  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.
