There are many results of the type 'an eigenfunction of a Schrödinger operator can not be too small at infinity'. Smallness in this context may mean superexponential decay, or compact support, vanishing of infinite order at a point, or other, even weaker, conditions.
I am interested in the following question: can an eigenfunction be supported in a set of finite measure?
The question is intentionally vague, I am curious about any result of this type, for operators of the form $H=(i\partial+A(x))^2+V(x)$ on $L^2(\mathbb{R}^n)$, under suitable assumption on the potentials $A(x)$ and $V(x)$. In particular I am interested in the case of unbounded potentials.
