共查询到20条相似文献,搜索用时 15 毫秒
1.
Haruya Mizutani 《偏微分方程通讯》2013,38(2):169-224
The present article is concerned with Schrödinger equations on non-compact Riemannian manifolds with asymptotically conic ends. It is shown that, for any admissible pair (including the endpoint), local in time Strichartz estimates outside a large compact set are centered at origin hold. Moreover, we prove global in space Strichartz estimates under the nontrapping condition on the metric. 相似文献
2.
Matthew D. Blair G. Austin Ford Sebastian Herr Jeremy L. Marzuola 《Journal of Geometric Analysis》2012,22(2):339-351
We prove Strichartz estimates with a loss of derivatives for the Schrödinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates on the polygon follow from those on Euclidean surfaces with conical singularities. We develop a Littlewood-Paley squarefunction estimate with respect to the spectrum of the Laplacian on these spaces. This allows us to reduce matters to proving estimates at each frequency scale. The problem can be localized in space provided the time intervals are sufficiently small. Strichartz estimates then follow from a recent result of the second author regarding the Schrödinger equation on the Euclidean cone. 相似文献
3.
We introduce the Besov space $\dot{B}^{0,L}_{1,1}$ associated with the Schrödinger operator L with a nonnegative potential satisfying a reverse Hölder inequality on the Heisenberg group, and obtain the molecular decomposition. We also develop the Hardy space $H_{L}^{1}$ associated with the Schrödinger operator via the Littlewood–Paley area function and give equivalent characterizations via atoms, molecules, and the maximal function. Moreover, using the molecular decomposition, we prove that $\dot{B}^{0,L}_{1,1}$ is a subspace of $H_{L}^{1}$ . 相似文献
4.
In this note we consider the Schrödinger equation on compact manifolds equipped with possibly degenerate metrics. We prove Strichartz estimates with a loss of derivatives. The rate of loss of derivatives depends on the degeneracy of metrics. For the non-degenerate case we obtain, as an application of the main result, the same Strichartz estimates as that in the elliptic case. This extends Strichartz estimates for Riemannian metrics proved by Burq-Gérard-Tzvetkov to the non-elliptic case and improves the result by Salort for the degenerate case. We also investigate the optimality of the result for the case on 𝕊3 × 𝕊3. 相似文献
5.
Clotilde Fermanian Kammerer 《偏微分方程通讯》2013,38(1):19-44
We consider a semi-classical Schrödinger operator with a matrix-valued potential presenting eigenvalue crossings on isolated points. We obtain estimates for the boundary values of the resolvent under a generalized non-trapping assumption. As a consequence, we prove the smoothing effect of this operator, derive Strichartz type estimate for the propagator and get an existence theorem for a system of non-linear Schrödinger equations. 相似文献
6.
There is a family of potentials that minimize the lowest eigenvalue of a Schrödinger operator under the constraint of a given L p norm of the potential. We give effective estimates for the amount by which the eigenvalue increases when the potential is not one of these optimal potentials. Our results are analogous to those for the isoperimetric problem and the Sobolev inequality. We also prove a stability estimate for Hölder’s inequality, which we believe to be new. 相似文献
7.
Georgi Vodev 《Annales Henri Poincare》2005,6(6):1179-1196
We prove time decay L1 → L∞ estimates for the Schr?dinger group eit(−Δ + V) for real-valued potentials
satisfying V (x) = O (|x|−δ), |x| ≫ 1, with δ > 5/2.
Communicated by Bernard Helffer
submitted 27/11/04, accepted 29/04/05 相似文献
8.
《分析论及其应用》2016,(1)
Let Hnbe the Heisenberg group and Q=2n +2 be its homogeneous dimension. In this paper, we consider the Schrdinger operator-?Hn+V, where ?Hn is the sub-Laplacian and V is the nonnegative potential belonging to the reverse Ho ¨lder class Bq1 for q1≥ Q/2. We show that the operators T1= V(-?Hn +V)-1 and T2=V1/2(-?Hn +V)-1/2 are both bounded from H1L(Hn) into L1(Hn). Our results are also valid on the stratified Lie group. 相似文献
9.
Laurent Bakri 《偏微分方程通讯》2013,38(1):69-91
On a closed manifold, we give a quantitative Carleman estimate on the Schrödinger operator. We then deduce quantitative uniqueness results for solutions to the Schrödinger equation using doubling estimates. Finally we investigate the sharpness of this results with respect to the electric potential. 相似文献
10.
