Suppose we have $u(r)=\sum_{\lambda=1}^{\infty} a_{\lambda} u_{\lambda}(r) \quad 0 \leq r \leq a$ in this article Introduction to R-matrix theory in atomic physics they say that $$\int_{0}^{a}\left[u_{\lambda} \frac{d^{2} u}{d r^{2}}-u \frac{d^{2} u_{\lambda}}{d r^{2}}\right] dr=\left[u_{\lambda}\frac{du}{d r}-u \frac{du_{\lambda}}{d r}\right]_{r=a}$$
using Green’s theorem. What is this Green’s theorem they are talking?
One can only speculate, but perhaps they mean the 1D case of Green's first identity $\int_U (f \Delta g+ \nabla f \cdot \nabla g) d V=\int_{\partial U} f \nabla g \cdot n dS$. In 1D this is (in the "indefinite" form) $\int (f g''+f'g') dx=fg'$, so that
$$\int fg''-gf''dx=\left(\int fg''+f'g' dx\right)-\left(\int gf'' +f'g'dx\right)=fg'-gf'$$
This is of course an overkill in 1D, but maybe they were specializing from a higher dimensional case.