The Bender-Dunne basis operators as Hilbert space operators

Abstract

The Bender-Dunne basis operators, ${\mathsf{T}}_{-m,n}=2^{-n}{\sum }_{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right){\mathsf{q}}^k{\mathsf{p}}^{-m}{\mathsf{q}}^{n-k}$ where $\mathsf{q}$ and $\mathsf{p}$ are the position and momentum operators respectively, are formal integral operators in position representation in the entire real line $\mathbb{R}$ for positive integers $n$ and $m$. We show, by explicit construction of a dense domain, that the operators ${\mathsf{T}}_{-m,n}$'s are densely defined operators in the Hilbert space $L^2(\mathbb{R})$.

Preprint. Please refer to the Journal of Mathematical Physics for the published version

References in corpus (1)

• Time evolution of non-Hermitian Hamiltonian systems

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