Quantum Particle on Dual Weight Lattice in Even Weyl Alcove

Even subgroups of affine Weyl groups corresponding to irreducible crystallographic root systems characterize families of single-particle quantum systems. Induced by primary and secondary sign homomorphisms of the Weyl groups, free propagations of the quantum particle on the refined dual weight lattices inside the rescaled even Weyl alcoves are determined by Hamiltonians of tight-binding types. Described by even hopping functions, amplitudes of the particle’s jumps to the lattice neighbours are together with diverse boundary conditions incorporated through even hopping operators into the resulting even dual-weight Hamiltonians. Expressing the eigenenergies via weighted sums of the even Weyl orbit functions, the associated time-independent Schrödinger equations are exactly solved by applying the discrete even Fourier–Weyl transforms. Matrices of the even Hamiltonians together with specifications of the complementary boundary conditions are detailed for the C2 and G2 even dual-weight models.


Introduction
The goal of this article is to formulate and exactly solve boundary-dependent quantum models of particle propagation on finite fragments of lattices that are interconnected with affine even Weyl groups of irreducible crystallographic root systems [2,29]. Induced by the constructed Hamiltonians of tight-binding type [32,45], the time-independent Schrödinger equations are resolved through the application of discrete even Fourier-Weyl transforms [17,18].
Representing a fundamental tool in solid state physics [39,43], discrete models of a quantum particle propagation are intensely applied to study mesoscopic structures of 2D and 3D materials [13,40], quantum dots [1,16,34], quantum billiards [14,30] and nanotubes [8,9]. Tight-binding models of electron propagation in lattice-structured materials, with the electron's hoppings to neighbouring positions of fixed degrees, provide explicit approximate forms of the wave functions and energy spectra [7,9,16,26,41]. Periodic boundary conditions serve as ubiquitously utilized assumptions which provide finite labellings of the electronic stationary states [39,43]. From a viewpoint of crystallographic root systems and associated affine Weyl groups, combining presupposed rescaled dual root lattice periodicity together with Weyl group (anti)symmetry produces descriptions of a quantum particle propagating on rescaled Weyl group invariant lattices inside rescaled closures of the Weyl alcoves and subjected to unique mixtures of Dirichlet and Neumann boundary conditions [4,5,19]. Replacing Weyl group symmetries with their even subgroup versions [18,29], boundary conditions of the currently considered quantum particle propagations on dual weight lattices constitute transitions between the dual-root periodic boundary conditions and Dirichlet/Neumann conditions. The even dual-weight Fourier-Weyl transforms of crystallographic root systems corresponding to the affine even Weyl groups [17,18] form the mathematical apparatus that provides an efficient description of the presented even dual-weight quantum dots.
Labelled by admissibly shifted weight lattices [10], the non-subgroup types of the Weyl orbit functions [27,28] admit discrete Fourier calculus on refined fragments of the shifted dual root, dual weight and weight lattices lying in closures of the Weyl alcoves [10,11,23]. Posing a greater challenge, the discrete transforms of the remaining non-trivial subgroup types have been developed with the point sets formed by subsets of the refined dual weight lattices only [17,18]. Distinguishing between the irreducible crystallographic root systems with one or two lengths of roots, there exist one/three types of affine even Weyl groups induced, as normal subgroups of the affine Weyl groups, by the underlying nontrivial sign homomorphisms [18]. Even Weyl alcoves, formed by the closures of the original Weyl alcoves along with their suitable reflections with respect to boundary walls and boundary reductions, form the fundamental domains of the affine even Weyl groups [18]. Pairs of sign homomorphisms determine in total three/ten types of generalized complex-valued Weyl orbit functions [17,18] which serve as the kernels of the discrete transforms as well as (renormalized) wave functions of the quantum particle. The primary sign homomorphisms determine the types of the affine even subgroups, while the secondary sign homomorphisms establish symmetries and antisymmetries of the even Weyl orbit functions. Since the affine Weyl groups are generated by reflections with respect to mirrors forming walls of the Weyl alcoves [24], the (anti)symmetries of the induced two/four non-subgroup Weyl orbit functions [10,11] subsequently enforce the Dirichlet and Neumann conditions on the alcove boundaries. On the other hand, inherent symmetries of the non-trivial subgroup Weyl orbit functions [18] produce diverse properties on the boundaries of the even Weyl alcoves which range from Dirichlet/Neumann to generalized periodic behaviours along with their uncommon combinations. The amplitudes of the particle's propagations to neighbouring positions are included in the Hermitian even hopping functions that are required to obey the symmetry of the even Weyl groups. Each non-zero value of the even dual-weight hopping function inside the fundamental domain of the even Weyl group induces the associated even hopping operator which approximates the particle's jumps to lattice neighbours of the corresponding degree as well as incorporates the boundary conditions into the resulting Hamiltonian. While the amplitude reflections from the boundary walls within the non-subgroup models are characterized via the sign χ -functions and coupling sets [4,5,19], the even sign χ -functions and even coupling sets also consistently encompass the generalized periodic as well as mixed boundary conditions. The energy spectra of the ensuing Hamiltonians of the tight-binding type are explicitly expressed through sums of products of the even dualweight hopping functions and even Weyl orbit functions. Similarly to the non-subgroup discrete Fourier-Weyl transforms and their one-dimensional archetypes [37], the discrete dual-weight orthogonality relations of the even Weyl orbit functions are weighted and normalized by the even dual-weight and weight counting functions [17,18]. Analogously to the dual-root and dual-weight propagation models in the rescaled Weyl alcoves [4,5], it appears that besides abstract relevance of the counting functions, measurable consequences affecting the particle's behaviour on the Neumann type boundaries of the rescaled even Weyl alcoves are inferred.
