Quantum Particle on Dual Weight Lattice in Weyl Alcove

: 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 .


Introduction
The goal of this article is to assemble families of tight-binding models [1][2][3] by describing a free non-relativistic quantum particle which propagates on shifted and rescaled dual weight lattices inside closures of scaled Weyl alcoves. Similarly to the recently developed dual-root lattice models [1], the boundary conditions of the presented discrete quantum billiards [4,5] are enforced by precisely positioned Dirichlet and Neumann walls.
The quantum billiard systems of multiform shapes that include 2D stadium [6], Hecke triangular domains [7], equilateral triangle on a spherical surface [8] and polygons [9] have been investigated. The quantum billiards on 2D triangles [10][11][12] and 3D Weyl chamber [13] realise the closest versions of the current simplex-shaped models. From the viewpoint of position restrictions, the discrete quantum billiards [4,14] involve the propagating particle on quantum dots [15][16][17]. Single particle properties of the quantum dots are commonly studied via the discrete tight-binding Hamiltonians [3,15,17]. Specifically for the studied class of tight-binding atomic lattice models [3,15], the atoms are coupled to neighbours of a predetermined fixed degree and possible overlaps among atom orbitals are neglected. The multivariate discrete Fourier transform provides a fundamental tool for deriving eigenenergies and momentum bases of these discrete tight-binding models. By utilising the underlying symmetries of the developed discrete quantum systems for implementing boundary conditions, it appears that generalised dual weight lattice Fourier-Weyl transforms [18][19][20] provide similarly crucial connections between position and momentum bases together with exact eigenenergies and time evolutions.
Representing a ubiquitous class of the discrete Fourier-Weyl transforms [21][22][23][24], the generalised dual-weight Fourier-Weyl transforms [18][19][20] form extensions of the classical trigonometric transforms [25] to the crystallographic root systems [26]. Symmetrised and signed by the actions of the Weyl groups together with the sign homomorphisms [27], the four types of Weyl orbit functions [19,20] that are characterised by labels from the weight lattices shifted by admissible shifts [18] serve as kernels of the generalised dual-weight

Weyl Groups and Invariant Shifted Lattices
The mathematical exposition and notations of this article are established in papers [19][20][21][22]. Each simple Lie algebra from the classical four series A n (n ≥ 1), B n (n ≥ 3), C n (n ≥ 2) and D n (n ≥ 4) and from the five exceptional cases E 6 , E 7 , E 8 , F 4 , G 2 determines the set ∆ = {α 1 , . . . , α n } of the simple roots [37,38]. For the simple Lie algebras B n , C n , F 4 and G 2 , the set ∆ is disjointly decomposed into the set ∆ s of the short simple roots and the set ∆ l of the long simple roots, The vectors of the set ∆ span the Euclidean space R n equipped with the standard scalar product ·, · . The simple roots α i , i ∈ {1, . . . , n}, rescaled by the length factor 2/ α i , α i , produce the set ∆ ∨ of dual simple roots α ∨ i . The dual basis for the basis of simple roots ∆ is formed by dual fundamental weights ω ∨ i , which are given by ω ∨ i , α j = δ ij , i, j ∈ {1, . . . , n}, and the dual basis corresponding to the basis of the dual simple roots ∆ ∨ consists of fundamental weights ω i satisfying ω i , α ∨ j = δ ij . The reflection r i , associated to each simple root α i ∈ ∆, is specified by the following standard formula, r i a = a − a, α ∨ i α i , a ∈ R n . The reflections r i , i ∈ {1, . . . , n} generate the finite Weyl group W. The action of the Weyl group W on the set of simple roots ∆ produces the entire root system Π = W∆ which contains a unique highest root ξ ∈ Π of the following form, ξ = m 1 α 1 + · · · + m n α n .
