Two quantization approaches to the Bateman oscillator model

Abstract

We consider two quantization approaches to the Bateman oscillator model. One is Feshbach-Tikochinsky's quantization approach reformulated concisely without invoking the $SU(1,1)$ Lie algebra, and the other is the imaginary-scaling quantization approach developed originally for the Pais-Uhlenbeck oscillator model. The latter approach overcomes the problem of unbounded-below energy spectrum that is encountered in the former approach. In both the approaches, the positive-definiteness of the squared-norms of the Hamiltonian eigenvectors is ensured. Unlike Feshbach-Tikochinsky's quantization approach, the imaginary-scaling quantization approach allows to have stable states in addition to decaying and growing states.

23 pages, final version published in Annals of Physics

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Cited by in corpus (13)

• A no-go result for the quantum damped harmonic oscillator
• Quantization of the damped harmonic oscillator based on a modified Bateman Lagrangian
• Weak pseudo-bosons
• Bateman Oscillators: Caldirola-Kanai and Null Lagrangians and Gauge Functions
• Quantum gravity effects in the infra-red: a theoretical derivation of the low energy fine structure constant and mass ratios of elementary particles
• A representation-theoretical approach to higher-dimensional Lie-Hamilton systems: The symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$
• Conformal symmetry in damped Pais-Uhlenbeck oscillator
• Square-integrable eigenfunctions in quantizing the Bateman oscillator model
• Quantization of nonlinear non-Hamiltonian systems
• Multiplication of distributions in a linear gain and loss system
• Algebraic analysis of non-Hermitian quadratic Hamiltonians
• Classical Hamiltonian Systems with Balanced loss and gain
• Reply to Comment on "A no-go result for the quantum damped harmonic oscillator"