Asymptotic behaviour of solutions of discretised unbounded positive symmetric and dirac operators on Hilbert Spaces

Abstract

Stability analysis has been investigated for deficiency indices and spectrum of sym- metric higher order differential operators. It has been established that deficiency indices and discrete spectrum are stable under bounded perturbations whereas sin- gular continuous spectrum is not stable even under finite rank perturbations. How- ever, little has been done to establish the stability of existence of positive self-adjoint operator extensions under unbounded perturbations. Similarly, asymptotic analysis for discretised Dirac operators with unbounded coefficients was limited. This study has added to existing knowledge on the related areas as well as analysed the stability of national income using asymptotics of the solutions. The main goal of this research was to analyse the stability of existence of positive self-adjoint operator extension of symmetric operators under unbounded perturbations and asymptotic behaviour of solutions of discrete Dirac operators. The specific objectives of this study were to: 1) Construct positive perturbed difference operators with unbounded potentials, 2) Analyse the necessary and sufficient conditions for the stability of existence of positive self-adjoint operator extension under unbounded perturbations, 3) Inves- tigate the spectral properties of self-adjoint operator extensions of the perturbed difference operators with unbounded coefficients, and 4) Examine the asymptotic behavior of solutions of the discretised Dirac system with unbounded potentials. In- ner products of Hilbert spaces was applied to obtain coefficients growth conditions for the construction of positive symmetric difference operators. Asymptotic sum- mation based on discretised Levinson theorem, and von Neumann theorem for the existence of self-adjoint operator extensions were used to determine the existence of the positive self-adjoint operator extensions of the positive symmetric operators. The M-matrix of square summable eigensolutions was constructed and its spectral measures applied to determine the spectral properties of the positive self-adjoint op- erator extensions. Finally, the matrix asymptotic summation was applied to obtain the asymptotic properties of the Dirac difference operator. Suppose L 1 and L 2 are fourth and second-order symmetric difference operators with L = L 1 –zL 2 , where z is a real constant, and if the leading coefficient of L 1 dominates the coefficients of L 2 , then L 1 and L are positive symmetric operators. Furthermore, the deficiency indices of L 1 and L are (n, n), 2 ≤ n ≤ 4, with each operator L 1 and L having positive self-adjoint operator extensions even if their coefficients are unbounded. The abso- lutely continuous spectra of H 1 and H, the positive self-adjoint operator extensions of L 1 and L respectively are subsets of (0, ∞) of spectral multiplicity two whenever the leading coefficient of L 1 is unbounded. The solutions of the Dirac system were the product of their transforming matrices and the direct product of the eigenval- ues matrix with asymptotic behavior of the solutions determined by the unbounded potentials. These results can be applied by economists to study the stability of national income via the Keynesian model. Due to the complexity in computations of roots of polynomials of higher degrees more than four, similar analysis could not be investigated for order six or more. It is thus recommended that order six or more could be analysed in future using numerical analysis techniques.