二电子量子点模型的再认识：束缚态及其它解析解和数值解
Two-electron quantum dot model revisited: bound states and other analytical and numerical solutions
摘要：
重新讨论了在二维平面空间中运动的双电子量子点模型。一般认为，通过求解双汇流Heun方程得到的模型解存在一定的局限性。特别是对以往的理论计算也作了一些修正。修正后的多项式解与Numerov方法的数值计算结果吻合较好。然后，新的解决方案，考虑$1/r$和$\ln r$类库仑势在（1+2）D，尚未获得，进行了数值讨论。特别是，我们能够计算的量子点本征函数为一个更大的频谱的外部谐波频率相比，以前的结果。还预言了这种平面系统在$l=0$情形下束缚态的存在性，并确定了相应的本征值。
英文摘要：
The model of a two-electron quantum dot, confined to move in a two dimensional flat space, is revisited. Generally, it is argued that the solutions of this model obtained by solving a biconfluent Heun equation have some limitations. In particular, some corrections are also made in previous theoretical calculations. The corrected polynomial solutions are confronted with numerical calculations based on the Numerov method, in a good agreement between both. Then, new solutions considering the $1/r$ and $\ln r$ Coulombian-like potentials in (1+2)D, not yet obtained, are discussed numerically. In particular, we are able to calculate the quantum dot eigenfunctions for a much larger spectrum of external harmonic frequencies as compared to previous results. Also the existence of bound states for such planar system in the case $l=0$ is predicted and the respective eigenvalues are determined.
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