The Bender-Dunne basis operators as Hilbert space operators

TL;DR

This paper rigorously constructs the Bender-Dunne basis operators as densely defined operators in the Hilbert space L^2(R), clarifying their mathematical properties in quantum mechanics.

Contribution

It provides the first explicit construction of dense domains for the Bender-Dunne basis operators as Hilbert space operators.

Findings

Operators are densely defined in L^2(R)

Explicit dense domain construction provided

Clarifies mathematical foundation of these operators

Abstract

The Bender-Dunne basis operators, $\mathsf{T}_{-m,n}=2^{-n}\sum_{k=0}^n {n \choose k} \mathsf{q}^k \mathsf{p}^{-m} \mathsf{q}^{n-k}$ 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 $L^2(\mathbb{R})$.