标题： 形状不变的有理延拓与与例外多项式相关的势能 显示英文标题

标题： Shape Invariant Rational Extensions And Potentials Related to Exceptional Polynomials

摘要： 本文中，我们证明了尝试构造已知形状不变势的形状不变扩展时，除了一个常数的平移外，还导出了著名的等谱位移变形技术。 利用这一点，我们构建了无穷组具有$X_m$异常多项式作为解的广义势。 这些势是现有形状不变势的有理扩展。 该方法通过径向振荡子势和三角Pöschl-Teller势进行了阐明。 对于径向振荡子势的情况，在已知的有理扩展之外，我们构建了两组无穷类别的有理扩展，似乎这些扩展研究得较少。 对于其中一个势，我们表明其解涉及第三种类型的异常Laguerre多项式。 给出了这一广义无穷势集及其相应解的显式表达式。 对于三角Pöschl-Teller势，我们的分析指出了可能存在多种超出文献中已知范围的有理扩展。 显示英文摘要

摘要： In this paper, we show that an attempt to construct shape invariant extensions of a known shape invariant potential leads to, apart from a shift by a constant, the well known technique of isospectral shift deformation. Using this, we construct infinite sets of generalized potentials with $X_m$ exceptional polynomials as solutions. These potentials are rational extensions of the existing shape invariant potentials. The method is elucidated using the radial oscillator and the trigonometric Pöschl-Teller potentials. For the case of radial oscillator, in addition to the known rational extensions, we construct two infinite sets of rational extensions, which seem to be less studied. For one of the potential, we show that its solutions involve a third type of exceptional Laguerre polynomials. Explicit expressions of this generalized infinite set of potentials and the corresponding solutions are presented. For the trigonometric Pöschl-Teller potential, our analysis points to the possibility of several rational extensions beyond those known in literature.