A Note on Weighted Estimates for the Schrödinger Operator

Varcárcel, Juan Antonio Barceló; Bennett, Jonathan; Carbery, Anthony; González, Alberto Ruiz; Bendaña, María de la Cruz Vilela

doi:10.5209/rev_rema.2008.v21.n2.16405

Cited by 6 publications

(2 citation statements)

References 10 publications

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“…At this point, it should be noted that this local smoothing estimate can be written in terms of a weighted L 2 setting as in (1). Indeed, if ρ is any function such that j∈Z ρ 2 L ∞ (|x|∼2 j ) < ∞, one has…”

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confidence: 99%

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On morawetz estimates with time-dependent weights for the Klein-Gordon equation

Kim¹,

Lee²,

Seo³

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - A

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confidence: 99%

“…One is spectral methods based on resolvent estimates, starting from Kato's work [6], and the other is to use Fourier restriction estimates from harmonic analysis (see e.g. [1,18] and references therein). A completely different approach was also recently developed by Ruzhansky and Sugimoto [17] using canonical transforms.…”

mentioning

confidence: 99%

On morawetz estimates with time-dependent weights for the Klein-Gordon equation

Kim¹,

Lee²,

Seo³

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - A

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From resolvent estimates to unique continuation for the Schrödinger equation

Seo

2016

Trans. Amer. Math. Soc.

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In this paper we develop an abstract method to handle the problem of unique continuation for the Schrödinger equation ( i ∂ t + Δ ) u = V ( x ) u (i\partial _t+\Delta )u=V(x)u . In general the problem is to find a class of potentials V V which allows the unique continuation. The key point of our work is to make a direct link between the problem and the weighted L 2 L^2 resolvent estimates ‖ ( − Δ − z ) − 1 f ‖ L 2 ( | V | ) ≤ C ‖ f ‖ L 2 ( | V | − 1 ) \|(-\Delta -z)^{-1}f\|_{L^2(|V|)}\leq C\|f\|_{L^2(|V|^{-1})} . We carry it out in an abstract way, and thereby we do not need to deal with each of the potential classes. To do so, we will make use of the limiting absorption principle and Kato H H -smoothing theorem in spectral theory, and employ some tools from harmonic analysis. Once the resolvent estimate is set up for a potential class, from our abstract theory the unique continuation would follow from the same potential class. Also, it turns out that there can be no dented surface on the boundary of the maximal open zero set of the solution u u . In this regard, another main issue for us is to know which class of potentials allows the resolvent estimate. We establish such a new class which contains previously known ones, and we will also apply it to the problem of well-posedness for the equation.

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