# Bound states for the Delta Function Well

In "Introduction to Quantum Mechanics" by Griffiths we discussed the delta potential well. They speak about bound and scattering states for $E<0$ and $E>0$ resp. But before that (Problem 2.2) we proved that $E$ must exceed the minimum value of $V(x)$. Clearly $V(x)$ has minimum value 0, How then, can there exist bound states?

The one-dimensional delta function well $V(x)= -\lambda \delta(x)$ can be constructed by taking $\delta(x)= \lim_{\epsilon\to 0} \delta_\epsilon(x)$, where $\delta_\epsilon(x)$ is a square potential of width $\epsilon$ and hight $1/\epsilon$. Thus the minimum value of $V(x)$ is $-\lambda/\epsilon$ which is heading to minus infinity.