incomplete inverse problems for the sturm-liouville type differential equation with the spectral boundary condition

In this study, we examine the inverse problem for the differential equation of the sturm-liouville type with the spectral boundary condition in the finite interval. using lieberman-hochstadt’s method, we show that if p(x) is prescribed on the half interval (π/2, π) then a single spectrum suffices to determine p(x) on (0,π). moreover, applying gesztesy-simon’s method, we demonstrate that if p(x) is assumed over the given segment [π/2(1 - θ), π] where θ∈ (0, 1), a finite number of the spectrum is enough to give p(x) on (0, π).