Decay estimates for four dimensional Schr\"odinger, Klein-Gordon and wave equations with obstructions at zero energy

TL;DR

This paper derives decay estimates for the Schrödinger, Klein-Gordon, and wave equations in four dimensions with zero energy obstructions, providing detailed low-energy expansions and conditions for optimal decay rates.

Contribution

It establishes precise low-energy expansions and decay estimates for these equations in the presence of zero energy resonances and eigenvalues, extending previous results to four dimensions.

Findings

Derived low-energy expansion for Schrödinger evolution with zero energy obstructions.

Established decay estimates for Klein-Gordon and wave equations under similar conditions.

Showed that orthogonality conditions can recover optimal decay rates despite zero energy eigenvalues.

Abstract

We investigate dispersive estimates for the Schr\"odinger 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.