F_{t}$ satisfies the $|t|^{-3/2}$- disper…
The initial value problem for the homogeneous Schrödinger equation is investigated for radially symmetric initial data with slow decay rates and not too wild oscillations. Our global wellposedness results apply to initial data for which Strichartz estimates fail.
