软核库仑势的能量和波函数
Energies and wave functions for a soft-core Coulomb potential
摘要：
对于V（r）= - \frac{Z}{(r^q+β^q)^{\frac{1}{q}}}的模型软库仑势族，当参数Z>0，β>0，\ge 1时，解析地证明了势和本征值E_{ν\ell}对每个参数都是单调的.势包络方法被应用于获得近似的解析估计的纯功率势的已知的精确谱。对于q =1的情形，用渐近迭代法求出了本征值E_{ν\ell}和相应的波函数的精确解析结果，它们用Z和β表示。证明建立了由截断势产生的核附近定标电子密度的一般表达式。通过对大量数值计算的分析，证明了态对[（ν，\ell），（ν '，\ell '）]之间的交叉由条件ν' \geq（ν+1）和\ell ' \geq（\ell +3）给出.指出了这些结果对于强激光场与原子相互作用的意义。讨论了软势和被限制在一个不可穿透的球体内的氢原子之间的能级交叉效应的差异。
英文摘要：
For the family of model soft Coulomb potentials represented by V(r) = -\frac{Z}{(r^q+β^q)^{\frac{1}{q}}}, with the parameters Z>0, β>0, q \ge 1, it is shown analytically that the potentials and eigenvalues, E_{ν\ell}, are monotonic in each parameter. The potential envelope method is applied to obtain approximate analytic estimates in terms of the known exact spectra for pure power potentials. For the case q =1, the Asymptotic Iteration Method is used to find exact analytic results for the eigenvalues E_{ν\ell} and corresponding wave functions, expressed in terms of Z and β. A proof is presented establishing the general concavity of the scaled electron density near the nucleus resulting from the truncated potentials for all q. Based on an analysis of extensive numerical calculations, it is conjectured that the crossing between the pair of states [(ν,\ell),(ν',\ell')], is given by the condition ν'\geq (ν+1) and \ell' \geq (\ell+3). The significance of these results for the interaction of an intense laser field with an atom is pointed out. Differences in the observed level-crossing effects between the soft potentials and the hydrogen atom confined inside an impenetrable sphere are discussed.
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