= Point interaction in a second-order differential equation
{title2=$L_\alpha y=y''+\alpha y(c)\delta_c$}
A point interaction concentrates a coefficient of a <linear differential equation> at a single interior point. On an <interval> $0<x<\ell$, with $0<c<\ell$, consider $L_\alpha y=y''+\alpha y(c)\delta_c$. The domain consists of <continuous functions> that are twice differentiable away from $c$, with the required one-sided derivatives. Multiplication of the <Dirac delta> by $y$ means the unambiguous point value $y(c)\delta_c$. It does not mean an undefined product of a delta with an arbitrary discontinuous function.
The equation is interpreted using <distributional derivatives>. Smooth forcing gives an ordinary equation on each side, with <transmission conditions for a delta coefficient> at the interaction. A <delta potential> in the one-dimensional <Schrödinger equation> is a physical example after choosing the corresponding sign and constants.
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