The Estimates L1-L∞ for the Reduced Radial Equation of Schrödinger (original) (raw)
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Journal of Functional Analysis, 2000
In this paper I prove a L p − Lṕ estimate for the solutions of the one-dimensional Schrödinger equation with a potential in L 1 γ where in the generic case γ > 3/2 and in the exceptional case (i.e. when there is a half-bound state of zero energy) γ > 5/2. I use this estimate to construct the scattering operator for the nonlinear Schrödinger equation with a potential. I prove moreover, that the low-energy limit of the scattering operator uniquely determines the potential and the nonlinearity using a method that allows as well for the reconstruction of the potential and of the nonlinearity.
Asymptotic Lower Bounds for a Class of Schrödinger Equations
Communications in Mathematical Physics, 2008
We shall study the following initial value problem: (0.1) i∂tu − ∆u + V (x)u = 0, (t, x) ∈ R × R n , u(0) = f, where V (x) is a real short-range potential, whose radial derivative satisfies some supplementary assumptions. More precisely we shall present a family of identities satisfied by the solutions to (0.1) that generalizes the ones proved in [12] and [21] in the free case. As a by-product of these identities we deduce some uniqueness results for solutions to (0.1), and a lower bound for the so called local smoothing which becomes an identity in a precise asymptotic sense.
Global existence and scattering for the inhomogeneous nonlinear Schr\"odinger equation
arXiv (Cornell University), 2021
In this paper we consider the inhomogeneous nonlinear Schrödinger equation i∂tu + ∆u = K(x)|u| α u, u(0) = u0 ∈ H s (R N), s = 0, 1, N ≥ 1, |K(x)| + |x| s |∇ s K(x)| |x| −b , 0 < b < min(2, N − 2s), 0 < α < (4 − 2b)/(N − 2s). We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted L 2-space for a new range α0(b) < α < (4 − 2b)/N. The value α0(b) is the positive root of N α 2 + (N − 2 + 2b)α − 4 + 2b = 0, which extends the Strauss exponent known for b = 0. Our results improve the known ones for K(x) = µ|x| −b , µ ∈ C and apply for more general potentials. In particular, we show the impact of the behavior of the potential at the origin and infinity on the allowed range of α. Some decay estimates are also established for the defocusing case. To prove the scattering results, we give a new criterion taking into account the potential K. Here u = u(t, x) ∈ C, t ∈ R, x ∈ R N , N ≥ 1, s = 0 or s = 1 and α > 0. The potential K is a complex valued function satisfying some hypothesis. In particular, K(x) = µ|x| −b and K(x) = µ(1 + |x| 2) − b 2 , b > 0, µ ∈ C, will be considered. Equation (1.1) with a constant function K, corresponds to the standard nonlinear Schrödinger equation. The case where K is non constant and bounded is considered in [33, 35]. The unbounded potential case is also treated in [11, 12, 41], where K(x) = |x| b. Here we consider (1.1) with potential having a decay like |x| −b at infinity and may be singular at the origin. This kind
Some dispersive estimates for Schrödinger equations with repulsive potentials
Journal of Functional Analysis, 2006
We prove the local smoothing effect for Schrödinger equations with repulsive potentials for n 3. The estimates are global in time and are proved using a variation of Morawetz multipliers. As a consequence we give sharp constants to measure the attractive part of the potential and its rate of decay, which turns out to be different whether dimension 3 or higher are considered. Also a notion of zero resonance arises in a natural way. Our smoothing estimate allows us to use Sobolev inequalities and treat nonradial perturbations.
On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation
Dynamics of Partial Differential Equations, 2004
We study the asymptotic behavior of large data radial solutions to the focusing Schrödinger equation iut +∆u = −|u| 2 u in R 3 , assuming globally bounded H 1 (R 3) norm (i.e. no blowup in the energy space). We show that as t → ±∞, these solutions split into the sum of three terms: a radiation term that evolves according to the linear Schrödinger equation, a smooth function localized near the origin, and an error that goes to zero in theḢ 1 (R 3) norm. Furthermore, the smooth function near the origin is either zero (in which case one has scattering to a free solution), or has mass and energy bounded strictly away from zero, and obeys an asymptotic Pohozaev identity. These results are consistent with the conjecture of soliton resolution. Contents 1991 Mathematics Subject Classification. 35Q55.
Scattering Theory for Radial Nonlinear Schrödinger Equations on Hyperbolic Space
Geometric and Functional Analysis, 2008
We study the long time behavior of radial solutions to nonlinear Schrödinger equations on hyperbolic space. We show that the usual distinction between short range and long range nonlinearity is modified: the geometry of the hyperbolic space makes every power-like nonlinearity short range. The proofs rely on weighted Strichartz estimates, which imply Strichartz estimates for a broader family of admissible pairs, and on Morawetz type inequalities. The latter are established without symmetry assumptions.
Dispersion estimates for fourth order Schrödinger equations
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 2000
Motivated by the study of the fourth-order nonlinear Schrödinger equations introduced by V. Karpman [4], we give dispersion estimates for the linear group associated to i ∂t + ∆ 2 ± ∆. 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
International Mathematics Research Notices, 2010
We consider the semiclassical (zero-dispersion) limit of the one-dimensional focusing Nonlinear Schrödinger equation (NLS) with decaying potentials. If a potential is a simple rapidly oscillating wave (the period has the order of the semiclassical parameter ε) with modulated amplitude and phase, the space-time plane subdivides into regions of qualitatively different behavior, with the boundary between them consisting typically of collection of piecewise smooth arcs (breaking curve(s)). In the first region the evolution of the potential is ruled by modulation equations (Whitham equations), but for every value of the space variable x there is a moment of transition (breaking), where the solution develops fast, quasi-periodic behavior, i.e., the amplitude becomes also fastly oscillating at scales of order ε. The very first point of such transition is called the point of gradient catastrophe. We study the detailed asymptotic behavior of the left and right edges of the interface between these two regions at any time after the gradient catastrophe. The main finding is that the first oscillations in the amplitude are of nonzero asymptotic size even as ε tends to zero, and they display two separate natural scales; of order O(ε) in the parallel direction to the breaking curve in the (x, t)-plane, and of order O(ε ln ε) in a transversal direction. The study is based upon the inverse-scattering method and the nonlinear steepest descent method.
Scattering theory for the Schrödinger equation with repulsive potential
Journal de Mathématiques Pures et Appliquées, 2005
We consider the scattering theory for the Schrödinger equation with −∆ − |x| α as a reference Hamiltonian, for 0 < α ≤ 2, in any space dimension. We prove that when this Hamiltonian is perturbed by a potential, the usual short range/long range condition is weakened: the limiting decay for the potential depends on the value of α, and is related to the growth of classical trajectories in the unperturbed case. The existence of wave operators and their asymptotic completeness are established thanks to Mourre estimates relying on new conjugate operators. We construct the asymptotic velocity and describe its spectrum. Some results are generalized to the case where −|x| α is replaced by a general second order polynomial.