General theory of regular biorthogonal pairs and its physical applications

In this paper we introduce a general theory of regular biorthogonal sequences and its physical applications. Biorthogonal sequences $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$ in a Hilbert space ${\cal H}$ are said to be regular if $Span\; \{ \phi_{n} \}$ and $Span\; \{ \psi_{n} \}$ are dense in ${\cal H}$. The first purpose is to show that there exists a non-singular positive self-adjoint operator T_{\mbox{f}} in ${\cal H}$ defined by an ONB \mbox{f} \equiv \{ f_{n} \} in ${\cal H}$ such that \phi_{n}=T_{\mbox{f}} f_{n} and \psi_{n}= T_{\mbox{f}}^{-1} f_{n}, $n=0,1, \cdots$, and such an ONB \mbox{f} is unique. The second purpose is to define and study the lowering operators A_{\mbox{f}} and B_{\mbox{f}}^{\dagger}, the raising operators B_{\mbox{f}} and A_{\mbox{f}}^{\dagger}, the number operators N_{\mbox{f}} and N_{\mbox{f}}^{\dagger} determined by the non-singular positive self-adjoint operator T_{\mbox{f}}. These operators connect with ${\it quasi}$-${\it hermitian \; quantum \; mechanics}$ and its relatives. This paper clarifies and simplifies the mathematical structure of this framework minimized the required assumptions.