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 $C_2$ and $G_2$.

中文翻译：

---

Weyl Alcove 双重晶格上的量子粒子

离散量子模型族描述了自由非相对论量子粒子在 Weyl 壁龛封闭内的重新缩放和移动的双权重晶格上传播。所提出的离散量子台球的边界条件由外尔壁龛边界上精确定位的狄利克雷和诺依曼壁强制执行。粒子传播到相邻位置的幅度由有限支持的复值双权重跳跃函数确定。离散双权重哈密顿量是作为专门构造的双权重跳跃算子的总和获得的。通过利用广义双加权傅立叶-外尔变换，时间无关的薛定谔方程的解与量子系统的本征能一起被精确求解。$C_2$ 和 $G_2$.