= Dirichlet Green function for a conjugated harmonic oscillator
{c}
{title2=$G(x,\xi)=-e^{a(x-\xi)}\sin(bx_<)\sin[b(L-x_>)]/[b\sin(bL)]$}
For $D=d/dx$ and $L_a=(D-a)^2+b^2$ on $(0,L)$, with $b>0$ and $\sin(bL)\ne0$, the substitution $y=e^{ax}v$ gives $L_ay=e^{ax}(v''+b^2v)$. Here $x_<=\min(x,\xi)$ and $x_>=\max(x,\xi)$. The displayed <Green function> vanishes at the endpoints, is continuous at $x=\xi$, and its first <derivative> has jump one, so $L_{a,x}G=\delta(x-\xi)$. It is the usual sine <Dirichlet Green function> conjugated by the exponential factor. At $bL\in\pi\mathbb Z\setminus\{0\}$ there is a homogeneous zero mode, so the inverse kernel does not exist without a <solvability condition> and a normalization.
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