论文标题
广义四分之一的非谐振荡器的多项式溶液
Polynomial Solutions of Generalized Quartic Anharmonic Oscillators
论文作者
论文摘要
本文介绍了通用对称四分之一振荡器的能量特征值问题的部分解决方案。该问题的代数化是通过根据nilpotent组的发电机来表达施罗辛格运营商来实现的，我们称之为四分之一。然后，可以看到能量特征值取决于该组的两个Casimir操作员的值。这种依赖性表现出从组发电机的缩放特性遵循的缩放定律。要求在特定谎言代数元素中产生多项式解决方案的电势对四个潜在参数构成限制，而其中只有两个免费。对于满足此类约束的潜力，至少可以通过纯粹的代数均值以封闭的分析形式获得至少一个能量特征值和相应的特征功能。通过我们的方法，我们扩展了通过更常见的SL（2，r）代数化在文献中获得的准精确解决的四分之一振荡器。最后，我们展示了广义四分之一振荡器问题的溶液如何为在特定非恒定电磁场中移动的带电粒子移动而产生解决方案。
This paper deals with the partial solution of the energy eigenvalue problem for generalized symmetric quartic oscillators. Algebraization of the problem is achieved by expressing the Schroedinger operator in terms of the generators of a nilpotent group, which we call the quartic group. Energy eigenvalues are then seen to depend on the values of the two Casimir operators of the group. This dependence exhibits a scaling law which follows from the scaling properties of the group generators. Demanding that the potential gives rise to polynomial solutions in a particular Lie algebra element puts constraints on the four potential parameters, leaving only two of them free. For potentials satisfying such constraints at least one of the energy eigenvalues and the corresponding eigenfunctions can be obtained in closed analytic form {by pure algebraic means. With our approach we extend the class of quasi-exactly solvable quartic oscillators which have been obtained in the literature by means of the more common sl(2,R) algebraization. Finally we show, how solutions of the generalized quartic oscillator problem give rise to solutions for a charged particle moving in particular non-constant electromagnetic fields.
