Bound state in potential less 0

Question

How to prove that there is a bound state in the potential $U(x) = -A e^{-a |x|}$, where for all $a \in \mathbb{R}$ and $A>0$. I heard that we can say something to the minimum of this form $ \left( \psi \right| H \left| \psi \right)$ for some vector of hilbert space, but that it will give us?

So i want to know, why if there is $\psi$ such that $\left( \psi \right| H \left| \psi \right) < 0$ then there is bound state?

Thank you!

Answer 1

The potential energy can have any reference energy without change in the outcome. Thus a negative energy is relative to the assumed reference energy. If you take a large enough negative reference, you will only have positive bound energy states.