Any homomorphism σ : W → U 2 from the Weyl group W to the multiplicative group U 2 = {±1} is called a sign homomorphism [19]. The identity 1 and the determinant σ e sign homomorphisms are, for any Weyl group W, defined on the generating reflections r i as follows, For the sets ∆ with two root-lengths (1), the short σ s and long σ l sign homomorphisms are determined on the generating reflections r i as follows, Four classical Weyl group invariant lattices [37] comprise the root lattice Q, the dual weight lattice P ∨ , the dual root lattice Q ∨ and the weight lattice P. These lattices are defined as the Z-spans of the following basis vectors of R n , Q =Zα 1 + · · · + Zα n , The dual weight lattice P ∨ is Z-dual to the root lattice Q, and the weight lattice is Z-dual to the dual root lattice Q ∨ . The orders of the quotient groups P/Q and P ∨ /Q ∨ equal the determinant c of the Cartan matrix C ij = α i , α ∨ j , The cone of the positive dual weights P ∨+ that comprises all points from the dual weight lattice P ∨ in the fundamental Weyl chamber, is explicitly given as the following [37], Since the fundamental Weyl chamber contains precisely one point of each W-orbit, the action of the Weyl group W on the cone P ∨+ produces the entire lattice P ∨ as follows, A vector ∈ R n , for which the Weyl group invariance condition holds as follows: constitutes an admissible shift of the weight lattice [18]. Similarly, a vector ∨ ∈ R n , such that the following is the case: represents an admissible shift of the dual weight lattice [18]. The equivalent admissible shifts induce identical shifted weight and dual weight lattices, respectively. The admissible shifts of the weight and dual weight lattices are listed up to equivalence in Table I of [18].

Signed Fundamental Domains
The affine Weyl group is the following semidirect product of the translation group Q ∨ and the Weyl group W, For any translation by the vector q ∨ ∈ Q ∨ and any w ∈ W, the affine Weyl group element z = T(q ∨ )w ∈ W aff acts canonically on R n as follows, The fundamental domain F ⊂ R n of W aff , which contains exactly one point from each W aff -orbit, forms a simplex explicitly described by the following, F = a 1 ω ∨ 1 + · · · + a n ω ∨ n | a 0 + m 1 a 1 + · · · + m n a n = 1, a i ≥ 0, i = 0, . . . , n .
The stabiliser Stab W aff (a) is the subgroup of W aff that comprises the elements stabilising a ∈ R n and the associated discrete ε-function ε : R n → N is defined by the following, The algorithm for the calculation of the coefficients ε(a) is detailed in ( [20], §3.7). The retraction homomorphism ψ : W aff → W and the mapping τ : W aff → Q ∨ are determined for any element z = T(q ∨ )w ∈ W aff by the following relations, The mapping τ induces, for any admissible shift of the weight lattice P, the dual shift homomorphism θ : W aff → U 2 given in [18] as follows, The dual shift homomorphism θ , restricted to the Weyl group W, evaluates as the following, For any sign homomorphism σ and any admissible shift of the weight lattice P, the homomorphism γ σ : W aff → U 2 is built from the retraction and dual shift homomorphisms as the following product, The values of γ σ on the generators of W aff are listed in Table II in [18].
