Quantum particle on dual root lattice in Weyl alcove

Abstract

Abstract 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.