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In this paper, the influence of the fractional dimensions of the L\'evy pathunder the Earth's gravitational field is studied, and the phase transitions ofenergy and wave functions are obtained: the energy changes from discrete tocontinuous and wave functions change from non-degenerate to degenerate whendimension of L\'evy path becomes from integer to non-integer. By analyzing thephase transitions, we solve two popular problems. First, we find an exotic wayto produce the bound states in the continuum (BICs), our approach only needs asimple potential, and does not depend on interactions between particles.Second, we address the continuity of the energy will become strong when themass of the particle becomes small. By deeply analyze, it can provide a way todistinguish ultralight particles from others types in the Earth's gravitationalfield, and five popular particles are discussed. In addition, we obtainanalytical expressions for the wave functions and energy in the Earth'sgravitational field in the circumstance of a fractional fractal dimensionalL\'evy path. Moreover, to consider the influence of the minimal length, weanalyze the phase transitions and the BICs in the presence of the minimallength. We find the phenomenon energy shift do not exist, which is a commonphenomenon in the presence of the minimal length, and hence such abovephenomena can still be found. Finally, relations between our results andexisting results are discussed.

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