Schrödinger operators on Lie groups with purely discrete spectrum

On a Lie group G, we investigate the discreteness of the spectrum of Schrödinger operators of the form L+V, where L is a subelliptic sub-Laplacian on G and the potential V is a locally integrable function which is bounded from below. We prove general necessary and sufficient conditions for arbitrary potentials, and we obtain explicit characterizations when V is a polynomial on G or belongs to a local Muckenhoupt class. We finally discuss how to transfer our results to weighted sub-Laplacians on G.