Symmetric extensions and spectral properties of the composite of higher order differential and difference operators on Hilbert Spaces

Abstract

The spectral theory of symmetric differential and difference equations with almost constant coefficients has been studied. The deficiency indices and spectral theory of commuting differential operators with bounded coefficients have been done exten- sively. However, the construction of symmetric composites and the spectral analysis of their self-adjoint extensions are lacking in the case of unbounded coefficients. In addition, this is lacking for the difference operators. This study extended the theory of the composites of differential operators with bounded coefficients to unbounded situations in addition to discrete settings. The main goal of this research was to ana- lyze the necessary and sufficient conditions for the existence of symmetric self-adjoint operator extensions of the composites for differential and difference operators with unbounded coefficients. The specific objectives were to: Construct commutative symmetric composites of differential and difference operators, analyze the necessary and sufficient conditions for the existence of symmetric self-adjoint extensions of the composites, evaluate the deficiency indices of the self-adjoint operator extensions of the composites, and examine the spectral properties of the self-adjoint operator extensions of the composites. Commutative symmetric composites were obtained by constructing appropriate comparison algebras for a specific self-adjoint opera- tor. Asymptotic integration and summation based on Levinson’s theorems were applied to achieve the optimum growth and decay conditions on the coefficients for the existence of self-adjoint operator extensions, then square integrability and summability of the eigen solutions were determined using the inner products and the number of square-integrable and summable solutions resulted into the deficiency indices. Finally, the M-matrix was used to determine the location of the spectrum and their spectral multiplicities. If T 1 and T 2 are symmetric operators of the same order belonging to the same group of comparison algebra, then T 1 commutes with T 2 whenever the first derivative or difference of the coefficients tends to zero in the limiting sense even if the coefficients are unbounded. The deficiency indices for the 4nth order composite T 1 T 2 lies between 2n and 4n both in the upper and lower half ′′ planes whenever f is unbounded and f or ∆ 2 f is either integrable or summable re- spectively. The spectrum of the self-adjoint extension of the composite T 1 T 2 is pure discrete whenever the eigen solutions are uniformly square-integrable or summable. However, absolutely continuous spectrum which is the whole real line was obtained whenever some solutions lost their square integrability or summability in the case of unbounded coefficients. The spectral multiplicity was equal to the number of so- lutions that lost their square integrability or summability. The achieved results are applicable in the film industries where a certain quality of image or sound is to be achieved. This purely depends on the wavelengths of the components of the white light in the electromagnetic spectrum which translates to the nature of the eigen solutions that form the spectrum. Discrete spectrum implied images of high preci- sion. In the future, research in designing programs that would assist in analysing spectral properties of operators of order six and above is highly recommended.