Possessing symmetries induced by the (affine) even Weyl groups, the even dual-weight quantum propagation models represent a novel class of exactly solvable discrete quantum systems of unique shapes and boundary properties. Most remarkably, simultaneous realizations of generalized Dirichlet, Neumann as well as (anti)periodic boundary conditions for the quantum particle propagation in the confining rescaled even Weyl alcoves emerge. Substantially extended by using summations of the even χ -functions over the even coupling sets, the comprehensive method for construction of the dual-weight boundary-dependent Hamiltonians appears to constitute a bridging approach that is suitable also for cases with periodic characteristics. Since the non-subgroup version of the current method has been successfully applied to obtain exact wave functions and energy spectra for electron propagation in triangular armchair and zigzag graphene quantum dots [19], similar relevance of the presented techniques in research of modern materials' electronic properties can be expected. Applied solely to the graphene quantum dots described in context of A 2 and G 2 root systems, the even subgroup method already conceivably adds up to 14 subgroup cases to 12 available non-subgroup shapes and boundary conditions. Additionally, the developed discrete even dual-weight quantum models potentially serve as a foundation for description of discrete space-time models [44], spin electronics [42], electronic transport [12], ultracold atoms in optical lattices [3], biexcitons [38] and quantum gates [33].
The paper is arranged as follows. In Section 2, pertinent properties of the root systems, Weyl group invariant lattices, sign homomorphisms and even Weyl groups, together with their fundamental domains, are specified. Section 3 establishes the crucial even counting functions as well as the even χ -functions. The dual-weight discretizations of the even Weyl orbit functions along with the exposition of the admissible even dual-weight hopping functions are included in Section 4. In Section 5, the Hamiltonians of the even dual-weight quantum models in conjunction with exact solutions of the corresponding time-independent Schrödinger equations are presented. Representing examples in the Euclidean plane, illustrating cases of the even dual-weight models associated with the root systems C 2 and G 2 are demonstrated in Section 6. Concluding remarks and further prospects are contained in Section 7.

Invariant Lattices
The mathematical exposition and notation of this article are based on papers [17,18,22,29]. The irreducible crystallographic root systems ⊂ R n of the four series A n (n ≥ 1), B n (n ≥ 3), C n (n ≥ 2) and D n (n ≥ 4) along with the five exceptional cases E 6 , E 7 , E 8 , F 4 and G 2 contain their associated subsets ⊂ of the simple roots [2]. Each set of the simple roots = {α 1 , . . . , α n }, ordered according to the notation in [22], forms a basis of the Euclidean space R n which is assumed to be equipped with the standard scalar product ·, · . The sets of indices I n and I n are introduced as . . , n}, In the case of the root systems with two different root lengths B n , C n , F 4 and G 2 , the set of the simple roots decomposes into a disjoint union which contains the set of the short simple roots s and the set of the long simple roots l , Constructed from the relations ω ∨ i , α j = δ ij , i, j ∈ I n , the dual fundamental weights ω ∨ 1 , . . . , ω ∨ n ⊂ R n provide the dual basis for the basis of the simple roots . An additional basis of the dual simple roots ∨ , induced by the basis , comprises the dual simple roots α ∨ 1 , . . . , α ∨ n ⊂ R n that are for each i ∈ I n expressed as the following rescaled simple roots, The basis ∨ forms a set of the simple roots of the dual root system ∨ . The dual basis of ∨ consists of the fundamental weights ω 1 , . . . , ω n ⊂ R n that are determined by the relations ω i , α ∨ j = δ ij , i, j ∈ I n . To each simple root α i ∈ , i ∈ I n corresponds a reflection r i ∈ O(R n ) satisfying r 2 i = 1 and determined for any a ∈ R n by the standard formula, Associated to the set of the simple roots , the set of reflections The root system contains a unique highest root ξ ∈ given in the basis as where the marks m i ∈ N are listed in [22,Tab. 1]. The marks induce the Coxeter numbers m = 1 + m 1 + · · · + m n that are summarized in [22,Tab. 2]. The dual root system includes the highest dual root η ∈ ∨ , given in the basis ∨ as where the dual marks m ∨ 1 , . . . , m ∨ n ∈ N are listed in [22,Tab. 1].
The dual weight lattice P ∨ is Z-dual to the root lattice Q and contains the entire dual root lattice, Q ∨ ⊂ P ∨ . Similarly, the weight lattice P is Z-dual to the dual root lattice Q ∨ and Q ⊂ P . Moreover, the orders of the quotient groups P /Q and P ∨ /Q ∨ coincide with the determinant c of the Cartan matrix C ij = α i , α ∨ j , i, j ∈ I n , c = det C = |P /Q| = P ∨ /Q ∨ .

Even Weyl Groups
The group of the sign homomorphisms comprises homomorphisms σ : W → U 2 from the Weyl group W to the multiplicative group U 2 = {−1, 1} [18,21]. For any irreducible crystallographic root system, the identity 1 and determinant σ e sign homomorphisms are specified by their values on the generating reflections r i , i ∈ I n as 1(r i ) = −σ e (r i ) = 1.
For the root systems with two different root-lengths which admit the decomposition of the simple roots (1), additional short σ s and long σ l sign homomorphisms are defined on the generating reflections r i as The values of the sign homomorphisms on the opposite involutions w 0 ∈ W , listed in Table 1, are calculated from formulas (3), (4) and (5). Table 1 The values of the sign homomorphisms σ e , σ s and σ l ∈ assigned to the opposite involutions Each sign homomorphism σ ∈ determines the even Weyl group W σ ⊂ W by the relation and any point a ∈ R n induces its associated stabilizer subgroup Stab W σ (a) ⊂ W σ . For the identity sign homomorphism σ = 1, the even Weyl group W 1 coincides with the Weyl group, W 1 = W . The closure of the fundamental Weyl chamber D ⊂ R n , described explicitly by forms a fundamental domain of the Weyl group W . The signed fundamental Weyl chamber D σ ⊂ D is induced by a sign homomorphism σ ∈ as The set of negative Weyl group generators S σ ⊂ S is introduced by the relation Because the stabilizer Stab W (a) ⊂ W is, for any a = a 1 ω ∨ 1 + · · · + a n ω ∨ n , generated by reflections r i , i ∈ I n for which a i = 0, the set D σ consists of all the points a ∈ D of the form a = a σ 1 ω ∨ 1 + · · · + a σ n ω ∨ n with coordinates a σ i restricted as follows, The fundamental domains of the even Weyl groups W σ are constructed in the following proposition.
