ap.analysis of pdes – Non-isolated ground state of a Schrödinger operator

Question. Is there a dimension $ d in mathbb {N} $ and a measurable function $ V: mathbb {R} ^ d to[0infty)$[0infty)$[0infty)$[0infty)$ such as the smallest spectral value $ lambda $ of the Schrödinger operator $ – Delta + V $ sure $ L ^ 2 ( mathbb {R} ^ d) $ is a proper value, but not an isolated point of the spectrum?

I would expect this to be known, but I could not give an example (neither myself nor by browsing scripts about Schrödinger operators).