 Repeated-root dynamic-boundary heat kernel

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 isEquivalently, $J_a={\int }_0^{\infty }r{e}^{ar}P(x+r,t)dr$ using the Heat Poisson kernel. This representation fixes the repeated-root sign and remains valid for either sign of $a$.