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

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 ℝd. 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]Hs(ℝd) = ((-Δ)s/2u, (-Δ)s/2v)ℝd 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.