Theorem 2.1 For any r σ ∈ S σ , the set D ∪ r σ D σ forms a fundamental domain of the even Weyl group W σ . Equivalently, the following two requirements are satisfied.
i) For any a ∈ R n , there exist a ∈ D ∪ r σ D σ and w ∈ W σ such that ii) If a, a ∈ D ∪ r σ D σ and a = wa for some w ∈ W σ , then a = a = wa.
Proof Since the fundamental Weyl chamber D represents the fundamental domain of the Weyl group W , for any a ∈ R n there exist a ∈ D and w ∈ W such that a = w a. If σ ( w) = 1, then w = w ∈ W σ and a = a. For σ ( w) = −1 and a ∈ D σ , it holds that w = wr σ ∈ W σ and a = r σ a ∈ r σ D σ . In the case of σ ( w) = −1 and a ∈ D \ D σ , there exists r ∈ S σ such that r a = a and σ ( wr) = 1. Therefore, it holds that w = wr ∈ W σ and a = a ∈ D \ D σ . If a, a ∈ D, then requirement ii) is satisfied because D is the fundamental Weyl chamber. For a, a ∈ r σ D σ , there exist b, b ∈ D σ such that a = r σ b and a = r σ b . The assumption a = wa guarantees that b = r σ wr σ b. Therefore, the assumption a = wa implies b = b from which it also holds that a = a. If a ∈ D and a ∈ r σ D σ , then it follows from the equalities a = wa = wr σ b with b ∈ D σ that a = b and wr σ ∈ Stab W (b). This forces the equality σ (Stab W (b)) = U 2 as well as subsequently contradicts definition (8) and thus, this case does not occur.
The cone of the positive dual weights P ∨+ and its signed subset P ∨σ ⊂ P ∨+ comprise the points from the dual weight lattice P ∨ belonging to the sets D and D σ , respectively, Since the fundamental domain of the even Weyl group D ∪ r σ D σ contains precisely one point from each W σ -orbit, the action of the even Weyl group W σ on the union of the cones P ∨+ ∪ r σ P ∨+ σ produces the entire dual weight lattice, The action of the opposite involution on the signed cone of the positive dual weights is determined in the following proposition.

Proposition 2.2 For any
Proof The opposite involution evaluations in the ω ∨ -basis (4) and (5) directly provide for p ∨ ∈ P ∨+ the following relation, Moreover, since for any stabilizing element w ∈ Stab W (−w 0 p ∨ σ ) it holds that w 0 ww 0 ∈ Stab W (p ∨ σ ), the sign homomorphism value σ (w) is directly determined from defining relations (8) and (12),

χ σ ,σ -and ε σ -functions
Determined by a sign homomorphism σ ∈ , the affine even Weyl group W aff σ is a semidirect product of the translation group Q ∨ and the even Weyl group W σ , For the identity sign homomorphism σ = 1, the affine even Weyl group coincides with the affine Weyl group W aff , W aff 1 = W aff = Q ∨ W . The retraction homomorphism ψ : W aff → W is determined for any element z = T (q ∨ )w ∈ W aff by the following relation, For any translation by a vector q ∨ ∈ Q ∨ and any w ∈ W σ , the affine even Weyl group and any point a ∈ R n induces associated stabilizer subgroup Stab W aff σ (a) of the group W aff σ . The closure of the fundamental Weyl alcove F ⊂ R n , described explicitly as the following simplex F = a 1 ω ∨ 1 + · · · + a n ω ∨ n a 0 + m 1 a 1 + · · · + m n a n = 1, forms a fundamental domain of the affine Weyl group W aff . The signed Weyl alcove F σ ⊂ F is, for each sign homomorphism σ ∈ , given by The even Weyl alcove F 1,σ ⊂ R n corresponding to the affine even Weyl group W aff σ ⊂ W aff , which contains exactly one point from each W aff σ -orbit, is for any fixed negative generator r σ ∈ S σ given as the following union [18, Prop. 3.1.1], Because F 1,σ is a fundamental domain of W aff σ , for any a ∈ R n there exist exactly one point a ∈ F 1,σ and some Considering the sign homomorphism σ ∈ inherent in the affine even Weyl group W aff σ as primary and bringing in a secondary sign homomorphism σ ∈ , the function χ σ ,σ : R n → {−1, 0, 1} is given for any a ∈ R n through relation (20) as Any two elements Since relation (20) also ensures that the stabilizers Stab W aff σ (a) and Stab W aff relation (22) guarantees that the even χ σ ,σ -function is well-defined.
To any pair of sign homomorphisms σ, σ ∈ corresponds the signed even Weyl alcove F σ ,σ ⊂ F 1,σ expressed as Moreover, the subset H σ ,σ of the even Weyl alcove F 1,σ , determined by produces the following disjoint decomposition of F 1,σ , Determined by the primary sign homomorphism σ ∈ , the counting ε σ -and h σ -functions, ε σ , h σ : R n → N, are given for any a ∈ R n by the relations Since it holds from [18, Proposition 3.  (19) is for any secondary sign homomorphism σ ∈ disjointly decomposed into its four subsets as follows, Since any points a ∈ R n with non-trivial stabilizers Stab W aff σ (a) do not lie in the interior of the even Weyl alcove F 1,σ , the sets H σ ,σ form subsets of the boundary sets B σ ,σ d , The four sets B σ ,σ d , B σ ,σ n , B σ ,σ p+ and B σ ,σ p− subsequently represent types of boundaries that affect propagation of a quantum particle.

