AI 中文总结
研究二维圆对称谐波、福克 - 达尔文型和四次非谐波振子的拉克斯对，发现二维各向同性谐波振子的相关情况，构建二次 + 四次各向同性非谐波振子等的拉克斯对和r矩阵，通过变量变换还适用于拉杰夫 - 兰肯模型。
AI 中文摘要
本文关注二维圆对称谐波、福克 - 达尔文型和四次非谐波振子的拉克斯对。二维各向同性谐波振子虽为双哈密顿量，但其递归算子未导出拉克斯对，通过谐波卡洛杰罗模型的极限也未得到。然而，该超可积谐波振子存在一个4×4块形式的拉克斯对及谱参数，给出两个对合的守恒模式能量和相应的动力学r矩阵。还发现了2×2拉克斯对，其谱参数给出满足非阿贝尔泊松代数的所有三个独立守恒量。接着构建了二次 + 四次各向同性非谐波振子的一族su(2)拉克斯对和r矩阵，并扩展到具有旋转能量的各向同性振子，通过变量变换，这些拉克斯对和r矩阵也适用于拉杰夫 - 兰肯模型。
英文摘要
This paper concerns Lax pairs for circularly symmetric harmonic, Fock-Darwin-type and quartic anharmonic oscillators in two dimensions. Although the 2d isotropic harmonic oscillator is bi-Hamiltonian, its recursion operator does not lead to a Lax pair, nor do we obtain such a pair by taking a limit of the harmonic Calogero model. On the other hand, we show that this superintegrable harmonic oscillator admits a $4 \times 4$ block-form Lax pair with spectral parameter giving two conserved mode energies in involution and a corresponding dynamical $r$-matrix. Interestingly, we also find $2 \times 2$ Lax pairs with spectral parameter that give all three independent conserved quantities satisfying a nonabelian Poisson algebra, thereby providing a simple example of a Lax pair whose conserved quantities are not all in involution. Next, we construct a family of $su(2)$ Lax pairs and $r$-matrices for the quadratic+quartic isotropic anharmonic oscillator. This is then extended to an isotropic oscillator with a rotational energy, which may be viewed as the Fock-Darwin oscillator with a quartic potential. With a change of variables, these Lax pairs and $r$-matrices also apply to the Rajeev-Ranken model, although its noncanonical Poisson structure is distinct from that of the anharmonic oscillator.
Comments21 pages
