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I was reading (Griffith's QM book) about the Bound states for delta-function potential of the form $-\alpha \, \delta(x)$ where $\alpha > 0$.

I feel a bit conceptually unclear. Few doubts I have are:

  1. For this question let me consider $V=\alpha\delta(x)$ where $\alpha > 0$. Now because of tunneling effect, we are only concerned with the magnitude of potential as $x \rightarrow \pm\infty$. It is said in the book that - The distinction is clearer in the quantum domain because the phenomenon of Tunneling allows the particle to "leak" through any finite potential barrier. But Delta-function is $\infty$ at $x=0$. So how is the particle able to tunnel through this barrier? Am I defining the finite potential barrier wrong?
  2. At a later point, he argues that the probability of passing through the potential even if $E < V_{max}$. So this is only when $E$ satisfies the condition $E > [V(\infty) \;or\; V(-\infty)]$ right?
  3. Since we are considering only the boundary conditions at $\pm \infty$, we only look for tunneling in scattering states is it? Since the bound state is normalizable and stationary, the wave function doesn't change at any time, and therefore we don't need to bother about any tunneling effects in the bound state's case. Is this reasoning sound?
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