h ∨σ M -functions
For any primary sign homomorphism σ ∈ , the dual affine even Weyl group is a semidirect product of the translation group Q and the even Weyl group W σ , For the identity sign homomorphism σ = 1, the dual affine even Weyl group coincides with the dual affine Weyl group W aff , For any translation by a vector q ∈ Q and any w ∈ W σ , the dual affine even Weyl group element y = T (q)w ∈ W aff σ acts canonically on R n as y · b = wb + q, b ∈ R n and any point b ∈ R n induces associated stabilizer subgroup Stab W aff The closure of the dual Weyl alcove F ∨ ⊂ R n , related with the dual affine Weyl group W aff , is a simplex explicitly described by The signed dual Weyl alcove F ∨σ is, for each primary sign homomorphism σ ∈ , given by The dual even Weyl alcove F ∨1,σ ⊂ R n of the dual affine even Weyl group W aff σ , which contains exactly one point from each W aff σ -orbit, is for any fixed negative generator r σ ∈ S σ given as the following union [18,Prop. 3 To any secondary sign homomorphism σ ∈ corresponds the signed dual even Weyl alcove The counting function h ∨σ M : R n → N is for any magnifying factor M ∈ N given by The algorithm for the calculation of the counting function h ∨1 M on the dual fundamental domain F ∨ is described in [22, §3.7] and, together with the W aff -invariance of the function

Discretized Even Weyl Orbit Functions
The kernels of the developed discrete transforms are formed by the most general form of the Weyl orbit functions σ ,σ b : R n → C [17,18,[27][28][29]35] that is induced by primary and secondary sign homomorphisms σ, σ ∈ and given for an argument a ∈ R n and a label b ∈ R n via the expression For both primary and secondary homomorphisms equal to the identity sign homomorphism, σ = σ = 1, the general form (38) yields the symmetric Weyl orbit functions [27] and, for σ = σ e and σ = 1, it specializes to the antisymmetric Weyl orbit functions [28], The choices σ = σ s , σ = 1 and σ = σ l , σ = 1 lead to the ϕ σ s -and ϕ σ l -functions [35] respectively, σ s ,1 = ϕ σ s , σ l ,1 = ϕ σ l .
Int J Theor Phys (2023) 62:65 The following combinations of the primary and secondary sign homomorphisms σ and σ produce the E-functions, e+ = 1,σ e = σ e ,σ e , s+ = 1,σ s = σ s ,σ s , (39) l+ = 1,σ l = σ l ,σ l and the remaining choices yield the mixed E-functions [17,18,29], Let the symbol * denote the complex conjugation. The duality, hermiticity and scaling symmetry represent essential properties that are common to all types of the even Weyl orbit functions (38) and determined by any pair of sign homomorphisms σ, σ ∈ as Product-to-sum decomposition formulas of specific products of the even Weyl orbit functions serve as foundations for solutions of the developed discrete quantum models. Product-to-sum identities of any functions σ ,σ b and the functions 1,σ b corresponding to the common primary homomorphism σ ∈ together with a label b ∈ R n and evaluated at any points a, a ∈ R n take the following form, For discretized labels from the weight lattice λ ∈ P , argument symmetries of the Weyl orbit functions σ ,σ λ with respect to the action (18) of the affine even Weyl group W aff σ appear. Governed by the secondary sign homomorphism σ ∈ , these argument symmetries are for any a ∈ R n and z ∈ W aff σ expressed as Argument symmetry identities (45) The argument symmetry and vanishing properties (45) and (46) are interlaced for any a ∈ F 1,σ and a ∈ W aff σ a via the even χ σ ,σ -function as The even dual-weight Fourier-Weyl transforms are constructed over the finite point sets consisting of rescaled dual weights together with the label sets of weights [18]. The notation of the points and labels from [18] is adjusted to emulate the notation introduced in [11]. For any pair of sign homomorphisms σ , σ ∈ , the finite even sets of points F σ ,σ P ∨ ,M comprise for any magnifying factor M ∈ N the rescaled dual weights belonging to the signed even Weyl alcove (24), The corresponding finite even sets of labels σ ,σ Q,M of Weyl orbit functions contain the weights inside the magnified signed dual even Weyl alcove (35), Crucially, numbers of elements of the even point and label sets match [18,Thm. 5 Discrete orthogonality of the even Weyl orbit functions over the even point set F σ ,σ P ∨ ,M is, for any labels from the even label sets λ, λ ∈ σ ,σ Q,M , formulated through the counting ε σand h ∨σ M -functions (27) and (36) Taking into account the associated Plancherel formulas [18], the following complementary discrete orthogonality relations are induced for any points a, a ∈ F σ ,σ P ∨ ,M , For any fixed ordering of the even label and point sets σ ,σ Q,M and F σ ,σ P ∨ ,M , the entries of the transform matrices I σ ,σ P ∨ ,M of the even dual-weight Fourier-Weyl transforms are, for any label λ ∈ σ ,σ Q,M and point a ∈ F σ ,σ P ∨ ,M , deduced from the discrete orthogonality relations (51) as Coinciding cardinalities (50) along with discrete orthogonality relations (51) guarantee that the transform matrices I σ ,σ P ∨ ,M are unitary.

Even Hopping Functions
Associated with any primary sign homomorphism σ ∈ , the even dual-weight hopping function of a given even dual-weight model is a fixed complex-valued function P ∨ σ : P ∨ → C that is determined on the dual weight lattice and assumed to have a finite support, Additionally, the admissible even dual-weight hopping function P ∨ σ is required to satisfy W σ -invariance and Hermiticity condition described, for any p ∨ ∈ P ∨ and w ∈ W σ , by The properties (56) and (57) imply the following elementary symmetries of the finite support of P ∨ σ , The dominant support supp + (P ∨ σ ) and signed dominant support supp + σ (P ∨ σ ) of the even dual-weight hopping function P ∨ σ comprise the dual weight lattice elements from the support supp(P ∨ σ ) that belong to the cone of the positive dual weights P ∨+ and to its signed subset P ∨+ σ from relations (11) and (12), respectively, Utilizing W σ -invariance condition (56) together with Theorem 2.1 yields that the resulting even dominant support of the form contains precisely one representative element of each W σ -orbit within the entire support supp(P ∨ σ ). Therefore, the action of the even Weyl group W σ on the points in supp 1,σ (P ∨ σ ) produces from relation (13) the following identity, As a result, W σ -invariance condition (56) and support conditions (54) and (55) guarantee that the even hopping function P ∨ σ suffices to determine for a finite number of dual weights from the even dominant support supp 1,σ (P ∨ σ ). The following theorem interlaces W σ -invariance (56) and Hermiticity condition (57) via the action of the opposite involution w 0 ∈ W .
