C. Maeve McCarthy, William Rundell, The inverse eigenparameter dependent Sturm-Liouville problem, Numerical Functional Analysis and Optimization, 24 no. 1-2 (2003) pp. 85-106.

Uniqueness of and numerical techniques for the inverse Sturm-Liouville problem with eigenparameter dependent boundary conditions will be discussed. We will use a Gel’fand-Levitan technique to show that the potential $q$ in

$$-u''+qu=\lambda u, 0<x<1, u(0)=0, (a\lambda +b)u(1)= (c\lambda +d)u'(1)$$

can be uniquely determined using spectral data. In the presence of finite spectral data, q can be reconstructed using a successive approximation method that involves solving a hyperbolic boundary value problem that arises in the Gel’fand-Levitan analysis. We also consider a shooting method where the right endpoint boundary condition is used in conjunction with a quasi-Newton scheme to recover the unknown potential, q.