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一类四阶微分方程的特殊函数

Special functions associated to a certain fourth order differential equation

Joachim Hilgert, Toshiyuki Kobayashi, Gen Mano, Jan Möllers

发表信息：
Ramanujan J. 26 (2011), no. 1, 1-34
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摘要：

我们发展了一个与$\mathbb{R}$上的某个四阶微分算子$\mathcal{D}_{μ,ν}$相关的“特殊函数“理论，它依赖于两个参数$μ,ν$ .对于具有$μ+ν\in2\mathbb{N}_0$的整数$μ,ν\geq-1$，该算子扩展为具有离散谱的$L^2(\mathbb{R}_+,x^{μ+ν+1}dx)$上的自伴算子。我们找到了本征函数的生成函数的封闭公式，并由此导出了本征函数的基本性质，如正交性、完备性、$L^2$ -范数、积分表示和各种递推关系.这四阶微分算子$\mathcal{D}_{μ,ν}$出现作为径向部分的卡西米尔行动在薛定谔模型的最小表示的组$O(p,q)$，和我们的“特殊函数“给$K$ -有限向量。

英文摘要：

We develop a theory of "special functions" associated to a certain fourth order differential operator $\mathcal{D}_{μ,ν}$ on $\mathbb{R}$ depending on two parameters $μ,ν$. For integers $μ,ν\geq-1$ with $μ+ν\in2\mathbb{N}_0$ this operator extends to a self-adjoint operator on $L^2(\mathbb{R}_+,x^{μ+ν+1}dx)$ with discrete spectrum. We find a closed formula for the generating functions of the eigenfunctions, from which we derive basic properties of the eigenfunctions such as orthogonality, completeness, $L^2$-norms, integral representations and various recurrence relations. This fourth order differential operator $\mathcal{D}_{μ,ν}$ arises as the radial part of the Casimir action in the Schrödinger model of the minimal representation of the group $O(p,q)$, and our "special functions" give $K$-finite vectors.

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