General theory of regular biorthogonal pairs and its physical applications

Abstract

In this paper we introduce a general theory of regular biorthogonal sequences and its physical applications. Biorthogonal sequences $\{{\varphi }_n\}$ and $\{{\psi }_n\}$ in a Hilbert space $\mathcal{H}$ are said to be regular if $Span\{{\varphi }_n\}$ and $Span\{{\psi }_n\}$ are dense in $\mathcal{H}$. The first purpose is to show that there exists a non-singular positive self-adjoint operator $T_{\mbox{$f$}}$ in $\mathcal{H}$ defined by an ONB $\mbox{$f$} \equiv \{ f_{n} \}$ in $\mathcal{H}$ such that $ϕ_{n}=T_{\mbox{$f$}} f_{n}$ and $ψ_{n}= T_{\mbox{$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 $quasi$-$hermitianquantummechanics$ and its relatives. This paper clarifies and simplifies the mathematical structure of this framework minimized the required assumptions.

23 pages

References in corpus (1)

• Non-self-adjoint hamiltonians defined by generalized Riesz bases

Cited by in corpus (5)

• Dynamics of finite dimensional non-hermitian systems with indefinite metric
• Biorthogonal vectors, sesquilinear forms and some physical operators
• Semi-regular biorthogonal pairs and generalized Riesz bases
• Generalized Riesz systems and orthonormal sequences in Krein spaces
• Some perturbation results for quasi-bases and other sequences of vectors