Self-Adjoint Time Operator in a Weighted Energy Space

We introduce a self-adjoint time operator $T_w=i\mathrm{\hslash }({\mathrm{\partial }}_E+\frac{1}{2}{\mathrm{\partial }}_E\ln w(E))$ 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 $\mid E\mid$), we prove that $T_w$ is essentially self-adjoint. A simple unitary conjugation carries $T_w$ back to $i\mathrm{\hslash }\mathrm{d}\mathrm{/}\mathrm{d}E$, which in turn leaves the Hamiltonian spectrum unbounded.