论文标题
涉及整体分数拉普拉斯和schrödinger运算符的PDE的广义HERMITE光谱方法
Generalised Hermite spectral methods for PDEs involving integral fractional Laplacian and Schrödinger operators
论文作者
论文摘要
在本文中，我们在任意维度上介绍了两个新的HERMITE多项式/功能（GHP/GHF）的新家族，并为PDE提供了具有积分分数laplacian（IFL）（IFL）（IFL）和/或Schrödinger的PDE的有效，准确的普遍的HERMITE光谱算法。 （1939年），GHP的第一个家庭（分别GHFS）在$ | \ e^{2μ} \ e^{ - | \ bx |^2} $（在$ \ \ bx |^{2μ} $中）是正交的。我们进一步定义了通过傅立叶变换与相应的GHF相互交织的连接，并且相对于内部产品$ [u，v] _ { ）_ {\ mathbb r^d} $与订单$ s> 0 $关联的IFL关联。因此，使用A-GHFS作为基础函数的光谱 - 盖尔金方法导致IFL的对角线刚度矩阵（已知这是众所周知的困难和离散昂贵的）。新的基础还可以找到有效而准确的与分数schrödinger操作员求解PDE：$（δ）^s +| \ bs x |^{2μ} $与$ s \ in（0,1] $和$μ> -1/2相对于与基础Schrödinger操作员相关的内部产品，它是针对原始溶液的奇异性量身定制的。
In this paper, we introduce two new families of generalised Hermite polynomials/functions (GHPs/GHFs) in arbitrary dimensions, and develop efficient and accurate generalised Hermite spectral algorithms for PDEs with integral fractional Laplacian (IFL) and/or Schrödinger operators in $\mathbb R^d.$ As a generalisation of the G. Szegö's family in 1D (1939), the first family of GHPs (resp. GHFs) are orthogonal with respect to $|\bx|^{2μ} \e^{-|\bx|^2}$ (resp. $|\bx |^{2μ}$) in $\mathbb R^d$. We further define adjoint generalised Hermite functions (A-GHFs) which have an interwoven connection with the corresponding GHFs through the Fourier transform, and which are orthogonal with respect to the inner product $[u,v]_{H^s(\mathbb R^d)}=((-Δ)^{s/ 2}u, (-Δ)^{s/2} v )_{\mathbb R^d}$ associated with the IFL of order $s>0$. Thus, the spectral-Galerkin method using A-GHFs as basis functions leads to a diagonal stiffness matrix for the IFL (which is known to be notoriously difficult and expensive to discretise). The new basis also finds efficient and accurate in solving PDEs with the fractional Schrödinger operator: $(-Δ)^s +|\bs x|^{2μ}$ with $s\in (0,1]$ and $μ>-1/2.$ Following the same spirit, we construct the second family of GHFs, dubbed as Müntz-type generalised Hermite functions (M-GHFs), which are orthogonal with respect to an inner product associated with the underlying Schrödinger operator, and are tailored to the singularity of the solution at the origin. We demonstrate that the Müntz-type GHF spectral method leads to sparse matrices and spectrally accurate to some Schrödinger eigenvalue problems.
