Comparative analysis of spectral theory of differential And difference operators on hilbert spaces

Abstract

There has been e ort to investigate the spectrum of di erence operators to parallel that of di erential operators. This has been done either through the subspace theory or direct de nition of the domain of the operator. Even though much has been done to compare the spectral theory of di erential and di erence operators of order two, few or limited comparative analysis exists beyond order two operators. In particular, no comparative analysis has been done for order six operators on Hilbert spaces and that of the fourth order has not been exhausted especially when the odd order coe cients are unbounded. Of importance is to compare the results obtained in di erential operators to those of their discrete counterparts if the two operators are of the same order under similar growth and decay conditions. The main aim of this study was to conduct a comparative analysis of spectral theory of higher order di erential and di erence operators on Hilbert spaces, when the odd order coe - cients are unbounded. The speci c objectives were; to evaluate and compare the de ciency indices of the second order di erential and di erence operators with unbounded odd order coe cients, discuss the spectrum of the fourth order di erential and di erence operators and nally apply asymptotic integration and summation to analyze the spectral properties of sixth order di erential and di erence operators on Hilbert spaces, with the third order coe cient unbounded. The comparative analysis was carried out by means of asymptotic integration and summation based on Levinson's and Levinson-Benzaid-Lutz theorems. For order two di erential operator with unbounded odd order coe cients, the absolutely continuous spectrum was the whole of the real line with spectral multiplicity as one. On the other hand, the spectrum of their discrete counterparts only consisted of eigenvalues under similar growth conditions. Similarly, order four di erential operator resulted into absolutely continuous spectrum with spectral multiplicity one whenever the third order coef- cient is unbounded while the spectrum of fourth order di erence operator under similar conditions is pure discrete. Finally, the absolutely continuous spectrum was found to be the whole real line in the case of order six di erential operator with sixth order di erence operator giving discrete spectrum when the third order coe cient is unbounded. Since spectral theory have wide applications in other elds like quantum mechanics, stability analysis of market prices as well as in epidemiology, the results obtained in this research are applicable in stability analysis of market prices because asymptotic integration and summation are perturbation processes. Due to complexity in computations and analysis of the roots of degree six polynomials, only three term sixth order operators were analyzed. In future, one can investigate the spectral properties of order six operators with all the coe cients taken as non-zero. This can be generalized to higher orders more than six.