Direct and inverse problems on spectral theory of Hill equation

ABSTRACTPhD ThesisDIRECT AND INVERSE PROBLEMS ON SPECTRAL THEORYOF HILL EQUATIONMünevver AYDINFırat UniversityInstitute of Applied ScienceDepatment of Mathematics2007, pages: 88This thesis essentially consist of four pourts.In the first chapter, some basic definitions and theorems are presented whichare commenly used in spectral theory of differential operations.In the second chapter, the stability, instability, conditional stability con-cepts, existance of periodic solutions and bounded case of this solutions andF (λ) with this concepts are examined for Hill equation. And, then the eigen-value and eigenfunction concepts for the t-periodic boundary value problem areinvestigated. After these, by using the Green function and the integral opera-tors method it is shown that the existance of countable many eigenvalues for theperiodic and antiperiodic boundary value problems. Furthermore, by provingthe necessary properties of the F (λ) function and the graph of this function isdrawed.In the third chapter, the solution of invers problem?s is examined on twopartialIy non coinciding spectra for Hill equation and Hochstadt Theorem isproved for periodic problems.In the fourth chapter, the generalized degeneracy of the invers problem ofthe fundemental integral equation is shown for Hill operator and in case of sym-metric potantial of single spectrum, the solution of invers problem are solved.Key Words: Sturm-Liouville equation, eigenvalue, eigenfunction, stability,instability, periodic function, Hill?s equation, kernel function, translation op-erator, general degeneracy, potential, spectrum, spectral function, normalizednumbers.1