Quantum anharmonic oscillator and its statistical properties

The simple quantum harmonic oscillator is an excellent model of many physical systems, e.g. electromagnetic field. The excitation particles or quanta of the electromagnetic field are called photons [1, 2] Moreover it applies for systems of atoms (ions, nuclei) in ideal crystals interacting via harmonic attractive force, etc. 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]. The harmonic potential energy operator is only an approximation for the real anharmonic potential energy operator of mutual interaction between atoms (ions, nuclei) in real crystals. Therefore the motivation of the present paper is an attempt for a more realistic description of quantum anharmonical systems.