Decay estimates for four dimensional Schrödinger, Klein-Gordon and wave equations with obstructions at zero energy

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 $$e^{itH}\chi (H)P_{ac}(H)=O(1/(\log t))A_0+O(1/t)A_1+O((t\log t{)}^{-1})A_2+O(t^{-1}(\log t{)}^{-2})A_3.$$ Here $A_0,A_1:L^1({\mathbb{R}}^n)\to L^{\infty }({\mathbb{R}}^n)$, 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.

arXiv commentUpdated references. To appear in Differential and Integral Equations