Since F is a fundamental domain of W aff , for any a ∈ R n there exist exactly one point a ∈ F and z[a] ∈ W aff satisfying the following, Employing relation (12), the function χ σ : R n → {−1, 0, 1} is introduced in [1] for any a ∈ R n by the following, To each sign homomorphism σ and admissible shift of the weight lattice corresponds a signed fundamental domain F σ ( ) ⊂ F given by the following, In particular, it holds that F 1 (0) = F and F σ e (0) = int(F). Two subsets H σ ( ), B σ ( ) ⊂ ∂F of the boundary ∂F are described by the following, The following σ, -dependent disjoint decompositions of the fundamental domain F and its boundary ∂F are obtained by specialising Proposition 2.7 from [18],

Signed Dual Fundamental Domains
The dual affine Weyl group is the following semidirect product of the translation group Q and the Weyl group W, For any translation by the vector q ∈ Q and any w ∈ W, the dual affine Weyl group element y = T(q)w ∈ W aff acts canonically on R n as follows, The dual fundamental domain F ∨ ⊂ R n of the dual affine Weyl group W aff is a simplex explicitly described below, The stabiliser Stab W aff (b) is the subgroup of W aff that consists of elements stabilising b ∈ R n and the associated discrete function h ∨ M : R n → N is for any magnifying factor M ∈ N provided by the following, The algorithm for the calculation of the discrete function h ∨ M is described in ( [20], §3.7). The dual retraction homomorphism ψ : W aff → W and the mapping τ : W aff → Q are determined for any element y = T(q)w ∈ W aff as follows, The mapping τ induces, for any admissible shift ∨ of the dual weight lattice P ∨ , the shift homomorphism θ ∨ : W aff → U 2 given in [18] as follows, For any sign homomorphism σ and any admissible shift ∨ of the dual weight lattice P ∨ , the homomorphism γ σ ∨ : W aff → U 2 is constructed from the dual retraction and shift homomorphisms as the following product, The values of γ σ ∨ on the generators of W aff are listed in Table II from [18]. The associated signed dual fundamental domains F σ∨ ( ∨ ) ⊂ F ∨ are of the following form,

Dual-Weight Discretization of Weyl Orbit Functions
The kernels of the developed discrete transforms are formed by the Weyl orbit functions ϕ σ b : R n → C [27,28,39] that are induced from the sign homomorphisms σ as (anti)symmetrised multivariate exponential functions and given for argument a ∈ R n and label b ∈ R n by the following expression, For the identity sign homomorphism σ = 1, restricting the summation in relation (27) to the Weyl group label orbit yields the C-functions [28], The correspondence of the C-functions with the symmetric Weyl orbit functions ϕ 1 is expressed by the following renormalisation, The fundamental properties of the Weyl orbit functions comprise the duality, hermiticity and scaling symmetry given by the following, where * denotes the complex conjugation. Product-to-sum decomposition formulas of any function ϕ σ b and the symmetric function ϕ 1 b are of the following form, The Weyl orbit functions ϕ σ λ with discretized labels λ ∈ + P gain additional symmetry with respect to the affine Weyl group W aff described for any a ∈ R n and z ∈ W aff by the following, Symmetry relation (34) implies that the functions ϕ σ λ take zero values on the boundary The χ σ -function allows the reformulation of the properties (34) and (35) for any a ∈ F and a ∈ W aff a as follows, For any sign homomorphism σ ∈ {1, σ e , σ s , σ l } and admissible shifts and ∨ of the weight and dual weight lattices, the finite sets of points F σ P ∨ ,M ( , ∨ ) contain the points from the shifted and rescaled dual weight lattice that belong to the signed fundamental The finite sets of labels Λ σ Q,M ( , ∨ ) of orbit functions consist of the labels from the shifted weight lattice that belong to the magnified signed dual fundamental domain The cardinalities of the point and label sets coincide ( [18], Thm. 3.4), Note that the notation of the finite sets of points and labels from [18] has been modified to reflect the notation introduced in [21].
