An application of the fixed point theorem to the inverse Sturm-Liouville problem

We consider Sturm-Liouville operators $-y^{\mathrm{\prime }\mathrm{\prime }}+v(x)y$ on $[0,1]$ with Dirichlet boundary conditions $y(0)=y(1)=0$. For any $1\le p<\mathrm{\infty }$, we give a short proof of the characterization theorem for the spectral data corresponding to $v\in L^p(0,1)$.