 Dirichlet resonance for a point interaction

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.