On Boundary Conditions of an Inverse Sturm–Liouville Problem

Source abstract

It is known that if the potential function $q( x )$ in a Sturm–Liouville problem is prescribed over the interval $(0,1/2 )$ and the boundary condition at the point zero is fixed, then a single spectrum suffices to determine $q( x )$ on the interval $(1/2,1)$ uniquely. Explicit examples are given that show the necessity of specifying the boundary condition at zero even if the boundary condition at one is given.