A UNIQUENESS THEOREM FOR INVERSE EIGENPARAMETER DEPENDENT STURM-LIOUVILLE PROBLEMS

We consider the regular Sturm-Liouville problem -y '' + qy = lambda y on [0, 1] with boundary conditions cos alpha y(0) + sin alpha y'(0) = 0 (a lambda + b)y(1) = (c lambda + d)y'(1) where, initially, ad - bc > 0, c not equal 0. We show that the eigenvalues and appropriately defi ned norming constants determine the potential q in the sense that two such problems with identical spectra and norming constants must have t he same potential. Exceptional cases in which ad - bc = 0, c = 0 are a lso discussed.