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.