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Journal of Mathematical Physics

Complex-potential description of the damped harmonic oscillator

Pavel Exner

Published May 1, 1983• 18 citations
Publisher

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.

DOI
10.1063/1.525840

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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.
Using the Caldirola-Kanai Hamiltonian for the quantum dissipative system, the author expresses the propagator as the modified Feynman path integral in the configuration space, which can then be evaluated for the damped harmonic oscillator by Montroll's method. The propagator of the damped harmonic oscillator can also be calculated beyond and at caustics with the help of Horvathy-Feynman formula. The new results are confirmed by investigating the classical paths joining two fixed end-point positions, Finally the author obtains the time-dependent wavefunctions from the propagator of the dynamical system.
The system of a two-dimensional damped harmonic oscillator is revisited in the extended phase space. It is an old problem that has already been addressed by many authors that we present here with some fresh points of view and carry on a whole discussion. We show that the system is singular. The classical Hamiltonian is proportional to the first-class constraint. We pursue with the Dirac’s canonical quantization procedure by fixing the gauge and provide a reduced phase space description of the system. As a result, the quantum system is simply modeled by the original quantum Hamiltonian.
The cases of under-, critical- and over-damping are treated for the quantum driven harmonic oscillator. Following a survey of the classical version, the quantum invariant operator for each case is constructed and their eigenvalues are evaluated. Using these eigenvalues and the Schrödinger equations, the wavefunctions are obtained for each case. From formal path integral theory, their propagators are evaluated and are checked with those obtained by the closed property formed by the complete set of wavefunctions.