Quantum damped oscillator II: Bateman's Hamiltonian vs. 2D Parabolic Potential Barrier

Abstract

We show that quantum Bateman's system which arises in the quantization of a damped harmonic oscillator is equivalent to a quantum problem with 2D parabolic potential barrier known also as 2D inverted isotropic oscillator. It turns out that this system displays the family of complex eigenvalues corresponding to the poles of analytical continuation of the resolvent operator to the complex energy plane. It is shown that this representation is more suitable than the hyperbolic one used recently by Blasone and Jizba.

arXiv comment15 pages

References in corpus (3)

• Quantum damped oscillator I: dissipation and resonances
• Quantum Mechanics of Damped Systems II. Damping and Parabolic Potential Barrier
• Bateman's dual system revisited: I. Quantization, geometric phase and relation with the ground-state energy of the linear harmonic oscillator

Cited by in corpus (9)

• Quantum Integrals of Motion for Variable Quadratic Hamiltonians
• On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator
• Quantization of the Damped Harmonic Oscillator Revisited
• Conservative perturbation theory for nonconservative systems
• Two quantization approaches to the Bateman oscillator model
• Dissipation and quantization for composite systems
• Canonical quantization of so-called non-Lagrangian systems
• Quantization of the damped harmonic oscillator based on a modified Bateman Lagrangian
• On the canonical quantization of the electromagnetic field and the emergence of gauge invariance