Absolutely Continuous Spectrum of Fourth Order Dierence Operators With Unbounded Coecients on the Hilbert Space `2(N)

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.

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Absolutely Continuous Spectrum of Fourth Order Di erence Operators With Unbounded Coecients on the Hilbert Space `2(N)