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arXiv · 1810.03678

On the Fourth order Schr\"odinger equation in four dimensions: dispersive estimates and zero energy resonances

William R. Green·Ebru Toprak
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Abstract

We study the fourth order Schr\"odinger operator $H=(-\Delta)^2+V$ for a decaying potential $V$ in four dimensions. In particular, we show that the $t^{-1}$ decay rate holds in the $L^1\to L^\infty$ setting if zero energy is regular. Furthermore, if the threshold energies are regular then a faster decay rate of $t^{-1}(\log t)^{-2}$ is attained for large $t$, at the cost of logarithmic spatial weights. Zero is not regular for the free equation, hence the free evolution does not satisfy this bound due to the presence of a resonance at the zero energy. We provide a full classification of the different types of zero energy resonances and study the effect of each type on the time decay in the dispersive bounds.

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William R. Green, Ebru Toprak. 2018-10-08. On the Fourth order Schr\"odinger equation in four dimensions: dispersive estimates and zero energy resonances. https://doi.org/10.1016/j.jde.2019.03.004

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