On the fourth order Schrödinger equation in three dimensions: dispersive estimates and zero energy resonances.(English)Zbl 1455.35212
Summary: We study the fourth order Schrödinger operator \(H=(-\Delta)^2+V\) for a short range potential in three space dimensions. We provide a full classification of zero energy resonances and study the dynamic effect of each on the \(L^1\to L^\infty\) dispersive bounds. In all cases, we show that the natural \(| t |^{-\frac{3}{4}}\) decay rate may be attained, though for some resonances this requires subtracting off a finite rank term, which we construct and analyze. The classification of these resonances, as well as their dynamical consequences differ from the Schrödinger operator \(-\Delta+V\).
MSC:
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 35B34 | Resonance in context of PDEs |
Cite
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