I think you can find an answer to your question in the book of Simon/Reed, "Methods of Mathematical Physics", vol.4 "Analysis of Operators". They have a chapter devoted to the question of the existence of nondegenerate ground states, chapter XIII.12.

A relevant theorem would be XIII.47, which says that the Schrödinger operator has a nondegenerate strictly positive ground state if the potential V is in $L^2_{loc}(\mathbb{R}^n)$ and $lim_{|x| \to \infty} V(x) = \infty$.

I think you can find an answer to your question in the book of Simon/Reed, "Methods of Mathematical Physics", vol.4 "Analysis of Operators". They have a chapter devoted to the question of the existence of nondegenerate ground states, chapter XIII.12.

One relevant theorem would be XIII.47, which says that the Schrödinger operator has a nondegenerate strictly positive ground state if the potential V is in $L^2_{loc}(\mathbb{R}^n)$ and $lim_{|x| \to \infty} V(x) = \infty$.

I don't think that there is a simple necessary condition on the potential, but only several sets of sufficient conditions, but could be wrong about that.