Discrete orthogonality relations of the Weyl orbit functions restricted to the finite point sets F σ P ∨ ,M ( , ∨ ) are for any labels λ, λ ∈ Λ σ Q,M ( , ∨ ) derived in ( [18], Thm. 4) as follows, The corresponding Plancherel formulas [18] yield the following complementary orthogonality relations for any points a, a ∈ F σ For any fixed ordering of the label and point sets Λ σ Q,M ( , ∨ ) and F σ P ∨ ,M ( , ∨ ), the unitary transform matrices I σ P ∨ ,M ( , ∨ ) of the generalised discrete dual-weight lattice Fourier-Weyl transforms are determined from the discrete orthogonality relations (39) by their entries for

Dual-Weight Dots
The dual-weight hopping function is realised by a fixed complex-valued function P ∨ : P ∨ → C that is defined on the points from the dual weight lattice P ∨ and required to have a finite support as follows, supp(P ∨ ) < +∞. (43) Crucially, the admissible dual-weight hopping function P ∨ is constrained to be Winvariant and Hermitian. Thus, for any p ∨ ∈ P ∨ and w ∈ W, the following holds, The following symmetries of the finite support of P ∨ are directly deduced from the properties (44) and (45), The dominant support supp + (P ∨ ) of the dual-weight hopping function P ∨ contains the dual weight lattice elements from the support supp(P ∨ ), which belong to the cone of the positive dual weights P ∨+ , The action of the Weyl group W on the cone of positive weights (3) guarantees that the action on the dominant support of P ∨ generates the entire support supp(P ∨ ) as follows, The W-invariance condition (44), together with support conditions (42) and (43), implies that the hopping function P ∨ suffices to define on a finite number of points from the dominant support p ∨ ∈ supp + (P ∨ ) for which the coordinates in ω ∨ -basis are nonnegative integers, p ∨ = (a 1 , . . . , a n ), a 1 , . . . , a n ∈ Z ≥0 .
Possible positions of a non-relativistic quantum particle are represented by the points P ∨ l,M ( ∨ ) of the shifted dual weight lattice that is scaled by a length factor l ∈ R together with the scaling factor M ∈ N, The particle jumps from the position at point Assuming that a fixed admissible dualweight hopping function P ∨ is given, any amplitude I M (x, x ) is determined by the following relation, By placing mirrors and barriers on the boundaries of the scaled fundamental domain lF, the positions of the quantum particle are further restricted to the dual-weight dot D σ P ∨ ,l,M ( , ∨ ) given as follows, The boundaries lB σ ( ) of lF in the boundary decomposition (18) represent perfect mirrors (Neumann walls) and the boundaries lH σ ( ) represent ideal barriers (Dirichlet walls).

Schrödinger Equations
The state determined by the vector from the position basis |a ∈ H σ P ∨ ,M ( , ∨ ), a ∈ F σ P ∨ ,M ( , ∨ ) represents the quantum particle positioned at la ∈ D σ P ∨ ,l,M ( , ∨ ). The counting formulas for the dimensions of the Hilbert spaces H σ P ∨ ,M ( , ∨ ) are as follows: Each dual weight from the dominant support of the admissible hopping function p ∨ ∈ supp + (P ∨ ) induces, for any two points a, a ∈ F σ P ∨ ,M ( , ∨ ), the corresponding dual-weight coupling set N p ∨ ,M (a, a ) as follows, The dual-weight hopping operator A σ , which incorporates interactions of the particle with the boundary walls via the χ-function summing over coupling sets N p ∨ ,M (a, a ), is determined by its matrix elements in the position basis as the following, The value of the hopping function at the origin P ∨ Hermiticity condition (45) enforces that E 0 is real-valued and represents the on-site energy of the particle at an arbitrary position of the dot D σ of the quantum particle propagating on the dual-weight dot D σ P ∨ ,l,M ( , ∨ ) is established as the sum of all dual-weight hopping operators, Representing time as the real-valued parameter t ∈ R, the time evolution of the state vectors |ψ(t) ∈ H σ P ∨ ,M ( , ∨ ) of the particle on the dual-weight dot D σ P ∨ ,l,M ( , ∨ ) is governed by the following standard Schrödinger equation,

