Inverse eigenvalue problems for Sturm-Liouville differential equations

We study the Sturm-Liouville differential equation {dollar}{dollar}{lcub}-{rcub}ysp{lcub}primeprime{rcub} + qy = lambda rho yeqno(1){dollar}{dollar}with Dirichlet boundary conditions y(0) = 0 = y(1). Here q is a real-valued square integrable function on the interval (0,1), {dollar}rho{dollar} is a positive {dollar}Hsp1{dollar} function on the interval (0,1) and {dollar}lambda{dollar} is a complex number. If a nontrivial solution y exists for some complex number {dollar}lambda{dollar}, then {dollar}lambda{dollar} is called a Dirichlet eigenvalue of q, and the solution y is called a corresponding Dirichlet eigenfunction of q. It is well-known that the set of all eigenvalues (Dirichlet spectrum) is infinite and discrete. We study the correspondence between the Dirichlet spectrum and the coefficient function q, taking {dollar}rho{dollar} to be a fixed function. We answer the following questions:; (1) Characterization Problem I: What are the sets of real numbers that are the Dirichlet spectra of some coefficient function q?; (2) Uniqueness Problem: Is the association between a coefficient function q and its Dirichlet spectrum unique? If the answer is negative for the question above, then what other complementary data in addition to the Dirichlet spectrum is needed to determine the coefficient function q uniquely? (These two sets are called the spectral data.); (3) Characterization Problem II: What are the sets of numbers that are the spectral data of some coefficient function q?; (4) Reconstruction Problem: How can the coefficient function q be obtained algorithmically from its spectral data?; (5) Isospectral Manifold: What is the geometry of the set of all functions that have the same Dirichlet spectrum?; The characterization problem has two parts: in what space does the spectral data for a coefficient function lie, and is every point in this space the spectral data for some function q?; Our results are generalizations of those in the book "Inverse Spectral Theory" by Poschel and Trubowitz. Using the results obtained for the differential equation (1), we draw conclusions on the Inverse Spectral Problems for several other differential equations including {dollar}{dollar}{lcub}-{rcub}ysp{lcub}primeprime{rcub} = lambda rho yquad {lcub}bf and{rcub}quad {lcub}-{rcub}(rho yspprime)spprime = lambda y{dollar}{dollar}with {dollar}rho{dollar} variable, and {dollar}{dollar}{lcub}-{rcub}(rho yspprime)spprime + qy = lambda yquad {lcub}bf and{rcub}quad {lcub}-{rcub}(rho yspprime)spprime + qy = lambda rho y{dollar}{dollar}with p fixed and q variable.