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

TL;DR

This paper explores how fractional dimensions of Levy paths in Earth's gravity influence quantum states, revealing new bound states in the continuum and methods to identify ultralight particles, considering effects of minimal length.

Contribution

It introduces a novel approach to produce BICs using simple potentials and analyzes phase transitions in fractional dimensions, aiding in ultralight particle detection.

Findings

Bound states in the continuum can be generated with simple potentials.

Energy transitions from discrete to continuous with fractional dimensions.

Minimal length effects do not cause energy shifts in these phenomena.

Abstract

In this paper, the influence of the fractional dimensions of the L\'evy 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 L\'evy 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 L\'evy 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.