Equivalence of inverse Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter

Three inverse problems for a Sturm–Liouville boundary value problem −y + qy = λy, y(0) cos α = y (0) sinα and y (1) = f (λ)y(1) are considered for rational f . It is shown that the Weyl m-function uniquely determines α, f , and q, and is in turn uniquely determined by either two spectra from different values of α or by the Prüfer angle. For this it is necessary to produce direct results, of independent interest, on asymptotics and oscillation.  2003 Elsevier Inc. All rights reserved.