Time Evolution
The orthonormal momentum bases |λ ∈ H σ P ∨ ,M ( , ∨ ), corresponding to the ordered label sets λ ∈ Λ σ Q,M ( , ∨ ), are given by the following relations: with the scalar products a|λ defined by the inverse of the unitary matrix I σ P ∨ ,M ( , ∨ ) of the generalised discrete dual-weight lattice Fourier-Weyl transform (41) as follows, In the following theorem, the vectors of the orthonormal momentum basis |λ ∈ H σ Theorem 1. The vectors (56) of the orthonormal basis |λ ∈ H σ P ∨ ,M ( , ∨ ), λ ∈ Λ σ Q,M ( , ∨ ) satisfy the time-independent Schrödinger Equation (58). The corresponding eigenenergies E σ P ∨ ,λ,M ( , ∨ ) are real-valued and determined for any admissible dual-weight hopping function P ∨ by summation of the C-functions (28) over the dominant support, Proof. Equivalent reformulation of the time-independent Schrödinger Equation (58) via coordinates in the position basis yields for any a ∈ F σ P ∨ ,M ( , ∨ ) the following condition, By utilising the hopping function W-invariance (44) together with the W-orbit decomposition (48) of its support supp(P ∨ ), the matrix elements of the Hamiltonian H σ P ∨ ,M ( , ∨ ) are, for any a, a ∈ F σ P ∨ ,M ( , ∨ ), calculated from each hopping operator elements (52) and summation (54) as follows, The scalar products a|λ that are for λ ∈ Λ σ Q,M ( , ∨ ) and a ∈ F σ P ∨ ,M ( , ∨ ) defined by relation (57), together with the matrix elements of the Hamiltonian (61), are substituted into the time-independent Schrödinger Equation (60) and produce the following equivalent relation, Vanishing property of the orbit functions (35) and decomposition of the fundamental domain (17) provide extension of the summation in relation (62) to the following points: and argument symmetry relation (36) guarantees the following simplification, Since the point set 1 M ( ∨ + P ∨ ) ∩ F contains exactly one point from each W aff -orbit of the refined shifted dual weight lattice 1 M ( ∨ + P ∨ ), the double summation in (63) actually represents the summation over the following set: and the reformulated time-independent Schrödinger equation (63) is further facilitated as follows, Focusing on the reformulation of the eigenenergies (59), duality formula (30), the scaling symmetry (32) and renormalisation property (29) of the symmetric orbit functions yield the following, The W-invariance of each term P ∨ (p ∨ )ϕ 1 λ p ∨ M , which is deduced from relations (10), (34) and (44), enables retracting the summation over W-orbits in eigenenergy expression (65) and permits using orbit-stabiliser theorem the following equivalent form, Consequently, the product-to-sum decomposition of the orbit functions (33) provides the following simplification, The W-invariance requirement (44) of the dual-weight hopping function P ∨ , together with the W-invariance (46) of the support supp(P ∨ ), further simplifies (67) as follows, Substituting the resulting expression (68) into the reformulation of the time-independent Schrödinger Equation (64) proves its original version (58).
