具有屏蔽核库仑场的一维类氢离子
One-dimensional hydrogenic ions with screened nuclear Coulomb field
AI总结：
本文在Born-Oppenheimer近似下研究了屏蔽核库仑场中单电子的一维Dirac哈密顿量谱，证明了其本质自伴性及有限束缚态特征值的存在，并发现特征函数与某动力系统的异宿鞍-鞍连接子存在一一对应关系，据此分析了束缚态数量随核电荷的变化。
摘要：
我们在Born-Oppenheimer近似（假设原子核固定在原点）下，研究处于点核静电势中单电子的一维空间Dirac哈密顿量谱。该势在大距离处被屏蔽，使其在空间无穷远处呈指数衰减至零。我们证明该哈密顿量本质自伴，本质谱具有通常的间隙$(-mc^2,mc^2)$，且该间隙内仅有有限个特征值，对应于系统的基态和激发态。我们发现该哈密顿量的特征函数与有限圆柱上某动力系统的异宿鞍-鞍连接子之间存在一一对应关系。我们利用这一对应关系研究束缚态数量如何随核电荷变化。
英文摘要：
We study the spectrum of the Dirac hamiltonian in one space dimension for a single electron in the electrostatic potential of a point nucleus, in the Born-Oppenheimer approximation where the nucleus is assumed fixed at the origin. The potential is screened at large distances so that it goes to zero exponentially at spatial infinity. We show that the hamiltonian is essentially self-adjoint, the essential spectrum has the usual gap $(-mc^2,mc^2)$ in it, and that there are only finitely many eigenvalues in that gap, corresponding to ground and excited states for the system. We find a one-to-one correspondence between the eigenfunctions of this hamiltonian and the heteroclinic saddle-saddle connectors of a certain dynamical system on a finite cylinder. We use this correspondence to study how the number of bound states changes with the nuclear charge.
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