Dispersive Estimates for higher dimensional Schr\"odinger Operators with threshold eigenvalues II: The even dimensional case

TL;DR

This paper derives dispersive estimates for the Schrödinger operator in even dimensions with zero energy eigenvalues, revealing how the evolution operator behaves over time and under orthogonality conditions.

Contribution

It provides new dispersive bounds and operator expansions for Schrödinger operators with zero eigenvalues in even dimensions, extending previous results to higher dimensions.

Findings

Dispersive estimates involve a rank one operator $F_t$ with decay rate $|t|^{2-rac{n}{2}}$.

Operator expansion includes terms $A_{-2}$, $A_{-1}$, and $A_0$ with specific decay rates.

Orthogonality conditions influence the vanishing of leading terms and the structure of the evolution operator.

Abstract

We investigate $L^1(\mathbb R^n)\to L^\infty(\mathbb R^n)$ dispersive estimates for the Schr\"odinger operator $H=-\Delta+V$ when there is an eigenvalue at zero energy in even dimensions $n\geq 6$. In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator $F_t$ satisfying $\|F_t\|_{L^1\to L^\infty} \lesssim |t|^{2-\frac{n}{2}}$ for $|t|>1$ such that $$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim |t|^{1-\frac{n}{2}},\,\,\,\,\,\text{ for } |t|>1.$$ With stronger decay conditions on the potential it is possible to generate an operator-valued expansion for the evolution, taking the form \begin{align*} e^{itH} P_{ac}(H)=|t|^{2-\frac{n}{2}}A_{-2}+ |t|^{1-\frac{n}{2}} A_{-1}+|t|^{-\frac{n}{2}}A_0, \end{align*} with $A_{-2}$ and $A_{-1}$ mapping $L^1(\mathbb R^n)$ to $L^\infty(\mathbb R^n)$ while $A_0$ maps weighted $L^1$ spaces to weighted $L^\infty$ spaces. The leading-order terms $A_{-2}$ and $A_{-1}$ are both finite rank, and vanish when certain orthogonality conditions between the potential $V$ and the zero energy eigenfunctions are satisfied. We show that under the same orthogonality conditions, the remaining $|t|^{-\frac{n}{2}}A_0$ term also exists as a map from $L^1(\mathbb R^n)$ to $L^\infty(\mathbb R^n)$, hence $e^{itH}P_{ac}(H)$ satisfies the same dispersive bounds as the free evolution despite the eigenvalue at zero.