Self-Adjoint Time Operator in a Weighted Energy Space

• URL: http://arxiv.org/abs/2504.17830v1
• Date: Thu, 24 Apr 2025 09:40:41 GMT
• Title: Self-Adjoint Time Operator in a Weighted Energy Space
• Authors: Radmir Kokoulin,
• Abstract summary: We prove that $T_w$ is essentially self-adjoint.<n>A simple unitary conjugation carries $T_w$ back to $ihbar,mathrmd/mathrmdE$, which in turn leaves the Hamiltonian spectrum.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce a self-adjoint time operator $T_w = i\hbar\bigl(\partial_E + \tfrac12\,\partial_E\ln w(E)\bigr)$ on the weighted energy space $L^2(\mathbb R,\,w(E)\,dE)$. Under mild conditions on the weight $w$ (positivity, local absolute continuity, and uniform bounds at large $\lvert E\rvert$), we prove that $T_w$ is essentially self-adjoint. A simple unitary conjugation carries $T_w$ back to $i\hbar\,\mathrm{d}/\mathrm{d}E$, which in turn leaves the Hamiltonian spectrum unbounded.

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