Classes of discrete quantum models that describe a free non-relativistic quantum particle propagating on rescaled and shifted dual root lattices inside closures of Weyl alcoves are constructed. Boundary conditions of the discrete quantum billiard systems on the borders of the Weyl alcoves are controlled by specific combinations of Dirichlet and Neumann walls that result from sign homomorphisms and admissible shifts inherent in generalized dual root lattice Fourier–Weyl transforms. The amplitudes of the particle’s jumps to neighbouring positions are controlled by a complex-valued dual root lattice hopping function with finite support. The solutions of the time-independent Schrödinger equation together with the eigenenergies of the quantum systems are explicitly determined. The matrix Hamiltonians and eigenenergies of the discrete models are exemplified for the rank two cases A 2 and C 2.

Sixteen types of the discrete multivariate transforms, induced by the multivariate antisymmetric and symmetric sine functions, are explicitly developed. Provided by the discrete transforms, inherent interpolation methods are formulated. The four generated classes of the corresponding orthogonal polynomials generalize the formation of the Chebyshev polynomials of the second and fourth kinds. Continuous orthogonality relations of the polynomials together with the inherent weight functions are deduced. Sixteen cubature rules, including the four Gaussian, are produced by the related discrete transforms. For the three-dimensional case, interpolation tests, unitary transform matrices and recursive algorithms for calculation of the polynomials are presented.

Families of discrete quantum models that describe a free non-relativistic quantum particle propagating on rescaled and shifted dual weight lattices inside closures of Weyl alcoves are developed. The boundary conditions of the presented discrete quantum billiards are enforced by precisely positioned Dirichlet and Neumann walls on the borders of the Weyl alcoves. The amplitudes of the particle’s propagation to neighbouring positions are determined by a complex-valued dual-weight hopping function of finite support. The discrete dual-weight Hamiltonians are obtained as the sum of specifically constructed dual-weight hopping operators. By utilising the generalised dual-weight Fourier–Weyl transforms, the solutions of the time-independent Schrödinger equation together with the eigenenergies of the quantum systems are exactly resolved. The matrix Hamiltonians, stationary states and eigenenergies of the discrete models are exemplified for the rank two cases C2 and G2.

Explicit links of the multivariate discrete (anti)symmetric cosine and sine transforms with the generalized dual-root lattice Fourier–Weyl transforms are constructed. Exact identities between the (anti)symmetric trigonometric functions and Weyl orbit functions of the crystallographic root systems A1 and Cn are utilized to connect the kernels of the discrete transforms. The point and label sets of the 32 discrete (anti)symmetric trigonometric transforms are expressed as fragments of the rescaled dual root and weight lattices inside the closures of Weyl alcoves. A case-by-case analysis of the inherent extended Coxeter–Dynkin diagrams specifically relates the weight and normalization functions of the discrete transforms. The resulting unique coupling of the transforms is achieved by detailing a common form of the associated unitary transform matrices. The direct evaluation of the corresponding unitary transform matrices is exemplified for several cases of the bivariate transforms.

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