Green’s function for the free particle system defined on a half-line($x \geq 0$) with $\delta$-function potential whose Hamiltonian is $$\label{toy1hamil1}
\hat{H} = \hat{H}_0^> - v \delta(x)$$ where $\hat{H}_0^>$ is a free particle Hamiltonian with the half-line constraint, [*i.e.*]{} $$\label{toy1hamil2}
\hat{H}_0^> = -\frac{1}{2} \partial_x^2
\hspace{2.0cm}
(x \geq 0).$$ Of course the main problem in this model is how to compute the fixed-energy amplitude for $\hat{H}_0^>$. Once this is completed, one can derive a fixed-energy amplitude for $\hat{H}$ by employing the Schulman procedure described in the previous section.

We start with a fixed-energy amplitude $\hat{G}_F[x, y; E]$ for free particle without any constraint $$\label{free1}
\hat{G}_F[x, y; E] = \frac{e^{-\sqrt{2E} |x - y|}}{\sqrt{2E}}.$$ Then, the fixed-energy amplitude for $\hat{H}_0^>$ can be computed as follows from $\hat{G}_F[x, y; E]$. First, we have to note that the fixed-energy amplitude for $\hat{H}_0^>$ is dependent on BC at $x = 0$ arising due to the half-line constraint. The usual Dirichlet or Neumann BCs at $x = 0$ are properly incorporated into the path-integral formalism using $\delta$- and $\delta^{\prime}$-function potentials with infinite coupling constant[@gros93; @gros95]; $$\begin{aligned}
\label{deldel}
\hat{G}_F^D[a, b; E]&=&\hat{G}_F[a, b; E] - 
\frac{\hat{G}_F[a, 0; E] \hat{G}_F[0, b; E]}{\hat{G}_F[0^+, 0; E]}
                                              \\   \nonumber
\hat{G}_F^N[a, b; E]&=&\hat{G}_F[a, b; E] 