Eigenfunctions with double exponential rate of localization

TL;DR

This paper constructs specific eigenfunctions and heat solutions in cylindrical domains that decay at a double exponential rate, advancing understanding of localization phenomena in PDEs.

Contribution

It introduces explicit constructions of eigenfunctions and heat solutions with double exponential decay, extending classical counterexamples in unique continuation theory.

Findings

Eigenfunctions with double exponential decay in cylindrical domains.

Heat solutions with double exponential decay in half-cylinders.

Connections to classical counterexamples in unique continuation.

Abstract

We construct a real-valued solution to the eigenvalue problem $-\text{div}(A\nabla u)=\lambda u$, $\lambda>0,$ in the cylinder $\mathbb{T}^2\times \mathbb{R}$ with a real, uniformly elliptic, and uniformly $C^1$ matrix $A$ such that $|u(x,y,t)|\leq C e^{-c e^{c|t|}}$ for some $c,C>0$. We also construct a complex-valued solution to the heat equation $u_t=\Delta u + B \nabla u$ in a half-cylinder with continuous and uniformly bounded $B$, which also decays with double exponential speed. Related classical ideas, used in the construction of counterexamples to the unique continuation by Plis and Miller, are reviewed.