The transition of energy and bound states in the continuum of fractional Schrödinger equation in gravitational field and the effect of the minimal length

In this paper, we study the fractional Schrödinger equation in the Earth's gravitational field. We firstly introduce a family of auxiliary functions to construct solutions to the fractional Schrödinger equation in the Planck length. These solutions include the particular solution obtained previously by using the classical “Fourier transform approach”. By analyzing the solutions, we find the transition phenomenon when the dimension of Lévy path changes from integer to non-integer: the energy changes from discrete to continuous and wave functions change from non-degenerate to degenerate. Then we study the effect of the minimal length on the fractional Schrödinger equation in the Earth's gravitational field and the solutions. We find that the presence of the minimal length brings a perturbation to the Hamiltonian in equation but it does not change the transition phenomenon. Based on these result, we prove the existence of bound states in the continuum (BICs) for the fractional quantum system in the Earth's gravitational field and compare our BICs with those previous ones. Moreover, we provide the energy characteristic of small mass particles.