Bound states in the continuum of fractional Schrödinger equation in the Earth's gravitational field and their effects in the presence of a minimal length: applications to distinguish ultralight particles

In this paper, the influence of the fractional dimensions of the Lévy path under the Earth's gravitational field is studied, and the phase transitions of energy and wave functions are obtained: the energy changes from discrete to continuous and wave functions change from non-degenerate to degenerate when dimension of Lévy path becomes from integer to non-integer. By analyzing the phase transitions, we solve two popular problems. First, we find an exotic way to produce the bound states in the continuum (BICs), our approach only needs a simple potential, and does not depend on interactions between particles. Second, we address the continuity of the energy will become strong when the mass of the particle becomes small. By deeply analyze, it can provide a way to distinguish ultralight particles from others types in the Earth's gravitational field, and five popular particles are discussed. In addition, we obtain analytical expressions for the wave functions and energy in the Earth's gravitational field in the circumstance of a fractional fractal dimensional Lévy path. Moreover, to consider the influence of the minimal length, we analyze the phase transitions and the BICs in the presence of the minimal length. We find the phenomenon energy shift do not exist, which is a common phenomenon in the presence of the minimal length, and hence such above phenomena can still be found. Finally, relations between our results and existing results are discussed.