AI 中文总结
本研究通过高阶Bethe-Ansatz方法求解GUP形式中的一维Coulomb问题，获得解析量子化条件，并发现形变对主量子数产生下界，弱形变区域外需谨慎解释。
AI 中文摘要
我们研究了由广义不确定性原理（GUP）的一种常用实现所生成的四阶薛定谔方程在正半轴上的二维Coulomb问题。该问题直接在位置空间中通过高阶Bethe-Ansatz构造处理，波函数表示为多项式乘以指数因子。由此产生的留数条件给出了解析量子化条件以及前三个束缚态的显式多项式解。我们识别出与普通Coulomb问题连续相连的分支，并证明其能量、衰变常数、多项式因子和Bethe-Ansatz根在形变消失极限下恢复普通半线Coulomb结果。在这个Coulomb相连的分支上，形变对主量子数产生一个较低的可容许性下界，而任意高的量子数仍然可容许。我们还讨论了形变强度的物理解释：对于普通微观系统，弱GUP区域是普朗克尺度动机模型中保守的预期，而中等和强区域在当前分析中主要是理论区域。由于微分方程在GUP参数的一阶处截断，弱形变区域之外的定量预测应谨慎解释。
英文摘要
We investigate the one-dimensional Coulomb problem on the positive half-line for a fourth-order Schrödinger equation generated by a commonly used realization of the Generalized Uncertainty Principle (GUP). The problem is treated directly in position space by a higher-order Bethe--Ansatz construction, with the wave function represented as a polynomial multiplied by an exponential factor. The resulting residue conditions yield an analytic quantization condition and explicit polynomial solutions for the first three bound states. We identify the branch that is continuously connected to the ordinary Coulomb problem and show that its energies, decay constants, polynomial factors, and Bethe--Ansatz roots recover the ordinary half-line Coulomb results in the vanishing-deformation limit. On this Coulomb-connected branch, the deformation produces a lower admissibility bound on the principal quantum number, while arbitrarily high quantum numbers remain admissible. We also discuss the physical interpretation of the deformation strength: for ordinary microscopic systems, the weak-GUP regime is the conservative expectation in Planck-scale motivated models, whereas intermediate and strong regimes are primarily theoretical regimes in the present analysis. Since the differential equation is truncated at first order in the GUP parameter, quantitative predictions outside the weak-deformation regime should be interpreted with care.
