Is there a proof that does not depend on Euclidean methods? Is this a proof? :

$V(g)$ can be written as $P+R$ where $P$ is non-negative and $R$ is $N$-bounded (and hence $(H_0+\lambda P)$-bounded). $H_0+\lambda P$ is essentially self-adjoint by this method. (These are short proofs depending on estimates on the kernels and $\phi < \sqrt N$).

By this theorem, $H_0+\lambda P$ has a gap for small $\lambda$. A gap is stable under the relatively bounded perturbation $\lambda R$, so $H_0+\lambda V$ has a gap for small $\lambda$.