Bounded solution structure of Schrodinger equation in the presence of the minimal length and its effect: Bound states in the continuum are universal

摘要

Bound states in the continuum (BICs) are generally considered unusual phenomena. In this work, first, 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; second, we provide a method to analyze the singular perturbation term's effect of the Schrodinger equation, which can determine whether the BICs are readily observed in systems. Using the first method, we find that a counterintuitive phenomenon: the BICs are universal phenomena under the effect of the minimal length. Several examples of typical linear and nonlinear potentials, i.e., the infinite potential well, linear potential, harmonic oscillator, Poschl-Teller potential, quantum bouncer, half oscillator, quantum bouncer in a closed court, harmonic oscillator plus Dirac delta function and Coulomb potential, are provided to show the BICs are universal. The wave functions and energy of the first three examples are provided. Using the second method, we find the reason for this phenomenon: 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. Three examples are discussed. And we provide the range of the deforming parameter beta of the minimal length that can make the BICs be readily observed. 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. (C) 2021 Elsevier B.V. All rights reserved.