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As we know a particle in attractive Dirac delta potential has discontinuity in the derivative of its wavefunction.

I have two questions in this regard:

  1. Can a second order differential equation be still defined when the first order derivative is discontinuous at some point?

  2. How can we justify $E<0$ for bound state when the particle in an attractive Dirac-delta potential will be confined to a single point?

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    $\begingroup$ A classical particle will be confined to a single point. A wavefunction describing a quantum particle will not. $\endgroup$ – probably_someone Mar 12 '18 at 5:47
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  1. Yes, see weak solutions.

  2. The wave function doesn't have to vanish in classically forbidden regions, cf. quantum tunnelling.

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