General theory of regular biorthogonal pairs and its physical operators

Abstract

In this paper, we introduce a general theory of regular biorthogonal sequences and its physical operators. Biorthogonal sequences {ϕn} and {ψn} in a Hilbert space H are said to be regular if Span {ϕn} and Span {ψn} are dense in H. The first purpose is to show that there exists a non-singular positive self-adjoint operator Tf in H defined by an orthonormal basis (ONB) f ≡ {fn} in H such that ϕn = Tffn and ψn=Tf−1fn, n = 0, 1, …, and such an ONB f is unique. The second purpose is to define and study the lowering operators Af and Bf†, the raising operators Bf and Af†, and the number operators Nf and Nf† determined by the non-singular positive self-adjoint operator Tf. These operators connect with quasi-Hermitian quantum mechanics and its relatives. This paper clarifies and simplifies the mathematical structure of this framework and minimizes the required assumptions.