The inverse Sturm–Liouville problem with mixed boundary conditions

Let $H\psi =-\psi ”+q\psi$, $\psi (0)=0$, $\psi ’(1)+b\psi (1)=0$ be a selfadjoint Sturm–Liouville operator acting in $L^2(0,1)$. Let $\lambda _n(q,b)$ and $\nu _n(q,b)$ denote its eigenvalues and the so-called norming constants, respectively. A complete characterization of all spectral data $(\{\lambda _n\}_{n=0}^{+\infty };\{\nu _n\}_{n=0}^{+\infty })$ corresponding to $(q;b)\in L^2(0,1)\times \mathbb {R}$ is given, together with a similar characterization for fixed $b$ and a parametrization of isospectral manifolds.