Ahmed A. Abdelhakim 《Archiv der Mathematik》2014,102(2):165-169
We find new necessary conditions for the estimate ${||u||_{L^{q}_{t} (\mathbb{R}; L^{r}_{x} (\mathbb{R}^{n}))} \lesssim\,||F||_{L^{{\tilde{q}}^{\prime}}_{t}(\mathbb{R};L^{{\tilde{r}}^{\prime}}_{x}(\mathbb{R}^{n}))}}$ , where u = u(t, x) is the solution to the Cauchy problem associated with the free inhomogeneous Schrödinger equation with identically zero initial data and inhomogeneity F = F(t, x). 相似文献
11.
We prove the existence and the uniqueness of a solution to the stochastic NSLEs on a two-dimensional compact riemannian manifold. Thus we generalize (and improve) a recent work by Burq et al. (J Nonlinear Math Phys 10(1):12–27, 2003) and a series of papers by de Bouard and Debussche, see e.g. de Bouard and Debussche (Commun Math Phys 205(1):161–181, 1999 and Stoch Anal Appl 21(1):97–126, 2003) who have examined similar questions in the case of the flat euclidean space. We prove the existence and the uniqueness of a local maximal solution to stochastic nonlinear Schrödinger equations with multiplicative noise on a compact d-dimensional riemannian manifold. Under more regularity on the noise, we prove that the solution is global when the nonlinearity is of defocusing or of focusing type, d?=?2 and the initial data belongs to the finite energy space. Our proof is based on improved stochastic Strichartz inequalities. 相似文献
12.
13.
We prove local smoothing estimates for the Schrödinger initial value problem with data in the energy space L 2(? d ), d ≥ 2 and a general class of potentials. In the repulsive setting we have to assume just a power like decay (1 + |x|)?γ for some γ > 0. Also attractive perturbations are considered. The estimates hold for all time and as a consequence a weak dispersion of the solution is obtained. The proofs are based on similar estimates for the corresponding stationary Helmholtz equation and Kato H-smooth theory. 相似文献
14.
VIJAY KUMAR SOHANI 《Proceedings Mathematical Sciences》2013,123(4):525-537
We obtain homogeneous Strichartz estimate for the Schrödinger propagator e $^{-itL_{\alpha}}$ for the Laguerre operator L α on ${\mathbb R}_+^n$ . We follow regularization technique as introduced in J. Funct. Anal. 224(2) (2005) 371–385. We also establish inhomogeneous Strichartz estimates for different admissible pairs. 相似文献
15.
The purpose of this Note is to prove sharp Strichartz estimates with derivative losses for the non-elliptic Schrödinger equation posed on the 2-dimensional torus. 相似文献
16.
In this paper we present analogues of the maximum principle and of some parabolic inequalities for the regularized time-dependent Schrödinger operator on open manifolds using Günter derivatives. Moreover, we study the uniqueness of bounded solutions for the regularized Schrödinger–Günter problem and obtain the corresponding fundamental solution. Furthermore, we present a regularized Schrödinger kernel and prove some convergence results. Finally, we present an explicit construction for the fundamental solution to the Schrödinger–Günter problem on a class of conformally flat cylinders and tori. 相似文献
17.
We prove a Poisson type formula for the Schrödinger group. Such a formula had been derived in a previous article by the authors, as a consequence of the study of the asymptotic behavior of nonlinear wave operators for small data. In this note, we propose a direct proof, and extend the range allowed for the power of the nonlinearity to the set of all short range nonlinearities. Moreover, H 1-critical nonlinearities are allowed. 相似文献
18.
Yu MIAO 《数学年刊B辑(英文版)》2007,28(5):555-570
In the present paper,the full range Strichartz estimates for homogeneous Schr(?)dinger equations with non-degenerate and non-smooth coefficients are proved.For inhomogeneous equation,the non-endpoint Strichartz estimates are also obtained. 相似文献
19.
FundamentalSolutionforOperatorLmontheHeisenbergGroupHnMaYulan(马玉兰)(DepartmentofMathematics,LanzhouUniversity,Lanzhou,Gansu,73... 相似文献
20.
P. K. Ratnakumar 《Journal of Fourier Analysis and Applications》2008,14(2):286-300
We establish a Strichartz type estimate for the Schrödinger propagator e it? for the special Hermite operator ? on ? n . Our method relies on a regularization technique. We show that no admissibility condition is required on (q,p) when 1≤q≤2. 相似文献