Optimal Recovery of Potentials or Weights for Sturm-Liouville Problems with One Given Eigenvalue

Driven by the application in many fields, the extremal problems in Sturm-Liouville theory and inverse spectral problems have been hot issues. In the study of classical inverse spectral problems, it is generally necessary to know two sets of full spectral information to uniquely determine the potential functions. In fact, one can only detect a finite number of eigenvalues. This thesis focuses on the quantitative expression of infimum of integral modulus for potentials or weights of Sturm-Liouville problems and the attainable function when one eigenvalue is given, so as to realize the optimal recovery of potentials (weights).