The Hermiticity conditions (31), (45) and (47) of both orbit functions ϕ σ λ and hopping function P ∨ guarantee from relation (66) the following calculation: For any initial state determined by the normalised vector |ψ(0) ∈ H σ P ∨ ,M ( , ∨ ): the normalised time-evolved state vector |ψ(t) ∈ H σ P ∨ ,M ( , ∨ ) is provided by the combination of the stationary states: The coordinates λ|ψ(0) are calculated from the position coordinates a|ψ(0) via the generalised dual-weight lattice Fourier-Weyl transform (41), The probability P σ, , ∨ P ∨ ,M (a, t) of finding the particle in the time-evolved state (71) at position la ∈ D σ P ∨ ,l,M ( , ∨ ) is determined as follows, Summing all time-dependent probabilities over the entire dual-weight dot D σ P ∨ ,l,M ( , ∨ ), while taking into account the normalisation condition of the initial state vector (69), substantiates the trapping of the quantum particle, The time-independent probability P σ, , ∨ P ∨ ,M [λ](a) of finding the particle in the stationary state (70) at position la ∈ D σ P ∨ ,l,M ( , ∨ ) is provided directly by the amplitude (57), 4. Dual Weight Lattice Models of C 2 and G 2 4.1. Case C 2 The root system C 2 admits, in addition to the trivial zero shifts, also the non-trivial shift = ω 2 /2 of the weight lattice and the shift ∨ = ω ∨ 1 /2 of the dual weight lattice [18]. Considering only the nearest and next-to-nearest neighbour coupling, the three non-zero dual-weight hopping function P ∨ values are specified by the zero energy level E 0 and two non-zero parameters A, B ∈ R, as follows, The energy functions E 1 , E 2 : F ∨ → R, which characterise the total eigenenergies of the quantum particle, are given for b ∈ F ∨ by the following relations, The 3D plots of energy functions E 1 and E 2 for the C 2 case are depicted in Figure 1. The total eigenenergies of the quantum particle propagating on the dots D σ P ∨ ,l,M ( , ∨ ) are determined for λ ∈ Λ σ Q,M ( , ∨ ) from relation (59) by the energy functions as follows, For the fixed scaling factor M = 5, the trivial admissible shift of the weight lattice = 0 and the admissible shift of the dual weight lattice ∨ = ω ∨ 1 /2, the point sets are determined in ω ∨ -basis as follows: The determinant σ e sign homomorphism dot D σ e P ∨ ,1,5 0, 1 2 ω ∨ 1 is depicted in Figure 2. Figure 2. The dot D σ e P ∨ ,1,5 0, 1 2 ω ∨ 1 of C 2 . The light blue triangle represents the fundamental domain F σ e (0). The boundary black dashed lines represent the Dirichlet walls H σ e (0). The point set F σ e P ∨ ,5 0, 1 2 ω ∨ 1 , representing possible positions of the quantum particle, is formed by the dark dots. The lines connecting the dots symbolise the nearest neighbour coupling characterised by the hopping operator A σ e ω ∨ 2 ,5 0, 1 2 ω ∨ 1 .
The matrices of the two dual-weight hopping operators (52) for the identity sign homomorphism, which is calculated in the position basis |a ordered as points from list (79), are of the following form: Calculated in the position basis |a ordered as points from list (80), the matrices of the two dual-weight hopping operators for the determinant sign homomorphism σ e are determined as follows: Calculated in the position basis |a ordered as points from list (81), the matrices of the two dual-weight hopping operators for the short sign homomorphism σ s are given as follows: The matrices of the two dual-weight hopping operators for the long sign homomorphism σ l , which are calculated in the position basis |a ordered as points from list (82), are of the following form: The four Hamiltonians of the quantum particle on the dual-weight dots D σ P ∨ ,l,5 0, 1 2 ω ∨ 1 are then given as follows, H σ e P ∨ ,5 0, 1 2 The set of (rounded) eigenenergies of the particle on the identity homomorphism dot D 1 Q ∨ ,l,5 0, 1 2 ω ∨ 1 is calculated from relation (78) in the ordering of the label set (83) as follows: For the short homomorphism dot D σ s Q ∨ ,l,5 0, 1 2 ω ∨ 1 , the set of rounded eigenenergies of the quantum particle in the ordering of the label set (85) is calculated as follows: and for the long homomorphism dot D σ l Q ∨ ,l,5 0, 1 2 ω ∨ 1 in the ordering of the label set (86) as the following: [λ] of finding the particle on the dots D σ P ∨ ,1,35 0, 1 2 ω ∨ 1 , σ ∈ {1, σ e , σ s , σ l } corresponding to several lower stationary states |λ are depicted in Figure 3.

