Nontensorial generalised hermite spectral methods for PDEs with fractional Laplacian and Schrödinger operators

¹ School of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China
² College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China
³ State Key Laboratory of Computer Science/Laboratory of Parallel Computing, Institute of Software, Chinese Academy of Sciences, Beijing 100190, China
⁴ Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Nanyang 637371, Singapore
⁵ Beijing Computational Science Research Center, Beijing, 100193, China

^* Corresponding author: lilian@ntu.edu.sg

Received: 23 December 2020
Accepted: 20 August 2021

Abstract

In this paper, we introduce two families of nontensorial generalised Hermite polynomials/functions (GHPs/GHFs) in arbitrary dimensions, and develop efficient and accurate spectral methods for solving PDEs with integral fractional Laplacian (IFL) and/or Schrödinger operators in ℝ ^d . As a generalisation of the G. Szegö’s family in 1D (1939), the first family of multivariate GHPs (resp. GHFs) are orthogonal with respect to the weight function |x|^2μe^-|x|2 (resp. |x|^2μ ) in ℝ^d. We further construct the adjoint generalised Hermite functions (A-GHFs), which have an interwoven connection with the corresponding GHFs through the Fourier transform, and are orthogonal with respect to the inner product [u,v]_H^s(ℝ^d) = ((-Δ)^s/2u, (-Δ)^s/2v)_ℝ^d associated with the IFL of order s > 0. As an immediate consequence, 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 remarkably efficient in solving PDEs with the fractional Schrödinger operator: (-Δ) ^s + | x |^2μ with s ∈ (0,1] and μ > −1/2 in ℝ ^d We construct the second family of multivariate nontensorial Müntz-type 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 solution to some Schrödinger eigenvalue problems.