Absolutely Continuous Spectrum of fourth Order Di_erence Operators With unbounded Coe_cients 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 de_ciency indices of fourth order di_erence operator generated by a fourth order di_erence 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 di_erence equation into _rst order, computation of the eigenvalues, proof of uniform dichotomy condition, calculating the de_ciency 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.