= Repeated-root dynamic-boundary heat kernel
{title2=$\mathcal L_tJ_a=e^{-x\sqrt p}/(\sqrt p-a)^2$}
For the <dynamic boundary condition for the heat equation> with $\alpha=2a$, $\beta=a^2$, the boundary resolvent denominator is $(\sqrt p-a)^2$. With the <principal square root>, its inverse kernel is
$$
J_a(x,t)=(1-ax+2a^2t)e^{-ax+a^2t}\operatorname{erfc}\!\left(\frac{x}{2\sqrt t}-a\sqrt t\right)+2a\sqrt{t/\pi}\,e^{-x^2/(4t)}.
$$
Equivalently, $J_a=\int_0^\infty r e^{ar}P(x+r,t)dr$ using the <Heat Poisson kernel>. This representation fixes the repeated-root sign and remains valid for either sign of $a$.
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