Question on number of eigenvalues of a Schrödinger operator

Question

Suppose I have a 1D Schrödinger operator on $L^2(\mathbb{R})$ $$ A = -\frac{d^2}{dx^2}+V(x) $$ where $V$ is real-valued, negative, smooth and satisfies $\lim_{|x|\rightarrow\infty}x^2V(x)=c$ for a constant $c$. What can be said about the number of eigenvalues $\lambda\leq 0$ and whether or not there are positive eigenvalues $\lambda>0$. In particular, are there finitely many non-positive eigenvalues and no positive ones? Are there references for this case?