Transmission conditions for a delta coefficient ($[y{]}_c=0,[y^{\prime }{]}_c+\alpha y(c)=J$)

For a point interaction in a second-order differential equation, the equation $y^{\prime \prime }+\alpha y(c){\delta }_c=f+J{\delta }_c$, with nonsingular continuous $f$, requiresThe distributional second derivative across a jump explains both conditions: there is no derivative-of-delta source to balance a value jump, and the coefficients of the delta source must agree. In particular, without a point source, $[y^{\prime }{]}_c=-\alpha y(c)$. With a unit point source at the same location, the condition is $[y^{\prime }{]}_c=1-\alpha y(c)$; one must combine the two contributions rather than impose two incompatible jump conditions.