论文标题
准确可溶解的非谐振荡器，退化正交多项式和painleve'II
Exactly solvable anharmonic oscillator, degenerate orthogonal polynomials and Painleve' II
论文作者
论文摘要
本文介绍了Shapiro和Tater在复杂平面中两组点之间的相似性上的猜想。一方面是$ t \ in \ mathbb {c} $的值，在复杂平面$$ \ frac {{\ rm d}^2 y} {{\ rm d} {{\ rm d}^2} x^2} x^2} x^4 + tx^2 + 2Jx y = withs y的频谱$$ \ frac {{\ rm d}^2 y} {{\ rm d}^2 y} {{\ rm d}^2 y} {特征值。另一侧是Vorob'ev-Yablonskii多项式的零集，即第二个Painlevé方程的理性解决方案的极点。在此过程中，我们指出了非谐波振荡器问题与某些变性正交多项式之间的令人惊讶而深厚的联系。
The paper addresses a conjecture of Shapiro and Tater on the similarity between two sets of points in the complex plane; on one side is the values of $t\in \mathbb{C}$ for which the spectrum of the quartic anharmonic oscillator in the complex plane $$\frac{{\rm d}^2 y}{{\rm d}x^2} - ( x^4 + tx^2 + 2Jx )y = Λy, $$ with certain boundary conditions, has repeated eigenvalues. On the other side is the set of zeroes of the Vorob'ev-Yablonskii polynomials, i.e. the poles of rational solutions of the second Painlevé equation. Along the way, we indicate a surprising and deep connection between the anharmonic oscillator problem and certain degenerate orthogonal polynomials.
