A ground state alternative for singular Schrödinger operators

Abstract

Let $\mathbf{a}$ be a quadratic form associated with a Schrödinger operator $L=-\nabla \cdot (A\nabla )+V$ on a domain $\Omega \subset {\mathbb{R}}^d$. If $\mathbf{a}$ is nonnegative on $C_0^{\infty }(\Omega )$, then either there is $W>0$ such that $\int W|u{|}^{2}dx\le \mathbf{a}[u]$ for all $C_0^{\infty }(\Omega ;\mathbb{R})$, or there is a sequence ${\varphi }_k\in C_0^{\infty }(\Omega )$ and a function $\varphi >0$ satisfying $L\varphi =0$ such that $\mathbf{a}[{\varphi }_k]\to 0$, ${\varphi }_k\to \varphi$ locally uniformly in $\Omega \setminus \{x_0\}$. This dichotomy is equivalent to the dichotomy between $L$ being subcritical resp. critical in $\Omega$. In the latter case, one has an inequality of Poincaré type: there exists $W>0$ such that for every $\psi \in C_0^{\infty }(\Omega ;\mathbb{R})$ satisfying $\int \psi \varphi dx\ne 0$ there exists a constant $C>0$ such that $C^{-1}\int W|u{|}^{2}dx\le \mathbf{a}[u]+C|\int u\psi dx{|}^2$ for all $u\in C_0^{\infty }(\Omega ;\mathbb{R})$.

14 pages