Fractional quantum oscillator and disorder in the vibrational spectra.

AbstractWe study the role of disorder in the vibration spectra of molecules and atoms in solids. This disorder may be described phenomenologically by a fractional generalization of ordinary quantum-mechanical oscillator problem. To be specific, this is accomplished by the introduction of a so-called fractional Laplacian (Riesz fractional derivative) to the Scrödinger equation with three-dimensional (3D) quadratic potential. To solve the obtained 3D spectral problem, we pass to the momentum space, where the problem simplifies greatly as fractional Laplacian becomes simply $$k^\mu $$ k μ , k is a modulus of the momentum vector and $$\mu $$ μ is Lévy index, characterizing the degree of disorde…