Quantum anharmonic oscillator and its statistical properties in the first quantization scheme

A family of quantum anharmonic oscillators is studied in any finite spatial dimension in the scheme of first quantization and the investigation of their eigenenergies is presented. The statistical properties of the calculated eigenenergies are compared with the theoretical predictions inferred from the Random Matrix theory. Conclusions are derived. 1 Motivation The quantum harmonic oscillator proved to be a fructuous model of many physical systems: quantum electromagnetic field or systems of atoms (ions, nuclei) in ideal crystals interacting via harmonic attractive force, etc. In the former case the excitation particles or quanta of the electromagnetic field are called photons [1, 2] whereas in the latter case the elementary excitation particles of vibrations of crystal lattice or quanta of the sound field are named phonons [3, 4]. Both of these quantum fields are bosonic ones [5, 6]. Also in the case of the interaction of the quantum electromagnetic field with the matter field via dipolar electrostatic interaction the quantum harmonic oscillator is hugely investigated. The harmonic potential energy is only an approximation for the real anharmonic potential energy of mutual interaction between atoms (ions, nuclei) in real crystals. Therefore the motivation of the present work is a more realistic description of quantum anharmonical systems. 2 Quantum harmonic oscillator in D = 1 spatial dimension Firstly: Our study commences to concentrate on the case of simple quantum harmonic oscillator in D = 1 spatial dimension. The dimensionless Cartesian coordinate is denoted