In this paper, we introduce a new family of orthogonal systems, termed as the M\"{u}ntz ball polynomials (MBPs), which are orthogonal with respect to the weight function: $\|x\|^{2\theta+2\mu-2} (1-\|x\|^{2\theta})^{\alpha}$ with the parameters $\alpha>-1, \mu>- 1/2$ and $\theta>0$ in the $d$-dimensional unit ball $x\in {\mathbb B}^d=\big\{x\in\mathbb{R}^d: r=\|x\|\leq1\big\}$. We then develop efficient and spectrally accurate MBP spectral-Galerkin methods for singular eigenvalue problems including degenerating elliptic problems with perturbed ellipticity and Schr\"odinger's operators with fractional potentials. We demonstrate that the use of such non-standard basis functions can not only tailor to the singularity of the solutions but also lead to sparse linear systems which can be solved efficiently.

翻译：本文引入了一类新的正交系统，称为 Müntz 球面多项式 (MBPs)，它们关于权函数 $\|x\|^{2\theta+2\mu-2} (1-\|x\|^{2\theta})^{\alpha}$ (参数为 $\alpha>-1, \mu>-1/2, \theta>0$) 在 $d$ 维单位球 $x\in {\mathbb B}^d=\big\{x\in\mathbb{R}^d: r=\|x\|\leq1\big\}$ 上正交。随后，我们针对奇异特征值问题（包括具有扰动椭圆性的退化椭圆问题和具有分式位势的 Schrödinger 算子）发展了高效且谱精度的 MBP 谱-Galerkin 方法。我们证明，使用此类非标准基函数不仅能够适应解的非奇异性，还能导出可高效求解的稀疏线性系统。