Application of Theorem 4.1 to the dual weights from the even dominant support (60) expresses admissibility conditions for each irreducible crystallographic root system. Utilizing explicit defining relations (7) and (10), the dual weights p ∨ ∈ supp + (P ∨ σ ) and p ∨ σ ∈ supp + σ (P ∨ σ ) are determined by the corresponding integer-valued coordinates in the ω ∨ -basis as p ∨ = (a 1 , . . . , a n ), a 1 , . . . , a n ∈ Z ≥0 , Because relations (15) and (14) imply that −w 0 p ∨ ∈ supp + (P ∨ σ ) and −r σ w 0 p ∨ σ ∈ r σ supp + σ (P ∨ σ ), the conditions in Theorem 4.1 force additional requirements on the values of the even hopping functions P ∨ σ on the representative elements of the W σ -orbits (66) and (67). According to the value of a given primary sign homomorphism on the opposite involution σ (w 0 ) ∈ U 2 , the following two cases are distinguished among irreducible crystallographic root systems.

Even Dual-Weight Hamiltonians
Given any irreducible crystallographic root system ⊂ R n , propagations of a nonrelativistic quantum particle on the rescaled dual weight lattices P ∨ l,M , scaled by any length factor l ∈ R together with the magnifying factor M ∈ N as form foundations of the constructed classes of quantum models. Assuming that a fixed admissible even dual-weight hopping function P ∨ σ corresponding to a primary sign homomorphism σ ∈ is supplied, the even dual-weight amplitude function I σ P ∨ ,l,M : P ∨ l,M × P ∨ l,M → C is determined for any x, x ∈ P ∨ l,M by the relation The current even dual-weight models of tight-binding type assume that the quantum particle propagates from the position on the rescaled dual weight lattice x ∈ P ∨ l,M to the position x ∈ P ∨ l,M with the amplitude I σ P ∨ ,l,M (x, x ) ∈ C per unit time. Characterizing points adjacent to a given point a ∈ 1 M P ∨ , the even dual-weight neighbourhood sets B σ According to W σ -invariance property of the even hopping function (56), the particle propagates with the same non-zero amplitude per time i P ∨ σ p ∨ from a point x = la to all neighbouring positions x ∈ P ∨ l,M such that x /l ∈ B σ p ∨ ,M (a). Incorporating a secondary sign homomorphism σ ∈ via the boundary sets (28)-(30), positions of the quantum particle are additionally confined to the domain lF σ ,σ by arranging boundaries of the Dirichlet, Neumann, periodic and antiperiodic types on the scaled boundary sets lB σ ,σ d , lB σ ,σ n , lB σ ,σ p+ and lB σ ,σ p− , respectively. The set of the quantum particle's resulting available positions is formed by the even dual-weight dot D σ ,σ P ∨ ,l,M , Any ordered even set of points F σ ,σ P ∨ ,M induces an orthonormal even dual-weight position basis |a; P ∨ , σ , σ , a ∈ F σ ,σ P ∨ ,M of the finite-dimensional complex Hilbert space H σ ,σ P ∨ ,M . The state determined by the vector from the even position basis |a; P ∨ , σ , σ ∈ H σ ,σ P ∨ ,M , a ∈ F σ ,σ P ∨ ,M represents the quantum particle present at the position in the even dual-weight dot la ∈ D σ ,σ P ∨ ,l,M . The counting formulas for the dimensions of the Hilbert spaces H σ ,σ are developed for σ = 1 in [18] and for σ = 1 in [21,22]. Each dual weight from the even dominant support p ∨ ∈ supp 1,σ (P ∨ σ ) induces for any two points a, a ∈ F σ ,σ P ∨ ,M the corresponding even dual-weight coupling set N σ p ∨ ,M (a, a ) ⊂ The even dual-weight hopping operator A σ ,σ p ∨ ,M : H σ ,σ P ∨ ,M → H σ ,σ P ∨ ,M is introduced via the even χ σ ,σ -function summing over the coupling sets N σ p ∨ ,M (a, a ) by prescribing its matrix elements in the even position basis as M (a,a ) χ σ ,σ (d) . (75) Any non-zero value of the even hopping function at the origin P ∨ σ (0) = −E 0 = 0 yields directly from definition (75) the following common diagonal form of the even dual-weight hopping operators A σ ,σ 0,M , Taking into account Hermiticity condition (57), E 0 stands for a real-valued parameter which represents the common on-site energy of the particle at an arbitrary position of the even dot D σ ,σ P ∨ ,l,M . The resulting even dual-weight Hamiltonian H σ ,σ P ∨ ,M : H σ ,σ P ∨ ,M → H σ ,σ P ∨ ,M of the quantum particle propagating on the even dual-weight dot D σ ,σ P ∨ ,l,M is postulated as the sum of all even dual-weight hopping operators (75),
Defined by transformation relation (78), the momentum vectors are demonstrated to satisfy the time-independent Schrödinger equation in the following theorem.
