A Spectral Gap Estimate and Applications

Abstract

We consider the Schrödinger 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 $w_V(y)=|I_y|,\text{ where }{I}_y:=\left\{x\in [a,b]:V(x)\le y\right\},$ then $${\lambda }_1\ge \frac{1}{250}\underset{y>\min V}{\min }\left(\frac{1}{w_V(y{)}^2}+y\right).$$ The result is sharp up to a universal constant if $\left\{x\in [a,b]:V(x)\le y\right\}$ 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 $u:\Omega \to \mathbb{R}$ be the first eigenfunction of the Laplacian $-\Delta$ on $\Omega$ with Dirichlet boundary conditions on $\partial \Omega$. We prove $$\Vert u{\Vert }_{L^{\infty }}\lesssim \frac{1}{{\rho }^{}}{\left(\frac{\rho }{D}\right)}^{1/6}\Vert u{\Vert }_{L^2},$$ which answers a question of van den Berg in the special case of two dimensions.