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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 $|bx|^{2mu} e^{-|bx|^2}$ (resp. $|bx |^{2mu}$) 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: $(-Delta)^s +|bs x|^{2mu}$ with $sin (0,1]$ and $mu>-1/2.$ Following the same spirit, we construct the second family of GHFs, dubbed as Muntz-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 Muntz-type GHF spectral method leads to sparse matrices and spectrally accurate to some Schr{o}dinger eigenvalue problems.
In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving integral fractional Laplacian in $mathbb{R}^d$, which is built upon two essential components: (i) the Dunford-Taylor formulation of the fractional Laplacian; and (ii
In this paper, we establish various maximal principles and develop the direct moving planes and sliding methods for equations involving the physically interesting (nonlocal) pseudo-relativistic Schr{o}dinger operators $(-Delta+m^{2})^{s}$ with $sin(0
This paper proposes and analyzes an ultra-weak local discontinuous Galerkin scheme for one-dimensional nonlinear biharmonic Schr{o}dinger equations. We develop the paradigm of the local discontinuous Galerkin method by introducing the second-order sp
In this paper, we provide a framework of designing the local discontinuous Galerkin scheme for integral fractional Laplacian $(-Delta)^{s}$ with $sin(0,1)$ in two dimensions. We theoretically prove and numerically verify the numerical stability and c
We study the inverse scattering for Schr{o}dinger operators on locally perturbed periodic lattices. We show that the associated scattering matrix is equivalent to the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph