A Spectral Gap Estimate and Applications

TL;DR

This paper establishes a lower bound on the first eigenvalue of a Schrödinger operator using sublevel set estimates and applies this to bound the maximum of the first Laplacian eigenfunction in convex planar domains.

Contribution

It introduces a sharp spectral gap estimate based on sublevel set measures and applies it to eigenfunction bounds in convex domains, answering a specific open question.

Findings

Lower bound on eigenvalue in terms of sublevel set measures.

Eigenfunction maximum bound depending on inradius and diameter.

Answer to van den Berg's question in 2D convex domains.

Abstract

We consider the Schr\"odinger operator $$-\frac{d^2}{d x^2} + V \qquad \mbox{on an interval}~~[a,b]~\mbox{with Dirichlet boundary conditions},$$ where $V$ is bounded from below and prove a lower bound on the first eigenvalue $\lambda_1$ in terms of sublevel estimates: if $ w_V(y) = |I_y|,\text{ where } I_y := \left\{ x \in [a,b]: V(x) \leq y \right\},$ then $$ \lambda_1 \geq \frac{1}{250} \min_{y > \min V}{\left(\frac{1}{w_V(y)^2} + y\right)}.$$ The result is sharp up to a universal constant if $\left\{ x \in [a,b]: V(x) \leq y \right\}$ is an interval for the value of $y$ solving the minimization problem. An immediate application is as follows: let $\Omega \subset \mathbb{R}^2$ be a convex domain with inradius $\rho$ and diameter $D$ and let $u:\Omega \rightarrow \mathbb{R}$ be the first eigenfunction of the Laplacian $-\Delta$ on $\Omega$ with Dirichlet boundary conditions on $\partial \Omega$. We prove $$ \| u \|_{L^{\infty}} \lesssim \frac{1}{\rho^{}} \left( \frac{\rho}{D} \right)^{1/6} \|u\|_{L^2},$$ which answers a question of van den Berg in the special case of two dimensions.