$L^p$-bounds for eigenfunctions of analytic non-self-adjoint operators with double characteristics

TL;DR

This paper establishes sharp $L^p$-bounds for eigenfunctions of certain non-self-adjoint semiclassical operators with doubly-characteristic symbols, improving previous results under less restrictive assumptions.

Contribution

It provides new sharp uniform $L^p$-bounds for eigenfunctions of non-self-adjoint operators with doubly-characteristic symbols, relaxing conditions on the quadratic approximation.

Findings

Bounds are sharp and uniform for low-lying eigenfunctions.

Results apply to operators with analytic symbols extending holomorphically.

Assumptions on the quadratic approximation are less restrictive than prior work.

Abstract

We prove sharp uniform $L^p$-bounds for low-lying eigenfunctions of non-self-adjoint semiclassical pseudodifferential operators $P$ on $\mathbb{R}^{n}$ whose principal symbols are doubly-characteristic at the origin of $\mathbb{R}^{2n}$. Our bounds hold under two main assumptions on $P$: (1) the total symbol of $P$ extends holomorphically to a neighborhood of $\mathbb{R}^{2n}$ in $\mathbb{C}^{2n}$, and (2) the quadratic approximation to the principal symbol of $P$ at the origin is elliptic along its singular space. Most notably, our assumptions on the quadratic approximation are less restrictive than those made in prior works, and our main theorem improves the already known results in the case when the symbol of $P$ is analytic.