Consider the following potential with $g>0$: $$V(x) = \frac{g}{x^2}-\frac{\Delta}{4}x^2.$$ For $\Delta=0$, the potential is bounded from below because it decreases monotonically to zero. For $\Delta<0$, it is also bounded from below but has a minimum at a finite $x$. For the first case the energy spectrum is continuous and for the second case it is discrete. My question is why is one of the spectrum continuous and the other is discrete if both operators are bounded from below?

I have seen some answers in this website that explains the spectrum from the point of view of functional analysis and operator theory. But they are dense in mathematics to me for I don't have much exposure to functional analysis. So, if you are going to use terms from operator theory it would be very helpful if you explain that term. Any physical reasoning is also very much appreciated.


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