Case G 2
The root system G 2 admits only the trivial shifts for both weight and dual weight lattices [18]. Considering only the nearest and next-to-nearest neighbour coupling, the three non-zero values of the dual-weight hopping function P ∨ are determined by the zero energy level E 0 and two non-zero parameters A, B ∈ R as follows, The energy functions E 1 , E 2 : F ∨ → R that characterise the total eigenenergies of the quantum particle are specified for b ∈ F ∨ by the following relations, The 3D plots of the energy functions E 1 and E 2 are depicted in Figure 4. Stemming from relation (59), the total eigenenergies are for λ ∈ Λ σ Q,M (0, 0) determined by the energy functions E 1 and E 2 as the following, Figure 3. The probability plots for D σ Q ∨ ,1,35 0, 1 2 ω ∨ 1 of C 2 . The dots display the probabilities (74) of finding the particle in the stationary states |(1, 1) , |(1, 2) and |(1, 3) over their respective positions from D σ Q ∨ ,1,35 0, 1 2 ω ∨ 1 , σ ∈ {1, σ e , σ s , σ l }. The red dots illustrate probabilities over the particle's positions on the Neumann walls. Dots D σ P ∨ ,l,M (0, 0) The four point sets (37) are for the fixed scaling factor M = 10 expressed in ω ∨ -basis as the following: and the label sets (38) are determined in ω-basis as the following: The long sign homomorphism dual-weight dot D σ l P ∨ ,1,10 (0, 0) is depicted in Figure 5. Figure 5. The dot D σ l P ∨ ,1,10 (0, 0) of G 2 . The light triangle depicts the fundamental domain F σ l (0) and its boundary black dashed lines correspond to the Dirichlet walls H σ l (0) and boundary black line to the Neumann wall B σ l (0). The point set F σ l P ∨ ,10 (0, 0), representing possible positions of the quantum particle, is formed by the dark dots. The lines connecting the dots symbolise the nearest neighbour coupling characterised by the hopping operator A σ l ω ∨ 1 ,10 (0, 0). Calculated in the position basis |a that is ordered as the points listed in expression (91), the two identity sign homomorphism dual-weight hopping operators (52) are determined as follows: Calculated in the position basis |a that is ordered as the points listed in expression (92), the two determinant sign homomorphism dual-weight hopping operators are given as the following: The two short sign homomorphism dual-weight hopping operators, which are calculated in the position basis |a ordered as the points from the list (93), are given as the following: The two long sign homomorphism dual-weight hopping operators, which are calculated in the position basis |a ordered as the points from the list (94), are given as the following: The Hamiltonians of the quantum particle on the dual-weight dots D 1 P ∨ ,l,10 (0, 0), D σ e P ∨ ,l,10 (0, 0), D σ s P ∨ ,l,10 (0, 0) and D σ l P ∨ ,l,10 (0, 0) are the sums (54) of the corresponding hopping operators, The set of (rounded) eigenenergies of the particle on the identity sign homomorphism dot D 1 P ∨ ,l,10 (0, 0) is computed from relation (90) in the ordering of the label set (95) as follows: and on the determinant sign homomorphism dot D σ e P ∨ ,l,10 (0, 0) in the ordering of the label set (96) as follows: The set of eigenenergies of the particle on the short sign homomorphism dot D σ s P ∨ ,l,10 (0, 0) in the ordering of the label set (97) is provided by the following: and on the long sign homomorphism dot D σ l P ∨ ,l,10 (0, 0) in the ordering of the label set (98) by the following: The probabilities P σ,0,0 P ∨ ,60 [λ] of finding the particle in several lower stationary states |λ at the positions of the four dual-weight dots a ∈ D σ P ∨ ,1,60 (0, 0), σ ∈ {1, σ e , σ s , σ l } are depicted in Figure 6.

