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

Zienkiewicz, Jacek

Studia Mathematica, Tome 122 (1997), p. 15-37 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $ʃ_{ℍ^{n}} | ϕ (x) | s u p_{0 < t \leq 1} {| e^{(\sqrt - 1) t Δ} f_{n} (x) |}^{2} d x \geq C_{n} ∥ f_{n} ∥_{W^{s}}^{2}, l i m_{n \to \infty} C_{n} = + \infty$ .

Publié le : 1997-01-01

Zbl 0869.22004

EUDML-ID : urn:eudml:doc:216357

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     author = {Jacek Zienkiewicz},
     title = {Initial value problem for the time dependent Schr\"odinger equation on the Heisenberg group},
     journal = {Studia Mathematica},
     volume = {122},
     year = {1997},
     pages = {15-37},
     zbl = {0869.22004},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv122i1p15bwm}
}

Zienkiewicz, Jacek. Initial value problem for the time dependent Schrödinger equation on the Heisenberg group. Studia Mathematica, Tome 122 (1997) pp. 15-37. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv122i1p15bwm/

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