发表机构
Brown University; Princeton University; CERMICS, École des Ponts, Institut Polytechnique de Paris; University of Vienna; Department of Mathematics, Columbia University（布朗大学; 普林斯顿大学; 巴黎高等师范学院 CERMICS，巴黎理工学院; 维也纳大学; 哥伦比亚大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明反演对称磁双阱的固定势猜想，构造特定单阱势，使得耦合常数趋于无穷时，双阱哈密顿量最低两特征值出现无穷多次精确简并，且基态宇称反复变化。
AI 中文摘要
我们证明了Fefferman、Shapiro和Weinstein关于反演对称磁双势阱的固定势猜想。我们构造了一个数$L_0>0$和一个光滑、紧支撑、非正、非径向的单阱势$v$，该势与耦合参数$\lambda$和阱位移均无关，使得对于每个固定的$L\ge L_0$，当$\lambda\to\infty$时，相关的磁双阱哈密顿量的两个最低特征值具有无穷多个精确简并。磁跳跃系数也沿一个趋于无穷的无限序列消失。该势由一个径向核心和两个小的、反射相关的微扰组成，这些微扰支撑在单侧二次尖点上。它们的对数平坦轮廓隔离了两个主要的跳跃贡献，我们在插入径向基态渐近后通过最速下降法对其进行评估。由此得到的渐近余弦表达式，连同偶-奇分裂的误差界，产生了无穷多个精确的特征值交叉和基态宇称变化。
英文摘要
We prove the fixed-potential conjecture of Fefferman, Shapiro, and Weinstein for inversion-symmetric magnetic double wells. We construct a number $L_0>0$ and one smooth, compactly supported, nonpositive, nonradial single-well potential $v$, independent of both the coupling parameter $λ$ and the well displacement, such that for every fixed $L\ge L_0$ the associated magnetic double-well Hamiltonian has infinitely many exact degeneracies of its two lowest eigenvalues as $λ\to\infty$. The magnetic hopping coefficient also vanishes along an infinite sequence tending to infinity. The potential consists of a radial core and two small, reflection-related perturbations supported on one-sided quadratic cusps. Their log-flat profiles isolate two dominant hopping contributions, which we evaluate by steepest descent after inserting the radial ground-state asymptotics. The resulting asymptotic cosine expression, together with error bounds for the even--odd splitting, yields infinitely many exact eigenvalue crossings and changes of ground-state parity.
