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A counterexample concerning nontangential convergence for the solution to the time-dependent Schrödinger equation
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
2007 (English)Independent thesis Advanced level (degree of Master (One Year)), 10 credits / 15 HE creditsStudent thesis
Abstract [en]

Abstract: Considering the Schrödinger equation $\Delta_x u = i\partial{u}/\partial{t}$, we have a solution $u$ on the form $$u(x, t)= (2\pi)^{-n} \int_{\RR} {e^{i x\cdot \xi}e^{it|\xi|^2}\widehat{f}(\xi)}\, d \xi, x \in \RR, t \in \mathbf{R}$$ where $f$ belongs to the Sobolev space. It was shown by Sjögren and Sjölin, that assuming $\gamma : \mathbf{R}_+ \rightarrow \mathbf{R}_+ $ being a strictly increasing function, with $\gamma(0) = 0$ and $u$ and $f$ as above, there exists an $f \in H^{n/2} (\RR)$ such that $u$ is continuous in $\{ (x, t); t>0 \}$ and $$\limsup_{(y,t)\rightarrow (x,0),|y-x|<\gamma (t), t>0} |u(y,t)|= + \infty$$ for all $x \in \RR$. This theorem was proved by choosing $$\widehat{f}(\xi )=\widehat{f_a}(\xi )= | \xi | ^{-n} (\log | \xi |)^{-3/4} \sum_{j=1}^{\infty} \chi _j(\xi)e^{- i( x_{n_j} \cdot \xi + t_j | \xi | ^a)}, \, a=2,$$ where $\chi_j$ is the characteristic function of shells $S_j$ with the inner radius rapidly increasing with respect to $j$. The purpose of this essay is to explain the proof given by Sjögren and Sjölin, by first showing that the theorem is true for $\gamma (t)=t$, and to investigate the result when we use $$S^a f_a (x, t)= (2 \pi)^{-n}\int_{\RR} {e^{i x\cdot \xi}e^{it |\xi|^a}\widehat{f_a}(\xi)}\, d \xi$$ instead of $u$.

Place, publisher, year, edition, pages
2007. , p. 27
Keywords [en]
Time-dependent Schrödinger equation, counterexample, nontangential convergence
National Category
Mathematics
Identifiers
URN: urn:nbn:se:vxu:diva-1082OAI: oai:DiVA.org:vxu-1082DiVA, id: diva2:204865
Uppsok
fysik/kemi/matematik
Supervisors
Examiners
Available from: 2007-02-13 Created: 2007-02-13 Last updated: 2010-03-10Bibliographically approved

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