The inverse problems of spectral and potential theory

ABSTRACTTHE INVERSE PROBLEMS OF SPECTRAL AND POTENTIALTHEORYErdal BAşSFırat UniversityGraduate School of Science and TechnologyDepartment of Mathematics2006, Page:90This thesis consists of four chapters.In the first chapter, some fundamental definitions and theorems are givenwhich will be used in the later chapters.In the second chapter, the potential function q (x) in a Sturm-Liouvilleequation is defined on the unit interval having of the singularity type q (x) =δxp +q0 (x) ( where δ is a constant, 1 < p < 2, q0 (x) L2 [0, π] .) Spectral analysisof singular Sturm-Liouville equation is given. At the end of the chapter, theproof of the Parseval equality is attained.In the third chapter, we give the solution of the inverse problem on twopartially non-coinciding spectra for the Sturm-Liouville operators with the pe-culiarity specially at zero. In particular in this case we obtain Hochstadt?stheorem concerning the structure of the difference q (x) − q (x) .˜In the fourth chapter, we prove the generalized degeneracy of the kernel oftranslation operator.It is also singular case we prove the generalized degeneracythe main integral equation inverse problem. In addition we obtain a new proofof the Hochstadt?s theorem which concerns the structure of the difference q (x)−q (x) .˜Key Words: Sturm-Liouville, Eigenvalue, Eigenfunction, Singular Prob-lem, Translation Operator, Generalized degeneracy, Potential, Spectrum, Nor-malized numbers, Spectral function.