Theorem 5.1 Let σ ,σ Q,M , M ∈ N be the even label sets (49) corresponding to the primary and secondary sign homomorphisms σ, σ ∈ and |λ; P ∨ , σ , σ ∈ H σ ,σ P ∨ ,M , λ ∈ σ ,σ Q,M be the vectors of the even momentum basis provided by relations (78) and (79). Then for any admissible even dual-weight hopping function P ∨ σ and even dual-weight Hamiltonian (77) the vectors |λ; P ∨ , σ , σ ∈ H σ ,σ P ∨ ,M , λ ∈ σ ,σ Q,M satisfy the time-independent Schrödinger equation The eigenenergies E σ ,σ P ∨ ,λ,M ∈ R are determined by the following weighted summation of the even 1,σ -functions (38) over the even dominant support (60), Proof The time-independent Schrödinger equation (81) is expressed in the even position basis and comparison of the resulting coordinates produces for any a ∈ F σ ,σ P ∨ ,M the condition Utilizing W σ -invariance (56) of the even hopping function P ∨ σ together with the W σ -orbit decomposition (61) of its support supp(P ∨ σ ), the matrix elements of the even Hamiltonian H σ ,σ P ∨ ,M are for any a, a ∈ F σ ,σ P ∨ ,M calculated from each even hopping operator elements (75) and summation (77) as a; P ∨ , σ , σ H σ ,σ P ∨ ,M a ; P ∨ , σ , σ The scalar products a; P ∨ , σ , σ |λ; P ∨ , σ , σ ∈ C that are for λ ∈ σ ,σ Q,M and a ∈ F σ ,σ P ∨ ,M defined by relation (79), together with the reformulated matrix elements of the Hamiltonian (84), are substituted into the time-independent Schrödinger equation (83) and produce the following equivalent relation (85) Vanishing property of the orbit functions (46) and decomposition of the even Weyl alcove (26) grant extension of the summation in relation (85) to the points a ∈ 1 M P ∨ ∩ F 1,σ and argument symmetry relation (47) guarantees the following simplification, Because the point set 1 M P ∨ ∩ F 1,σ contains exactly one point from each W aff σ -orbit of the refined dual weight lattice 1 M P ∨ , the double summation in (86) actually represents the summation over the set Therefore, the reformulated time-independent Schrödinger equation (86) is additionally reduced as follows, Deconstructing the form of the eigenenergies (82), duality formula (41) and the scaling symmetry (43) initially produce the equality M , which is deduced from relations (45) and (56), allows retracting the summation over W σ -orbits in eigenenergy expression (88) and provides using the orbit-stabilizer theorem the following equivalent form, Consequently, the product-to-sum decomposition formulas of the orbit functions (44) produce the evaluation The W σ -invariance requirement (56) of the even dual-weight hopping function P ∨ σ along with the subsequent W σ -invariance (58) of the support supp(P ∨ σ ) further reinterpret relation (90) as Substituting simplified formula (91) into the reformulation of the time-independent Schrödinger equation (87) proves the starting statement (81).

Case C 2
Since the root system C 2 comprises roots of two different lengths, the set of simple roots decomposes to s = {α 1 } and l = {α 2 } and six different types of E-functions (39) and (40) occur [17]. Focusing on the short even Weyl group W σ s by fixing in defining relation (6) the primary sign homomorphism σ = σ s , the choices of the secondary sign homomorphism σ = 1 and σ = σ e represent two distinct types of point and label sets (48) and (49) as well as produce the corresponding s+ -and s− -functions from expressions (38). Since the opposite involution of C 2 is of the form w 0 = −1 and according to Table 1 it holds that w 0 ∈ W σ s , the admissible even dual-weight hopping functions P ∨ σ s are real-valued. Considering only the nearest and next-to-nearest neighbour coupling in the resulting two types of short dual-weight dots (73), the admissible even dual-weight hopping function P ∨ σ s is, according to determining relations (68), given by the on-site energy E 0 ∈ R and three non-zero parameters A, B, D ∈ R as The short energy functions E σ s 1 , E σ s 2 , E σ s 3 : F ∨1,σ s → R are defined for b ∈ F ∨1,σ s by the relations The plots of the short energy functions (97), evaluated on the short dual even Weyl alcove F ∨1,σ s , are shown in Fig. 1. Specializing eigenenergy relation (82), the eigenenergies E σ ,σ s P ∨ ,λ,M of the quantum particle propagating on the short dots D σ ,σ s P ∨ ,l,M are given for λ ∈ σ ,σ s Q,M as Fig. 1 The plots of the short energy functions E σ s 1 , E σ s 2 and E σ s 3 of C 2 . The light grey triangle, placed at the zero energy level, represents closure of the dual Weyl alcove F ∨ . The reflection of the signed dual Weyl alcove r 1 F ∨σ s is depicted as the dark grey triangle Utilizing the explicit description of the point and label sets from [17], the even point sets F 1,σ s P ∨ ,4 and F σ e ,σ s P ∨ ,4 are expressed in the ω ∨ -basis as F 1,σ s P ∨ ,4 = (0, 0), 0, 1 4 , 0, 1 2 , 0, 3 4 , (0, 1) , 1 4 The entire boundaries of the scaled signed even Weyl alcoves lF 1,σ s and lF σ e ,σ s , which surround the short dual-weight dots D 1,σ s P ∨ ,1,4 and D σ e ,σ s P ∨ ,1,4 , are formed by the scaled boundary sets lB 1,σ s n and lB σ e ,σ s d of the Neumann and Dirichlet types, respectively. The short dot D 1,σ s P ∨ ,1,4 is depicted in Fig. 2.   Fig. 2 The short dual-weight dot D 1,σ s P ∨ ,1,4 of C 2 . The grey square depicts the short even Weyl alcove F ∪ r 1 F σ s of the affine even Weyl group W aff σ s . The 13 numbered dark blue nodes reproduce the even point set F 1,σ s P ∨ ,4 and represent possible positions of the quantum particle. The nodes labelled by the matching number lie in the same W aff σ s -orbit. The blue lines connecting the nodes indicate the nearest neighbour coupling characterized by the even hopping operator A 1,σ s ω ∨ 2 ,4 . The red and green arrows highlight the next-to-nearest neighbour coupling of the third node that is described simultaneously by the even hopping operators A 1,σ s ω ∨ 1 ,4 and A 1,σ s Ordering the even position basis |a; P ∨ , 1, σ s , a ∈ F 1,σ s P ∨ ,4 as the point set (99), the matrix of the nearest neighbour even hopping operator A 1,σ s ω ∨ 2 ,4 is calculated from defining relation (75) as are given as Ordering the even position basis |a; P ∨ , σ e , σ s , a ∈ F σ e ,σ s P ∨ ,4 as the point set (100), the matrix of the nearest neighbour even hopping operator A σ e ,σ s ω ∨ 2 ,4 is calculated as The even Hamiltonians H 1,σ s P ∨ , 4 and H σ e ,σ s P ∨ ,4 of the quantum particle propagating on the even dual-weight dots D 1,σ s P ∨ ,l,4 and D σ e ,σ s P ∨ ,l,4 , respectively, are calculated as the sums of the even hopping operators (77),  The probabilities P 1,σ s P ∨ ,60 [λ] and P σ e ,σ s P ∨ ,60 [λ] of finding the particle at available positions on the short dots D 1,σ s P ∨ ,1,60 and D σ e ,σ s P ∨ ,1,60 are plotted in Fig. 3.

