Constructing a Green function for Sturm - Liouville operator

Question

I can't find Green function for the Sturm - Liouville operator : $$ L = - \frac{d²}{dx²} + 1 $$

with condition : $v(0) = v(1) = 0$

I would appreciate any help

Thanks in advance

Answer 1

The Green function is the integral kernel of the solution to $Lf=g$, where $f\in\mathcal{D}(L)$ means that $f$ is twice differentiable with $f(0)=f(1)=0$. If $\varphi,\psi$ are solutions of $Lf=0$ with conditions $$ \varphi(0)=0,\;\varphi'(0)=1 \\ \psi(1)=0,\;\psi'(1)=1, $$ then a solution of $Lf=g$ can be constructed using variation of parameters, and found to be $$ f = \frac{\psi(x)}{w}\int_0^xg(y)\varphi(y)dy+\frac{\varphi(x)}{w}\int_x^1g(y)\psi(y)dt, \\ \varphi(x)=\sinh(x),\;\; \psi(x)=\sinh(x-1) $$ where $w=\psi'(x)\varphi(x)-\psi(x)\varphi'(x)$ is the Wronskian. $w$ is a constant in $x$ in this case because $$ w'=\psi''\varphi-\psi\varphi''=\psi\varphi-\psi\varphi=0. $$