Self-Adjoint Time Operator in a Weighted Energy Space

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.