Is there an example of a Schrödinger operator $-\Delta + V$ in $\mathbb R^3$ with a negative, smooth, compactly supported potential $V$, finitely many negative eigenvalues and no zero eigenvalue?
In general, it seems to be a difficult problem to determine if such an operator with a specific potential has no zero eigenvalue and I just found results for long range potentials, that is, potentials which decay slower than $-\frac{1}{x^2}$. However, I guess it should be quite plausible that such potentials should exist.
In the Spherical cavity, zero eigenvalues occur for certain values of increasing well depth where new eigenvalues come into existence.