Inverting Time-Dependent Harmonic Oscillator Potential by a Unitary Transformation and a New Class of Exactly Solvable Oscillators

A time-dependent unitary (canonical) transformation is found which maps the Hamiltonian for a harmonic oscillator with time-dependent real mass and real frequency to that of a generalized harmonic oscillator with time-dependent real mass and imaginary frequency. The latter may be reduced to an ordinary harmonic oscillator by means of another unitary (canonical) transformation. A simple analysis of the resulting system leads to the identification of a previously unknown class of exactly solvable time-dependent oscillators. Furthermore, it is shown how one can apply these results to establish a canonical equivalence between some real and imaginary frequency oscillators. In particular it is shown that a harmonic oscillator whose frequency is constant and whose mass grows linearly in time is canonically equivalent with an oscillator whose frequency changes from being real to imaginary and vice versa repeatedly. The solution of the Schrödinger equation, Hψ = iψ̇, for a harmonic oscillator with time-dependent mass m and frequency ω, i.e., H(t) = 1 2m(t) p + m(t)ω(t) 2 x , (1) has been the subject of continuous investigation since late 1940’s, [1, 2, 3, 4]. The main reason for the interest in this problem is its wide range of application in the description of physical systems. Although by now there exist dozens of articles on the subject, a closed analytic expression for the time-evolution operator is still missing. Recently, Ji and Kim [4] showed that using the Lewis-Riesenfeld method [2] one can construct an invariant operator in terms of the (two independent) solutions of the classical dynamical equations: d dt [m(t) d dt xc(t)] +m(t)ω (t)xc(t) = 0 , (2) and therefore reduce the solution of the Schrödinger equation to that of Eq. (2), for ω(t),m(t) ∈ IR. The case where the frequency ω is imaginary has been considered only in the time-independent case [5]. The purpose of this note is to study the implications of the recently developed method of adiabatic unitary transformation of the Hilbert space [6] for this problem. The basic idea of this method is to use the inverse of the adiabatically approximate time-evolution operator to transform to a moving frame. This transformation has proven to lead to some interesting results for the system consisting of a magnetic dipole in a changing magnetic field. The analogy between the dipole system and the time-dependent harmonic oscillator is best described in terms of the relation between their dynamical groups, namely SU(2) and SU(1, 1), [7]. E-mail: [email protected]