+ z \sinh \sqrt{2E} L}
                                                   \\   \nonumber
\nu(z, w)&=& \frac{(1 - z) w}{(1 - z) \cosh \sqrt{2E} L + z \sinh \sqrt{2E} L}.\end{aligned}$$ It is interesting to note the following special cases; $$\begin{aligned}
\label{toy2special}
\hat{G}_0^{DD}[a, b; E]&=&
\frac{\cosh \sqrt{2E} (L - |a - b|) - \cosh \sqrt{2E} ((a + b) - L)}
     {\sqrt{2E} \sinh \sqrt{2E} L}
                                           \\   \nonumber
\hat{G}_0^{NN}[a, b; E]&=&
\frac{\cosh \sqrt{2E} (L - |a - b|) + \cosh \sqrt{2E} ((a + b) - L)}
     {\sqrt{2E} \sinh \sqrt{2E} L}\end{aligned}$$ where the superscript $DD$(or $NN$) stands for Dirichlet-Dirichlet(or Neumann-Neumann) BCs at $x = 0$ and $x = L$. Similar results to Eq.(\[toy2special\]) are found at Ref.[@gros98]. Bound state energy spectrum is obtained from poles of $\hat{G}_0^{DD}$ and $\hat{G}_0^{NN}$ which indicates $B_n^{DD} = B_n^{NN} = n^2 \pi^2 / 2 L^2$ where $n$ is integer. Another interesting case is $\xi_1 = \xi_2 = 1/2$ case where $\hat{G}_0^{Box}$ is simply reduced to the free particle case without any constraint, [*i.e.*]{} $e^{-\sqrt{2E} |a - b|} / \sqrt{2E}$.

Inserting Eq.(\[toy2box2\]) into Eq.(\[toy2schul2\]) and subsequently Eq.(\[toy2schul1\]) we get the final form of $\hat{G}_{\delta}^{Box}$ for the Hamiltonian $\hat{H}_{\delta}^{Box}$. Since final expression is too long, we do not describe it explicitly in this paper. Instead we will consider 