On the maximal operator associated with the free Schrödinger equation

Sichun Wang

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Displaying similar documents to “On the maximal operator associated with the free Schrödinger equation”

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We establish a Carleman type inequality for the subelliptic operator $ℒ = Δ_{z} + | x |^{2} \partial_{t}^{2}$ in $ℝ^{n + 1}$ , $n \geq 2$ , where $z \in ℝ^{n}$ , $t \in ℝ$ . As a consequence, we show that $- ℒ + V$ has the strong unique continuation property at points of the degeneracy manifold ${(0, t) \in ℝ^{n + 1} | t \in ℝ}$ if the potential $V$ is locally in certain $L^{p}$ spaces.

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In this work we study the integrability of a two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fourth degree. We give sufficient conditions for integrability in polar coordinates. Finally we establish a conjecture about the independence of the two classes of parameters which appear in the system; if this conjecture is true the integrable cases found will be the only possible ones.

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Javier Chavarriga, Jaume Giné (1997)

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In this work we study the integrability of two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fifth degree. We give a simple characterisation for the integrable cases in polar coordinates. Finally we formulate a conjecture about the independence of the two classes of parameters which appear on the system; if this conjecture is true the integrable cases found will be the only possible ones.

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A. C. Zaanen (1940)

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Necessary and sufficient conditions for a system of eigenfunctions and associated functions of a Sturm-Liouville operator with discontinuous coefficients to possess the basis property.

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Initial value problem for the time dependent Schrödinger equation on the Heisenberg group

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Let L be the full laplacian on the Heisenberg group $ℍ^{n}$ of arbitrary dimension n. Then for $f \in L^{2} (ℍ^{n})$ such that ${(I - L)}^{s / 2} f \in L^{2} (ℍ^{n})$ , s > 3/4, for a $ϕ \in C_{c} (ℍ^{n})$ we have $ʃ_{ℍ^{n}} | ϕ (x) | s u p_{0 < t \leq 1} {| e^{(\sqrt - 1) t L} f (x) |}^{2} d x \leq C_{ϕ} ∥ f ∥_{W^{s}}^{2}$ . On the other hand, the above maximal estimate fails for s < 1/4. If Δ is the sublaplacian on the Heisenberg group $ℍ^{n}$ , then for every s < 1 there exists a sequence $f_{n} \in L^{2} (ℍ^{n})$ and $C_{n} > 0$ such that ${(I - L)}^{s / 2} f_{n} \in L^{2} (ℍ^{n})$ and for a $ϕ \in C_{c} (ℍ^{n})$ we have $ʃ_{ℍ^{n}} | ϕ (x) | s u p_{0 < t \leq 1} {| e^{(\sqrt - 1) t Δ} f_{n} (x) |}^{2} d x \geq C_{n} ∥ f_{n} ∥_{W^{s}}^{2}, l i m_{n \to \infty} C_{n} = + \infty$ .