Connection between quantum systems involving the fourth Painlevé transcendent and k-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial
Connection between quantum systems involving the fourth Painlevé transcendent and k-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial
复制标题
量子系统之间的联系第四 Painlevé 超越和涉及与 Hermite 例外正交多项式相关的振荡器的谐波的 k 步有理扩展
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
C. Quesne
中科院分区：
文献类型：
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作者：
I. Marquette;C. Quesne
这种通信的目的是指出在二阶超对称量子力学和三阶阶梯算符的背景下得到的涉及第四Painleve超越PIV的一维量子哈密顿量与称为谐振子的k步有理扩张的量子系统家族之间的联系，并与多指标Xm1，m2，…有关利用广义Hermite和Okamoto多项式的第四类Painleve方程的有理数解，在哈密顿等价的水平上建立了这些精确可解模型之间的联系。我们还讨论了达布-克拉姆和Krein-Adler超荷的超对称结构的不同组合所得到的不同阶梯算符、它们的零模和相应的能量。这些结果将证明和澄清在以前的论文中观察到的针对特定案例的关系。
The purpose of this communication is to point out the connection between a 1D quantum Hamiltonian involving the fourth Painleve transcendent PIV, obtained in the context of second-order supersymmetric quantum mechanics and third-order ladder operators, with a hierarchy of families of quantum systems called k-step rational extensions of the harmonic oscillator and related with multi-indexed Xm1,m2,…,mk Hermite exceptional orthogonal polynomials of type III. The connection between these exactly solvable models is established at the level of the equivalence of the Hamiltonians using rational solutions of the fourth Painleve equation in terms of generalized Hermite and Okamoto polynomials. We also relate the different ladder operators obtained by various combinations of supersymmetric constructions involving Darboux-Crum and Krein-Adler supercharges, their zero modes and the corresponding energies. These results will demonstrate and clarify the relation observed for a particular case in previous papers.