Conclusions
• The developed one-particle dual-weight discrete quantum billiard systems describe the non-relativistic quantum particle propagating on the dots D σ P ∨ ,l,M ( , ∨ ), which comprise finitely-many positions located inside the scaled closure of the Weyl alcove lF ⊂ R n . The precise arrangements (15) and (16) of the Dirichlet and Neumann walls lH σ ( ) and lB σ ( ) that realise the quantum trapping of the particle and coincide with the dual-root billiards [1] constitute the boundaries of the simplex lF. Any predetermined admissible dual-weight hopping function P ∨ , which encodes the amplitude propagation to the neighbouring positions, directly provides explicit formulas for the eigenenergies of the systems (59) via its Fourier transform by the symmetric Weyl orbit sums over its finite dominant support supp + (P ∨ ). The vectors of the orthonormal momentum basis, |λ ∈ H σ P ∨ ,M ( , ∨ ), λ ∈ Λ σ Q,M ( , ∨ ), determined independently on the dual-weight hopping function P ∨ by their explicit form (56) constitute solutions of the time-independent Schrödinger equation (58). The time evolution of the dual-weight quantum systems from any normalised initial state vector given in the position basis |a , a ∈ F σ P ∨ ,M ( , ∨ ) is exactly determined (71) .

•
The presence of the affine Weyl group orbits of the target positions a ∈ F σ P ∨ ,M ( , ∨ ) in the coupling sets (51) represent the first essential symmetry component for implementing the interactions enforced by the boundaries of lF. Secondly, the addition of the sign χ-function (13) values over the affine-reflected positions W aff a in the coupling set N p ∨ ,M (a, a ) counts the number and type of amplitude reflections between the source position a ∈ F σ P ∨ ,M ( , ∨ ) and the target position a ∈ F σ P ∨ ,M ( , ∨ ). The χ-function generalises the sign functions from [40] that are necessary for describing the Galois symmetries of Weyl orbit functions. The square roots of the stabiliser ε-functions (7) present as factors in defining relation of the dual-weight hopping operators A σ p ∨ ,M ( , ∨ ) matrix elements in the position basis (52) and manifest a direct consequence of the weighted discrete orthogonality relations (39). The ε-function subsequently straightforwardly regulates the probabilities (74) of finding the particle in a stationary state on the Neumann walls of the simplex lF. The Neumann boundary effect, which is observed similarly in dual-root models [1], is pointedly evident in Figures 3 and 6. • Considering an electron as the quantum particle propagating in the current discrete quantum systems, a novel class of the tight-binding models [3] with the electron propagating among atoms positioned at the points of the dual-weight dot D σ P ∨ ,l,M ( , ∨ ) is obtained. The hopping integrals [15] between the coupled neighbouring positions in the atomic lattice, which might be estimated from theoretical considerations and/or fine-tuned experimentally, directly enter the present models as the values of the dualweight hopping function P ∨ . Analogously to the dual-root models, the physical interpretation of the dual-weight models coincides with the inductively developed electron propagation in a crystal lattice [41]. Since the dual-weight Fourier-Weyl transforms of the current one-dimensional A 1 model of a linear crystal specialise to the four types I-IV of the discrete cosine and sine transforms [18,25], the current stationary state vectors represent boundary-dependent (anti)symmetric alternatives to the periodic exponential solutions [41]. Moreover, the discrete Hamiltonian approach used for dual-weight and dual-root models produces strictly defined boundary-dependent forms of the energy spectra (59). • Similarly to the dual-root models, the dual-weight models employ the generalised dual-weight Fourier-Weyl transforms (41) to construct the momentum basis (56) together with the stationary states (70) and time-evolutions (71). The utilisation of the weight lattice transforms [23] as well as the dual-weight E-transforms [42] for the description of analogous discrete quantum systems deserves further study. Potentially resulting in the generalisation of the present models to the prominent honeycombtype (pseudo)lattices [17,32], the intricate composition of the weight and root lattice transforms demands a specific construction of extension coefficients of the extended Weyl orbit functions [43]. The calculation of the extension coefficients is determined by the desired form of product-to-sum decomposition formulas (33), which characterise the coupling of the considered (pseudo)lattice model. Since the extended Weyl orbit function approach potentially represents alternative description to the (pseudo)spinor wavefunctions approach [17], the Fourier-Weyl transforms induced by the extended Weyl orbit functions, together with the discrete symmetry analysis of the associated quantum systems, deserve further study.