六次增长的有理无单值势函数
On the rational monodromy-free potentials with sextic growth
摘要：
研究了无穷远处具有六次增长的有理势$V(x)$，使得相应的一维\Sch方程在复域内对所有的谱参数值都没有单值性.我们详细研究了这类势的子类，它可以由著名的拟精确可解势类$$V= x^6 - νx^2 +\frac{l(l+1)}{x^2}.$$通过达布变换构造出来。我们证明，与二次增长的情形相比，存在具有拟有理本征函数的monodromy-free势，但不能由这种构造给出。讨论了相应代数簇之间的关系，给出了具有六次位势的外场中Calogero-Moser问题的一些初等解。
英文摘要：
We study the rational potentials $V(x)$, with sextic growth at infinity, such that the corresponding one-dimensional \Sch equation has no monodromy in the complex domain for all values of the spectral parameter. We investigate in detail the subclass of such potentials which can be constructed by the Darboux transformations from the well-known class of quasi-exactly solvable potentials $$V= x^6 - νx^2 +\frac{l(l+1)}{x^2}.$$ We show that, in contrast with the case of quadratic growth, there are monodromy-free potentials which have quasi-rational eigenfunctions, but which can not be given by this construction. We discuss the relations between the corresponding algebraic varieties, and present some elementary solutions of the Calogero-Moser problem in the external field with sextic potential.
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