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According to Griffith's Quantum Mechanics, "$E$ must exceed the minimum value of $V(x)$, for every normalizable solution to the time independent Schroedinger equation"

As an example, there is no acceptable solution to the time-independent Schroedinger equation for the infinite square well with $E = 0$ or $E < 0$.

For the Dirac delta function potential, however, there are both bound states with $E < 0$ and scattering states with $E > 0$ allowed. This is also true for the finite square well.

Where in the process of solving the T.I.S.E. can one determine if states with $E < V_0$ are allowed?

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The key here is that E must exceed the minimum of $V(x)$.

If one has a delta function well then the minimum is $V_0 = -\infty$ while for a delta function barrier the minimum is $V_0 = 0$

Hence for the delta function well, one can have bound states with $E<0$ or one can have scattering states with $E>0$.

For the delta function barrier the necessary condition is $E>0$, hence only scattering states are allowed.

The reasoning is similar for any finite potential well or barrier.

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