Deficiency Indices and Spectra of Fourth Order Differential Operators with Unbounded Coeffcients on a Hilbert space

Master's Thesis from the year 2016 in the subject Mathematics - Algebra, , language: English, abstract: In this thesis, using asymptotic integration, we have investigated the asymptotic of the eigensolutions and the deficiency indices of fourth order differential operators with unbounded coefficients as well as the location of absolutely continuous spectrum of self-adjoint extension operators. We have mainly endeavored to compute eigenvalues of fourth order differential operators when the coefficients are unbounded, determine the deficiency indices of such differential operator and the location of the absolutely continuous spectrum of the self-adjoint extension operator together with their spectral multiplicity. Results obtained for deficiency indices were in the range (2, 2) = defT = (4, 4) under different growth and decay conditions of co-efficients. The concept of unbounded operators provides an abstract framework for dealing with differential operators and unbounded observable such as in quantum mechanics. The theory of unbounded operators was developed by John Von Neumann in the late 1920s and early 1930s in an effort to solve problems related to quantum mechanics and other physical observables. This has provided the background on which other scholars have developed their work in differential operators. Higher order differential operators as defined on Hilbert spaces have received much attention though there still lays the problem of computing the eigenvalues of these higher order operators when the coefficients are unbounded.

Autorentext

As Achiando Rodgers Onyango, i was born on 23rd March, 1981 in Kasembe village, Kagan ward in Rangwe sub-county, Homa-bay county. I attended Siburi primary school then proceded to Mbita High school for my O'level. From there i passed and i proceeded to Maseno University for my Bachelor degree in Education majoring in Mathematics in 2002 and I graduated in 2006 with First Class Honours. Owing to lack of scholarship i could not proceed with my masters immediately, so i looked for job and I successfully secured a job as a teacher by the Teacher Service commission of Kenya from 2008 October to-date. I then enrolled for my masters of science inpure mathematics in 2011 but owing to the numerous responsibilities and financial challenges i completed it in 2016.