Past exam of the mathematics course of the University of Cambridge / 2002 / ib / Paper 1 / 18D / a / Solution

The attractive potential has $\Delta >0$. Its time-independent Schrödinger equation isIntegrate from $-\epsilon$ to $\epsilon$. Continuity makes the delta term equal to $-{\hslash }^2\chi (0)/(2M\Delta )$, while the integral of $E\chi$ tends to zero as the interval shrinks. The second-derivative term integrates to the difference of first derivatives. HenceThis expresses the point interaction through a derivative jump while the wavefunction remains continuous.