Decay of eigenfunctions of elliptic PDE's

TL;DR

This paper investigates how eigenfunctions of self-adjoint higher order elliptic operators decay exponentially on , establishing algebraic criteria for decay rates, ruling out super-exponential decay, and providing refined bounds.

Contribution

It introduces algebraic criteria for decay rates and rules out super-exponential decay for eigenfunctions of elliptic operators, with refined exponential bounds.

Findings

Critical decay rates are algebraically determined.

Super-exponentially decaying eigenfunctions do not exist.

Refined exponential upper bounds are established.

Abstract

We study exponential decay of eigenfunctions of self-adjoint higher order elliptic operators on $\R^d$. We show that the possible critical decay rates are determined algebraically. In addition we show absence of super-exponentially decaying eigenfunctions and a refined exponential upper bound.