Generalised Hermite spectral methods for PDEs involving integral fractional Laplacian and Schr\"{o}dinger operators

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 Schr\"{o}dinger operators in $\mathbb R^d.$ As a generalisation of the G. Szeg\"{o}'s family in 1D (1939), the first family of GHPs (resp. GHFs) are orthogonal with respect to $|\bx|^{2\mu} \e^{-|\bx|^2}$ (resp. $|\bx |^{2\mu}$) 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 $[u,v]_{H^s(\mathbb R^d)}=((-\Delta)^{s/ 2}u, (-\Delta)^{s/2} v )_{\mathbb R^d}$ 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 Schr\"{o}dinger operator: $(-\Delta)^s +|\bs x|^{2\mu}$ with $s\in (0,1]$ and $\mu>-1/2.$ Following the same spirit, we construct the second family of GHFs, dubbed as M\"untz-type generalised Hermite functions (M-GHFs), which are orthogonal with respect to an inner product associated with the underlying Schr\"{o}dinger operator, and are tailored to the singularity of the solution at the origin. We demonstrate that the M\"untz-type GHF spectral method leads to sparse matrices and spectrally accurate to some Schr\"{o}dinger eigenvalue problems.