The Cauchy Problem for the Nonlinear Schrödinger Equation on a Compact Manifold

Burq, Nicolas; Gérard, Patrick; Tzvetkov, Nikolay

doi:10.2991/jnmp.2003.10.s1.2

Cited by 33 publications

(58 citation statements)

References 22 publications

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“…In 2007 Burq, Gérard and Tzvetkov [4] produced results giving L p estimates of the restriction of eigenfunctions of the Laplace-Beltrami operator on a compact manifold to a submanifold. This work directly extends these results using techniques found in Koch-Tataru-Zworski [9] and Burq-Gérard-Tzvetkov [2] to move them into the more general semiclassical setting.…”

mentioning

confidence: 66%

“…Interpolating between (2) and (3) gives us better L p estimates than those given by Theorem 1.7. Consequently we can ignore regions where p(x, ξ) is bounded away from zero.…”

Section: Semiclassical Analysismentioning

confidence: 80%

“…In proving the semiclassical version for the full manifold estimates both Koch, Tataru and Zworski [9] and Burq, Gérard and Tzvetkov [2] used the assumption (A1) along with the implicit function theorem to write p(x, ξ) as…”

Section: Semiclassical Analysismentioning

confidence: 99%

“…Note that R n−1 is a time slice in M and X = R k−1 is a time slice in Y . The Strichartz assumptions modified to include a semiclassical parameter h (see Koch-Tataru-Zworski [9] and BurqGérard-Tzvetkov [2]) are that,…”

Section: Extended Strichartz Estimatesmentioning

confidence: 99%

“…A more detailed discussion of Semiclassical Analysis can be found in [9], [5] and [2], however for the reader's convenience the main definitions and results used in this paper are provided in Section 1.…”

mentioning

confidence: 99%

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SemiclassicalL^pEstimates of Quasimodes on Submanifolds

Tacy

2010

Communications in Partial Differential Equations

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Let P = P (h) be a semiclassical pseudodifferential operator on a Riemannian manifold M . Suppose that u(h) is a localised, L 2 normalised family of functions such that P (h)u(h) is O(h) in L 2 , as h → 0. Then, for any submanifold Y ⊂ M , we obtain estimates on the L p norm of u(h) restricted to Y , with exponents that are sharp for h → 0. These results generalise those of Burq, Gérard and Tzvetkov [4] on L p norms for restriction of Laplacian eigenfunctions. As part of the technical development we prove some extensions of the abstract Strichartz estimates of Keel and Tao [7].Let P = P (h) be a semiclassical pseudodifferential operator on a Riemannian manifold M . We will assume that P has a real principal symbol, and that its full symbol is smooth in the semiclassical parameter h. Other more technical assumptions on P are given in Definition 1.6. We prove estimates for approximate solutions u = u(h) to the equation P (h)u(h) = 0. As usual in semiclassical analysis we assume that u(h) is defined at least for a sequence h n tending to zero.Our precise definition of approximate solution, or quasimode, is thatThis definition is natural with respect to localisation: if, and χ is a pseudodifferential operator of order zero (with a symbol smooth in h), then P (χu) is also O L 2 (h). We will make the assumption that u(h) can be localised, see Definition 1.3, and therefore will be able to reduce the problem to one of local analysis.Given a submanifold Y of M , we estimate the L p norm of the restriction of u to Y , assuming the normalisation condition u L 2 (M ) = 1. These estimates are of the form u L p (Y ) ≤ Ch −δ where δ depends on the dimension n of M , the dimension k of Y and p (except for one case where there is a logarithmic divergence) -see Theorem 1.7. In every case the exponent δ(n, k, p) given by Theorem 1.7 is optimal. Figure 1 shows the exponent δ for a hypersurface and, for comparison, the L p estimates over the whole manifold (Sogge [11] for spectral clusters and Koch-Tataru-Zworski [9] for semiclassical operators). The potential growth/concentration of the quasimodes of a semiclassical operator is of great interest due to the connection to Quantum Mechanics. It is from Quantum Mechanics that we get the important set of motivating examples,here ∆ g is the (positive) Laplace-Beltrami operator associated with the metric g. We can transition between this picture and the usual eigenfunction picture of Quantum Mechanics by dividing the eigenfunction equationwhere P is as in (1) with a potential term of h 2 V 0 (x) − 1. Therefore the higher eigenvalue asymptotics of eigenfunctions of Quantum Mechanical systems corresponds to the h → 0 limit in semiclassical analysis. When V 0 (x) = 0 this problem reduces to estimating the size of Laplacian eigenfunctions restricted to a submanifold. A complete set of estimates for Laplacian eigenfunctions on compact manifolds is given by Burq, Gérard and Tzvetkov [4].A

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mentioning

confidence: 66%

“…Interpolating between (2) and (3) gives us better L p estimates than those given by Theorem 1.7. Consequently we can ignore regions where p(x, ξ) is bounded away from zero.…”

Section: Semiclassical Analysismentioning

confidence: 80%