离散谐振子：Mathieu函数和一类新的广义Hermite多项式
The discretised harmonic oscillator: Mathieu functions and a new class of generalised Hermite polynomials
摘要：
我们提出了一个一般的，渐近解的离散谐振子。相应的薛定谔方程与马蒂厄微分方程，即量子摆的薛定谔方程正则共轭。因此，除了给出孤立约瑟夫结或超导单电子晶体管（SSET）的哈密顿量的显式解外，我们还得到了Mathieu函数的渐近表示。我们解决了离散谐振子变换成一个无限的代数方程组，后来被证明是满足所获得的解决方案的无限维矩阵本征值问题。该方法定义了一类新的广义Hermite多项式，它们是耦合参数的显函数，在耦合常数为零的极限下趋于普通Hermite多项式.多项式作为厄米矩阵的特征向量的一部分变得正交，因此，解的指数部分不能被排除。我们已经阐明了解的一般结构，包括关于量子数和展开阶的结构。给出了渐近解的三个领先阶的显式证明，并给出了特征向量关于范数渐近收敛的证明。从更实际的角度来看，我们可以估计所需的努力，以改善已知的解决方案和准确性的特征向量。所应用的方法可以被推广，以适应几个变量。
英文摘要：
We present a general, asymptotical solution for the discretised harmonic oscillator. The corresponding Schrödinger equation is canonically conjugate to the Mathieu differential equation, the Schrödinger equation of the quantum pendulum. Thus, in addition to giving an explicit solution for the Hamiltonian of an isolated Josephon junction or a superconducting single-electron transistor (SSET), we obtain an asymptotical representation of Mathieu functions. We solve the discretised harmonic oscillator by transforming the infinite-dimensional matrix-eigenvalue problem into an infinite set of algebraic equations which are later shown to be satisfied by the obtained solution. The proposed ansatz defines a new class of generalised Hermite polynomials which are explicit functions of the coupling parameter and tend to ordinary Hermite polynomials in the limit of vanishing coupling constant. The polynomials become orthogonal as parts of the eigenvectors of a Hermitian matrix and, consequently, the exponential part of the solution can not be excluded. We have conjectured the general structure of the solution, both with respect to the quantum number and the order of the expansion. An explicit proof is given for the three leading orders of the asymptotical solution and we sketch a proof for the asymptotical convergence of eigenvectors with respect to norm. From a more practical point of view, we can estimate the required effort for improving the known solution and the accuracy of the eigenvectors. The applied method can be generalised in order to accommodate several variables.
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