Isomorphic Hamiltonian dynamical systems being nonisomorphic unlike quantum systems (Q2782065)

On the phase space \(\,M=T^*(R^+)^n\), \(\,n\geq 1\), the Hamiltonian system with the function NEWLINE\[NEWLINE H = \sum\limits_{k=1}^n H_k = \sum\limits_{k=1}^n \frac12 \bigg(y_k^2 + \lambda^2 x_k^2 + \frac{\mu_k^2}{x_k^2}\bigg)\tag{1} NEWLINE\]NEWLINE is considered, where \(\lambda\) and \(\mu_k\) are real positive constants, \(\,x_k>0\,\) and \(y_k\), \(\,k=1,\dots,n\), are the canonical coordinates. It is established that for this system the Konstant-Souriau geometric quantization procedure does not yield a correct result in terms of quantum mechanics. This is due to the fact that this dynamical system and the system with the interaction potential \(\lambda^2x^2\) (harmonic oscillator) are smoothly isomorphic (up to the linear transformation).