 Solution

Let $x_{<}=\min (x,x^{\prime })$ and $x_{>}=\max (x,x^{\prime })$. A homogeneous solution vanishing at the left wall is $u(x)=\sin [k(x+a/2)]$; one vanishing at the right wall is $v(x)=\sin [k(a/2-x)]$. Their Wronskian is $u{v}^{\prime }-u^{\prime }v=-k\sin ka$. Continuity and the source derivative jump $G_{+}^{\prime }-G_{-}^{\prime }=-2m/{\hslash }^2$ therefore giveThis is exactly the printed form because $x_{<}=(x+x^{\prime }-|x-x^{\prime }|)/2$ and $x_{>}=(x+x^{\prime }+|x-x^{\prime }|)/2$, with a minus sign from reversing the argument of the second sine.

The true poles occur at $k_n=n\pi /a$, $n=1,2,\dots$, corresponding toThe apparent zero at $k=0$ is removable: the numerator and $k\sin ka$ both vanish quadratically. Negative values of $k_n$ duplicate the same energies, rather than describing extra levels. The boundary conditions and source jump uniquely determine the inverse whenever $E$ is not an eigenvalue.