Two quantization approaches to the Bateman oscillator model

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