Dirichlet half-line Green function for minus d2 plus 1 ($G(x,\xi )=\sinh (x_{<})e^{-x_{>}}$)

On the positive half-line, the Green function for $-d^2/d{x}^2+1$ with a zero Dirichlet boundary condition at the origin and decay at infinity isIt is continuous, has derivative jump $G_x(\xi +,\xi )-G_x(\xi -,\xi )=-1$, and is nonnegative. It is the negative of the Dirichlet half-line Green function for d2 minus 1; the sign of the leading differential term matters.

For $\mu >0$, the decaying solution of $-y^{\prime \prime }+y=1_{[0,\mu ]}$ on the positive half-line with $y(0)=0$ follows by integrating the Dirichlet half-line Green function for minus d2 plus 1:Both $y$ and $y^{\prime }$ are continuous at $\mu$, whereas $y^{\prime \prime }$ has the jump required by the bounded step in the forcing. Its common value there is $(1-e^{-\mu }{)}^2/2$.