Formula for Green function of Sturm-Liouville problem.

Question

I'm trying to remember exact formula for Green function of Sturm-Liouville problem for operator $L\::y\longmapsto (p(t)y')' + q(t)y$, left condition $D_0$ and right condition $D_1$ (that looks like $a_0y'(0) + b_0y(0)$ and $a_1y'(1) + b_1y(1)$), when we have $h_0$ and $h_1$ such that $$Lh_0 = 0, \;\; D_0h_0 = 0$$ $$Lh_1 = 0, \;\; D_1h_1 = 0$$ My notes tell me that Green function in that case is $$ h_0(x)h_1(y) ;\; x < y$$ $$ h_1(x)h_0(y) ;\; x < y$$ multiplied on $$(h_0(x)h_1'(x) - h_1(x)h_0'(x)) \frac{1}{p(x)}$$ I'm in doubt if my notes are correct and would like to ask someone to verify them, because I can't find any information about this formula at all.

Answer 1

You can apply this classical theorem to get the Green function that you want:

Theorem [Variation of Parameters]: Let $a(x),b(x),c(x)$ be continuous real or complex functions on an open interval $(r,s)$, and suppose $a(x)$ does not vanish on $(r,s)$. Let $f, g$ be solutions of $$ ay''+by'+cy=0 $$ with a non-vanishing Wronskian $W(f,g)$. Let $h$ be a complex continuous function on $[a,b]$. Then $y$ is solution of $$ ay''+by'+cy=h $$ iff there are complex constants $A$, $B$ such that $$ y = Af+Bg +g\int f\frac{1}{aW(f,g)} hdx - f\int g\frac{1}{aW(f,g)}hdx. $$