Bound states in the continuum are universal under the effect of minimal length

Abstract: Bound states in the continuum (BICs) are generally considered unusual phenomena. In this work, we provide a method to analyze the spatial structure of particle's bound states in the presence of a minimal length, which can be used to find BICs. It is shown that the BICs are universal phenomena under the effect of the minimal length. Several examples of typical potentials, i.e., infinite potential well, linear potential, harmonic oscillator, quantum bouncer and Coulomb potential, et al, are provided to show the BICs are universal. The wave functions and energy of the first three examples are provided. A condition is obtained to determine whether the BICs can be readily found in systems. Using the condition, we find that although the BICs are universal phenomena, they are often hardly found in many ordinary environments since the bound continuous states perturbed by the effect of the minimal length are too weak to observe. The results are consistent with the current experimental results on BICs. In addition, we reveal a mechanism of the BICs. The mechanism explains why current research shows the bound discrete states are typical, whereas BICs are always found in certain particular environments when the minimal length is not considered.