= Transmission conditions for a delta coefficient
{title2=$[y]_c=0,\quad[y']_c+\alpha y(c)=J$}
For a <point interaction in a second-order differential equation>, the equation $y''+\alpha y(c)\delta_c=f+J\delta_c$, with nonsingular continuous $f$, requires
$$
[y]_c=0,\qquad [y']_c+\alpha y(c)=J.
$$
The <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, $[y']_c=-\alpha y(c)$. With a unit point source at the same location, the condition is $[y']_c=1-\alpha y(c)$; one must combine the two contributions rather than impose two incompatible jump conditions.
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