Complex-potential description of the damped harmonic oscillator

Abstract

The multidimensional damped harmonic oscillator is treated by means of a non-self-adjoint Hamiltonian with complex potential. The propagator referring to the evolution semigroup is evaluated from the Lie–Trotter formula. The one-dimensional case is discussed in detail with the following results: (a) the nondamped limit gives the correct propagator including the Maslov phase factor, (b) for some initial conditions, the classical limit of the solution can differ from the behavior of the classical damped oscillator, the difference being negligible in the case of weak damping, and (c) the point spectrum of the considered pseudo-Hamiltonian is found.