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 $Ω\subset \mathbb{R}^d$. If $\mathbf{a}$ is nonnegative on $C_0^{\infty}(Ω)$, then either there is $W>0$ such that $\int W|u|^2 dx\leq \mathbf{a}[u]$ for all $C_0^{\infty}(Ω;\mathbb{R})$, or there is a sequence $ϕ_k\in C_0^{\infty}(Ω)$ and a function $ϕ>0$ satisfying $Lϕ=0$ such that $\mathbf{a}[ϕ_k]\to 0$, $ϕ_k\toϕ$ locally uniformly in $Ω\setminus\{x_0\}$. This dichotomy is equivalent to the dichotomy between $L$ being subcritical resp. critical in $Ω$. In the latter case, one has an inequality of Poincaré type: there exists $W>0$ such that for every $ψ\in C_0^\infty(Ω;\mathbb{R})$ satisfying $\int ψϕdx \neq 0$ there exists a constant $C>0$ such that $C^{-1}\int W|u|^2 dx\le \mathbf{a}[u]+C|\int u ψdx|^2$ for all $u\in C_0^\infty(Ω;\mathbb{R})$.