Point interaction in a second-order differential equation ($L_{\alpha }y=y^{\prime \prime }+\alpha y(c){\delta }_c$)

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 $L_{\alpha }y=y^{\prime \prime }+\alpha y(c){\delta }_c$. 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.

The boundary value problem $y^{\prime \prime }+\alpha y(c){\delta }_c=f$ on $(0,\ell )$ with Dirichlet boundary conditions has a unique solution for every continuous forcing and prescribed endpoint values exactly when $\alpha \ne {\alpha }_{\ast }:=\ell /(c(\ell -c))$. The left-normalized homogeneous solution iswhere $(t{)}_{+}=\max (t,0)$. Continuity at $c$ and the transmission conditions for a delta coefficient give this formula. The final boundary condition fixes the initial slope unless the displayed endpoint value is zero. This is a one-dimensional kernel obstruction, analogous to the Fredholm alternative.

At resonance the normalized null function isFor $y(0)=a$ and $y(\ell )=b$, the compatibility condition isIndeed, integrate $\varphi L_{\alpha }y-y{L}_{\alpha }\varphi$ on both sides of $c$. The internal terms cancel by the matching derivative jumps, leaving $[\varphi y^{\prime }-{\varphi }^{\prime }y{]}_0^{\ell }$. This equals the right-hand side. The initial-slope construction also proves sufficiency: when compatible there is a one-parameter family $y_0+t\varphi$; otherwise no solution exists.

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.