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

TL;DR

This paper establishes dispersive estimates for higher-dimensional Schrödinger operators with zero-energy eigenvalues in odd dimensions, revealing conditions under which the evolution behaves similarly to the free case despite the eigenvalue.

Contribution

It provides new dispersive bounds and operator expansions for Schrödinger operators with zero eigenvalues in odd dimensions, including orthogonality conditions that nullify leading order terms.

Findings

Dispersive estimates with decay rates depending on dimension

Existence of operator-valued expansions for the evolution

Orthogonality conditions eliminate leading order terms

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 and $n\geq 5$ is odd. 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 $$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim |t|^{1-\frac{n}{2}},\qquad\textrm{ for } |t|>1.$$ With stronger decay conditions on the potential it is possible to generate an operator-valued expansion for the evolution, taking the form $$ e^{itH} P_{ac}(H)=|t|^{2-\frac{n}{2}}A_{-2}+ |t|^{1-\frac{n}{2}} A_{-1}+|t|^{-\frac{n}{2}}A_0, $$ with $A_{-2}$ and $A_{-1}$ finite rank operators 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 $A_{-1}$ 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.