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May/June 2017 Decay estimates for four dimensional Schrödinger, Klein-Gordon and waveequations with obstructions at zero energy
William R. Green, Ebru Toprak
Differential Integral Equations 30(5/6): 329-386 (May/June 2017).

Abstract

We investigate dispersive estimates for the Schrödinger operator $H=-\Delta +V$ with $V$ is a real-valued decaying potential when there are zero energy resonances and eigenvalues in four spatial dimensions. If there is a zero energy obstruction, we establish the low-energy expansion \begin{align*} e^{itH}\chi(H) P_{ac}(H) & =O(\frac 1{\log t}) A_0+ O(\frac 1 t )A_1+O((t\log t)^{-1})A_2 \\ & + O(t^{-1}(\log t)^{-2})A_3. \end{align*} Here, $A_0,A_1:L^1(\mathbb R^4)\to L^\infty (\mathbb R^4)$, while $A_2,A_3$ are operators between logarithmically weighted spaces, with $A_0,A_1,A_2$ finite rank operators, further the operators are independent of time. We show that similar expansions are valid for the solution operators to Klein-Gordon and wave equations. Finally, we show that under certain orthogonality conditions, if there is a zero energy eigenvalue one can recover the $|t|^{-2}$ bound as an operator from $L^1\to L^\infty$. Hence, recovering the same dispersive bound as the free evolution in spite of the zero energy eigenvalue.

Citation

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William R. Green. Ebru Toprak. "Decay estimates for four dimensional Schrödinger, Klein-Gordon and waveequations with obstructions at zero energy." Differential Integral Equations 30 (5/6) 329 - 386, May/June 2017.

Information

Published: May/June 2017
First available in Project Euclid: 18 March 2017

zbMATH: 06738553
MathSciNet: MR3626580

Subjects:
Primary: 35Q41, 42B20

Rights: Copyright © 2017 Khayyam Publishing, Inc.

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Vol.30 • No. 5/6 • May/June 2017
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