Abstract
Sturm-Liouville operators and Jacobi matrices have so far been developed
in parallel for many years. A result in one eld usually leads
to a result in the other. However not much in terms of spectral theory
has been done in the discrete setting compared to the continuous
version especially in higher order operators. Thus, we have investigated
the deciency indices of fourth order dierence operator generated by
a fourth order dierence equation and located the absolutely continuous
spectrum of its self-adjoint extension as well as the spectral multiplicity
using the M-matrix. The results are useful to mathematicians and can be
applied in quantum mechanics to calculate time dilation and length contraction
as used in Lorentz-Fotzgeralds transformations. The study has
been carried out through asymptotic summation as outlined in Levinson
Benzaid Lutz-theorem. This involved: reduction of a fourth order dierence
equation into rst order, computation of the eigenvalues, proof of
uniform dichotomy condition, calculating the deciency indices and locating
absolutely continuous spectrum. In this case we have found the
absolutely continuous spectrum to be the whole set of real numbers of
spectral multiplicity one.
EVANS, M (2021). Absolutely Continuous Spectrum of Fourth Order Di erence Operators With Unbounded Coecients on the Hilbert Space `2(N). Afribary. Retrieved from https://afribary.com/works/absolutely-continuous-spectrum-of-fourth-order-di-erence-operators-with-unbounded-coe-cients-on-the-hilbert-space-2-n
EVANS, MOGOI "Absolutely Continuous Spectrum of Fourth Order Di erence Operators With Unbounded Coecients on the Hilbert Space `2(N)" Afribary. Afribary, 07 May. 2021, https://afribary.com/works/absolutely-continuous-spectrum-of-fourth-order-di-erence-operators-with-unbounded-coe-cients-on-the-hilbert-space-2-n. Accessed 19 Apr. 2024.
EVANS, MOGOI . "Absolutely Continuous Spectrum of Fourth Order Di erence Operators With Unbounded Coecients on the Hilbert Space `2(N)". Afribary, Afribary, 07 May. 2021. Web. 19 Apr. 2024. < https://afribary.com/works/absolutely-continuous-spectrum-of-fourth-order-di-erence-operators-with-unbounded-coe-cients-on-the-hilbert-space-2-n >.
EVANS, MOGOI . "Absolutely Continuous Spectrum of Fourth Order Di erence Operators With Unbounded Coecients on the Hilbert Space `2(N)" Afribary (2021). Accessed April 19, 2024. https://afribary.com/works/absolutely-continuous-spectrum-of-fourth-order-di-erence-operators-with-unbounded-coe-cients-on-the-hilbert-space-2-n