De ciency indices and spectra of Fourth order Differential Operators with Unbounded Coefficients on a Hilbert space

Abstract

The concept of unbounded operators provides an abstract framework for

dealing with dierential 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 eort

to solve problems related to quantum mechanics and other physical ob-

servables. This has provided the background on which other scholars

have developed their work in dierential operators. Higher order dieren-

tial operators as dened on Hilbert spaces have received much attention

though there still lies the problem of computing the eigenvalues of these

higher order operators when the coecients are unbounded. In this the-

sis,using asymptotic integration, we have investigated the asymptotics of

the eigensolutions and the deciency indices of fourth order dierential

operators with unbounded coecients as well as the location of absolutely

continuous spectrum of self-adjoint extension operators. We have mainly

endeavuored to compute eigenvalues of fourth order dierential operators

when the coecients are unbounded, determine the deciency indices of

such dierential operator and the location of the absolutely continuous

spectrum of the self-adjoint extension operator together with their spec-

tral multiplicity. Results obtained for deciency indices was in the range

(2; 2) defT (4; 4) under dierent growth and decay conditions of co-

ecients. In addition, the absolutely continuous spectrum is either half

or full line of spectral multiplicity 1 or 2 depending on the integrability