UNIQUENESS THEOREMS FOR STURM-LIOUVILLE OPERATORS WITH INTERIOR TWIN-DENSE NODAL SET

We study Inverse problems for the Sturm-Liouville operator with Robin boundary conditions. We establish two uniqueness theorems from the twin-dense nodal subset W-S ([1-epsilon/2, 1/2]), 0 < epsilon <= 1, together with parts of either one spectrum, or the minimal nodal subset {x(n)(1)}(n=1)(infinity) on the interval [0, 1/2]. In particular, if one spectrum is given a priori, then the potential q on the whole interval [0, 1] can be uniquely determined by W-S ([1-epsilon/2, 1/2]) for any S and arbitrarily small epsilon.