Determining the Potential of a Sturm-liouville Operator from Its Dirichlet and Neumann Spectra

y(0) cosα+ y′(0) sinα = 0, y(1) cosβ + y′(1) sinβ = 0 (2) y(0) cosα+ y′(0) sinα = 0, y(1) cos γ + y′(1) sin γ = 0 (3) with sin(γ − β) = 0, then q(x) is uniquely determined. Notice that this theorem does not include the case of Dirichlet (boundary conditions y(0) = y(1) = 0) and Neumann (boundary conditions of y′(0) = y′(1) = 0) spectra. Borg [1], Levinson [11], Isaacson, McKean and Trubowitz [8] among others demonstrated that the spectrum given by one boundary condition does not determine the operator. The dynamical behavior of solutions to Hill’s Operator (the 1-D Schrödinger or Sturm-Liouville Operator with periodic potential) is determined by the properties of the associated Floquet discriminant function [12]. Its and Matveev [10], Gelfand [5], Gelfand and Levitan [6], McKean [15], Garnett [4], Trubowitz [17], and Buslaev and Faddeev [2] illustrate that for periodic potentials the periodic, anti-periodic, and Dirichlet spectra determine the potential.