Two quantization approaches to the Bateman oscillator model

TL;DR

This paper compares two quantization methods for the Bateman oscillator model, highlighting how the imaginary-scaling approach resolves energy spectrum issues and enables stable states, unlike the Feshbach-Tikochinsky method.

Contribution

It introduces a reformulated Feshbach-Tikochinsky quantization and applies the imaginary-scaling approach to the Bateman oscillator, demonstrating advantages in stability and spectrum boundedness.

Findings

Imaginary-scaling quantization overcomes unbounded energy spectrum issues.

Both approaches ensure positive-definite squared-norms of eigenvectors.

Imaginary-scaling allows stable states alongside decaying and growing states.

Abstract

We consider two quantization approaches to the Bateman oscillator model. One is Feshbach-Tikochinsky's quantization approach reformulated concisely without invoking the ${\mathit{SU}(1,1)}$ 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.