Iterative construction of the quasi-time operator (QTO) for the Hamiltonian H = p2/2μ + fqn in anti-standard form

Abstract

The problem of finding self-adjoint time operators canonically conjugate to the Hamiltonian can be traced back from the early exploratory paper of Schrödinger in seeking various ways to satisfy the principle of relativity by treating the space-time variables (x, y, z, t) on a common ground as a four multiplicative self-adjoint operators. Realizing that time can not be consistently treated as self-adjoint in the formalism, Schrödinger himself left the idea. In earlier independent works of Heisenberg and Dirac they propose to use the symmetric non-self-adjoint time operators that satisfy formally the Heisenberg commutation relation in problems involving dynamical concepts of time.
It was only lately realized that the above proposals of Heisenberg and Dirac make sense in the context of Positive-Operator-Valued-Measure. Furthermore, the former attempt of Schrödinger is being revitalized in the context of localization of events in space-time in the quantum domain. Lately, Lewis and his collaborators seek similar operators in constructing operator invariants for perturbative purposes. Bender, in an earlier paper, used time operators in solving for the exact solutions to various operator differential equations such as the harmonic and anharmonic oscillators. Now it appears that soon there will be a need to determine time operators for much complicated systems. In this paper we are going to present a refinement of the method earlier proposed by Galapon in determining time operators, which in this work we shall refer to as quasitime-operators (QTO), and solve exactly the QTO for the Hamiltonian H = p²/2 + fqⁿ (μ = 1) constant f and for positive integer n.