Case G 2
Because the root system G 2 contains roots of two different lengths, the set of simple roots decomposes to s = {α 2 } and l = {α 1 } and six distinct types of E-functions (39) and (40) exist [17]. Considering only the long even Weyl group W σ l by substituting the primary sign homomorphism σ = σ l in defining relation (6), the choices of the secondary sign homomorphism σ = 1 and σ = σ e produce two different types of even point and label sets (48) and (49) as well as yield the corresponding l+ -and l− -functions from expressions (38). The opposite involution of G 2 is of the form w 0 = −1 and according to Table 1 it holds that w 0 / ∈ W σ s . Considering only the nearest and next-to-nearest neighbour coupling in the resulting two types of long dual-weight dots (73), the admissible even dual-weight hopping function P ∨ σ l is according to determining relations (69) given by the on-site energy level E 0 ∈ R and two non-zero parameters D ∈ R and I = A + iB ∈ C with A, B ∈ R as The long energy functions E σ l 1 , E σ l 2 : F ∨1,σ l → R are defined for b ∈ F ∨1,σ l by the relations The plots of the long energy functions E σ l 1 , E σ l 2 , evaluated on the long dual even Weyl alcove F ∨1,σ l , are for B = 0 shown in Fig. 4.
Resolving eigenenergy relation (82), the eigenenergies E σ ,σ l P ∨ ,λ,M of the quantum particle propagating on the long dots D σ ,σ l P ∨ ,l,M are given for λ ∈ σ ,σ l Q,M as The open line segments with endpoints 1 2 lω ∨ 1 , 1 3 lω ∨ 2 and 1 3 lω ∨ 2 , 1 2 lr 1 ω ∨ 1 form the boundaries of periodic types lB 1,σ l p+ and lB σ e ,σ l p+ partly surrounding the long dual-weight dots D 1,σ l P ∨ ,1,7 and D σ e ,σ l P ∨ ,1,7 , respectively. Except for the vertex 1 3 lω ∨ 2 which represents the Neumann type boundary for both cases, the remaining boundaries of the domains lF 1,σ s and lF σ e ,σ s are formed by the boundary sets lB 1,σ l n and lB σ e ,σ l d of the Neumann and Dirichlet types, respectively. The long dot D 1,σ l P ∨ ,1,7 is shown in Fig. 5.

Fig. 5
The long dual-weight dot D 1,σ l P ∨ ,1,7 of G 2 . The grey kite depicts the long even Weyl alcove F ∪ r 1 F σ l of the affine even Weyl group W aff σ l . The 12 numbered dark blue nodes reproduce the even point set F 1,σ l P ∨ ,7 and represent possible positions of the quantum particle. The nodes labelled by the matching number lie in the same W aff σ l -orbit. The blue and red arrows illustrate the nearest neighbour coupling characterized by the even hopping operators A 1,σ l ω ∨ 1 ,7 and A 1,σ l r 1 ω ∨ 1 ,7 . The green arrows highlight next-to-nearest neighbour coupling of the second node that is described in general by the even hopping operator A 1,σ l ω ∨ 2 ,7 Ordering the even position basis |a; P ∨ , 1, σ l , a ∈ F 1,σ l P ∨ ,7 as the point set (107), the matrices of the nearest neighbour even hopping operators A 1,σ l ω ∨ 1 ,7 and A 1,σ l r 1 ω ∨ 1 ,7 are calculated from defining relation (75) as and the matrix of the next-to-nearest neighbour even hopping operator A 1,σ l ω ∨ 2 ,7 is given as Ordering the even position basis |a; P ∨ , σ e , σ l , a ∈ F σ e ,σ l P ∨ ,7 as the point set (108), the matrices of the nearest neighbour even hopping operators A σ e ,σ l ω ∨ The even Hamiltonians H 1,σ l P ∨ ,7 and H σ e ,σ l P ∨ ,7 of the quantum particle propagating on the even dual-weight dots D 1,σ l P ∨ ,l,7 and D σ e ,σ l P ∨ ,l,7 , respectively, are calculated as the sums of the even hopping operators (77), The set of rounded eigenenergies E 1,σ l P ∨ ,λ,7 , λ ∈ 1,σ l Q,7 , ordered as the labels from the list (109), is determined from relation (82)  The probabilities P 1,σ l P ∨ ,78 [λ] and P σ e ,σ l P ∨ ,78 [λ] of finding the particle at available positions on the long dots D 1,σ l P ∨ ,1,78 and D σ e ,σ l P ∨ ,1,78 are plotted in Fig. 6. Fig. 6 The probability plots for the long dual-weight dots D 1,σ l P ∨ ,1,78 and D σ e ,σ l P ∨ ,1,78 of G 2 . The small dots display the probabilities (95) of finding the particle in the momentum states |λ; P ∨ , 1, σ l and |λ; P ∨ , σ e , σ l , labelled by λ = (0, 1), (1, 1) and (1, 2), over their respective positions from D 1,σ l P ∨ ,1,78 (upper row) and D σ e ,σ l P ∨ ,1,78 (bottom row). Affected by the non-constant values of the counting ε σ l -function (27), the red dots illustrate the calculated probabilities over the available positions on the Neumann type boundaries B 1,σ l n and B σ e ,σ l n of the even Weyl alcove F 1,σ l

Concluding Remarks
• Induced by the primary sign homomorphism σ ∈ of the initial irreducible crystallographic root system ⊂ R n , structural symmetries of the presented particle propagation quantum models are governed by the finite even Weyl groups (6) together with their infinite affine extensions (16). The amplitudes of propagation on the rescaled dual weight lattice (70) are embedded within the predetermined admissible even hopping functions P ∨ σ together with the resulting amplitude functions (71). As immediate consequences of the even Weyl group invariance admissibility condition (56), the even hopping functions are according to Theorem 2.1 restricted to their even dominant support (60) and the particle propagates with identical amplitudes per time to the adjacent positions described by the even neighbourhood sets (72). The Hermiticity conditions (57) are interlaced with the even Weyl group invariance to produce via the opposite involutions directly applicable admissibility conditions reformulations stemming from Theorem 4.1. The even χ σ ,σ -functions (21), which act as generalized sign functions on the underlying Euclidean space by means of the secondary sign homomorphism σ ∈ , characterize the types of boundaries of the even Weyl alcoves (28)- (30) as well as the (anti)symmetries of the even Weyl orbit functions (47).
