Eigenfunctions of linear combinations of creation and annihilation operators

Eigenfunctions of the linear combinations of Bose creation c⁺ and annihilation c operators are considered. The operators are classified into c, c⁺, and (c + c⁺) types. It is shown that the eigenfunctions of c-type operators are normalized but not orthogonal, while the eigenfunctions of (c + c⁺)-type operators are orthogonal and are normalized to the -function. As an illustration, several examples of (c + c⁺)-type operators are considered, their eigenfunctions are found, and their orthogonality and completeness are proved.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 13–19, September, 1975.