Construction of spectral decomposition for non-self-adjoint friedrichs model operator - Cheremnikh, Evhen; Ivasyk, Halyna; Alieksieiev, Vladislav; Kuchma, Mariia; Brodyak, Oksana (Eastern-European Journal of Enterprise Technologies, 2018)
The spectral decomposition for the non-self-adjoint Friedrichs model is given and a generalization of the well-known Weyl function in non-self-adjoint cases is given. It is found that for the non-self-adjoint Friedrichs model, an arbitrary space element can be presented as a linear combination of the operator's eigenelements corresponding to the points of the spectrum. The spectral decomposition, that is, the representation of an arbitrary space element through its own functions, is constructed, which indicates the completeness of its eigenfunctions. This is done taking into account the spectral features (i.e., eigenvalues on a continuous spectrum) of the non-self-adjoint operator of the Friedrichs model. This model serves as an important tool for finding the solution of ordinary differential equations after the application of the corresponding Fourier transform. A general method for constructing the spectral decomposition (that is, not bound only to the Friedrichs model) is proposed, which is based on the concept of so-called branching of the resolvent and which can be used for arbitrary non-self-adjoint operators, as well as for self-directed operators. It is proved that under the conditions of the existence of the maximal operator, the resolvent allows separation of the branch. Sufficient conditions for the existence of the Weyl function m(ζ) for the operator of the non-self-adjoint Friedrichs model are given and formulas for its calculation through the resolvent are obtained. It is shown that the Weyl function m(ζ) for the self-directed operator coincides with the classical Weyl function in the case of the Sturm-Liouville operator on the semiaxis. Two examples are given in which we find the generalized Weyl function m(ζ) for the non-self-adjoint Friedrichs model