• The finite sets of the particle's available positions (73) are formed by fragments of the rescaled dual weight lattices (70) inside the rescaled even Weyl alcoves (19). The weight sets (49) label the orthogonal even Weyl orbit functions as well as the even momentum bases (78) which are given through the inverse matrices of the even Fourier-Weyl transforms (79). Assigned to each value from the even dominant supports of the even hopping functions (60), the ε σ -weighted matrix elements of the even hopping operators (75) are given in the even position bases via summing of the even χ σ ,σfunctions over the even coupling sets (74). The vectors of the even momentum bases satisfy according to Theorem 5.1 the time-independent Schrödinger equations (81), while the corresponding eigenenergies are given exactly through sums of the even Weyl orbit functions (82). Observing a generalization of the Neumann boundary effect from the non-subgroup models [4,5,19], the time-independent probabilities (95) of finding the particle in the momentum states (78) at given positions inside the dualweight dots (73) are on the Neumann type boundaries (29) affected by the ε σ -function factors (27). • Demonstrating application of the even subgroup method for the Euclidean plane root systems with two different root lengths, the short dots D 1,σ s P ∨ ,l,M and D σ e ,σ s P ∨ ,l,M of C 2 admit purely the Dirichlet and Neumann walls, respectively. The long dots D 1,σ l P ∨ ,l,M of G 2 contain Neumann and periodic boundary types, while the long dots D σ e ,σ l P ∨ ,l,M combine Dirichlet, Neumann as well as the periodic boundary types. The dominant even support values of the admissible even hopping functions describing the nearest and next-to-nearest neighbour coupling are given by relations (96) and (104). The sums of matrices comprising the associated even hopping operators expressed in the even position bases yield the matrix forms of the even dual-weight Hamiltonians (103) and (111). The C 2 and G 2 energy functions of the short and long dots (97) and (105), plotted over the dual even Weyl alcoves in Figs. 1 and 4, provide through their exact discretizations (98) and (106) the numerical evaluations of the eigenenergies (82). Manifesting the subgroup version of the Neumann boundary effect, decreased probabilities of finding the particle over positions inside the Neumann type boundaries are visualized by the red dots in Figs. 3 and 6.
• Since the even Weyl group W σ e associated with the root system A 1 equals the trivial identity subgroup, the resulting affine even Weyl group specializes to the group of the dual root lattice shifts, W aff σ e = Q ∨ . Therefore, the current (rescaled) model of the A 1 even dual-weight dots corresponds to the tight-binding description of particle propagation in the one-dimensional crystal with even numbers of atoms and periodic boundary conditions [15,39]. Moreover, replacing in the current method the even affine subgroups of the affine Weyl groups by the dual root lattice shift subgroups Q ∨ ⊂ W aff conceivably leads through a further modification of the presented formalism to description of the quantum particle on (dual) weight and root lattices subjected to dual root lattice periodic boundary conditions. Even though the discrete Fourier analysis necessary for the description of periodic models based on crystallographic root systems has already been partially achieved [31,36], a fully general mathematical exposition of the (dual) root and weight Fourier transforms together with a compatible Hamiltonian reformulation of the current models deserve further study.
• The honeycomb lattice, which serves as a foundation for study of graphene and graphene-like 2D materials [6,16], admits both additive and subtractive constructions through (shifted) root and weight lattices of the root system A 2 [19]. Exploiting two specific subtractive formations of the honeycomb lattice [20], even subgroup adaptation of the armchair and zigzag honeycomb Fourier-Weyl transforms crucially depends on exact formulation of the even (dual)-root Fourier-Weyl transforms. Thus, classification of the even admissible shifts of the (dual) root and weight lattices as well as the form of the subsequent generalized even Fourier-Weyl transforms and particle propagation models pose directly applicable open problems. Moreover, representing real-valued variants of the discrete Fourier-Weyl transforms, the dual-root, even dualweight as well as the honeycomb Hartley-Weyl transforms have been developed [11,18,20]. The current Hamiltonian formulations potentially produce through the even dual-weight Hartley-Weyl transforms their real-valued subgroup modifications which might be utilized for description of longitudinal vibrations of 1D and 2D lattices [25].
Acknowledgments The authors gratefully acknowledge support from the Czech Science Foundation (GAČR), Grant No. 19-19535S.
Funding Open access publishing supported by the National Technical Library in Prague.
Data Availability Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Conflict of Interests
The authors have no conflicts to disclose.
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