What can we qualitatively say about the spectrum of a Hamiltonian operator if the spectrum is not bounded from below but bounded from above?

e.g. $$\mathcal{H}=-\frac{d}{dx^2}-\frac{\lambda}{x^2}$$

This potential has a center of attraction at the origin.

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    $\begingroup$ This inverse square potential is discussed in this Phys,SE post. $\endgroup$ – Qmechanic Nov 3 '19 at 3:56