Quantization in the neighborhood of classical solutions in the \(N\) particle problem and superfluidity (Q1804143)

The authors consider the stationary Schrödinger equation for a collection of pairwise interacting bosons in an external field. Certain asymptotic eigenvalues and eigenfunctions are obtained in the limit \(N \to \infty\), \(\varepsilon \to 0\), \(\varepsilon N \to \alpha\), where \(N\) is the number of particles, \(\varepsilon\) the coupling coefficient for the potential and \(\alpha\) a positive real constant. These solutions are actually stationary solutions of the corresponding Hartree equation which is obtained by approximating the \(N\)-particle wave function by a product of \(N\) copies of a one-particle wave function and then optimizing the expectation value of the energy over the class of such wave functions. For zero external field and the one-particle function \(f\) of the form constant \(\times\) \(\exp (ipx)\), the result coincides with Bogoliubov's well-known result in superfluidity. Certain similar phenomena also arise in other cases considered by the authors.