An algorithm to check stability

Due to the main results of the thesis, we can now give an algorithm which checks if (G, x0, V) is stable with respect tok · k_R, see Definition 4.8. The algorithm for the stability with respect tok · k_R,0,0 is analogous.

Algorithm 4.56. Given is a discrete groupG<E(d) and its associated groupsF,S and setT, see Definition 2.6, some pointx0∈R^d such that the mapG →R^d,g7→g·x0 is injective, and an interaction potentialV, see Definition 4.1. Since the algorithm is numeric and by (H3), we may assume thatV has finite support.

(i) Check if χGx0 is a critical point of the configurational energy E, e. g. by computing the derivative ∂gV(y0) for all g ∈ suppV, see Definition 4.1, the vector e_V, see Definition 4.11 and checking if e_V = 0, see Corollary 4.16.

(ii) Determine the derivative∂g∂hV(y₀) for all g, h∈suppV, see Defi-nition 4.1. Then compute the functionfV by computingfV(g) for all g ∈ ({id} ∪suppV)⁻¹({id} ∪suppV), see Definition 4.11 and Remark 4.12(ii).

(iii) Determine a setRwith Property 2, see Definition 3.5. Fix a bijec-tionϕ:R → {0, . . . ,|R| −1}. Thus the map

ψ:Uiso(R),→R^d|R|, u7→(u(ϕ⁻¹(0)), . . . , u(ϕ⁻¹(|R| −1)))^T, which maps a function to a column vector, is an embedding, where U_iso(R) is defined in Definition 3.1. By Proposition 3.28 and the Gram-Schmidt process, we can determine an orthonormal basis{b1, . . . , b_n}ofψ(U_iso(R)), wheren= dim(U_iso(R)). LetBbe the d|R|-by-nmatrix (b₁, . . . , b_n). The matrixI_d|R|−BB^Tis the orthogonal projection matrix with kernel ψ(U_iso(R)). Now we can determine the function g_R, i. e. the matrix g_R(g) for all g ∈ R, see Defini-tion 4.43 and Remark 4.44.

(iv) Determine a representation setRofT Fd/∼, e. g. with Lemma 2.36, where∼is the equivalence relation defined in Definition 2.32. For all ρ ∈ R determine the space group Gρ, see Definition 2.38 and Definition 4.53, with, e. g., Proposition 2.39, and determine a rep-resentation set (or a fundamental domain, see Remark 4.55(i))Kρ

ofR^d2/Gρ.

(v) Fix a complete set of representatives of the cosets of T F in G.

Thus the induced representation Ind(χkρ) is well-defined for all ρ∈Rand k∈Kρ, see Definition 2.29 and Definition 2.23. For all ρ ∈ R and k ∈ Kρ the matrices fV

V

(Ind(χkρ)) and^Vg_R(Ind(χkρ)) can be computed with Definition 2.59. For all ρ ∈R and all but finitely many k∈ Kρ, the matrix^Vg_R(Ind(χkρ)) has full rank and thus the real number λmin(fV

V

(Ind(χkρ)), gR

V

(Ind(χkρ)) can easily be computed, see Definition 4.49 and Remark 4.50(ii). Due to the upper semicontinuity, see Lemma 4.52, by Theorem 4.54 we can compute the extended real numberλ_a.

(vi) The triple (G, x₀, V) is stable with respect to k · k_R if and only if χ_Gx0 is a critical point of Eandλa>0, see Definition 4.8.

In the following two examples, we investigate the stability of a triple (Gi, xi, Vi) for alli∈I, where I is a suitable index set. The figures are generated with the programming language Python, seehttps://github.

com/Toymodel-Nanotube/for the source code.

Example 4.57. A suitable toy model for the investigation of stability is an atom chain.

Leta > 0 be the scale factor,t = ta = (I₂, ae₂)∈ E(2) and G =Ga = hti<E(2) analogously to Definition 2.63 and Definition 2.63.

We define the interaction potentialV =Va, see Definition 4.1 and Re-mark 4.2(iv), by

V_a(y) =v₁(ky(t_a)k) +v₂(ky(t²_a)k), where

v₁: (0,∞)→R, r7→r⁻¹²−r⁻⁶ is the Lennard-Jones potential and

v2: (0,∞)→R, r7→8r⁻⁶. Letx0= 02. By Lemma 4.6 for alla >0 we have

E(χ_Gx0) =V(y0) =a⁻¹²−7 8a⁻⁶,

where E = E_a is the configurational energy and y₀ = y_0,a = (g·x₀− x0)_g∈Ga. We define

a^∗:= arg min

a∈(0,∞)

E(χGx0) = ⁶ r16

7 ≈1.1477.

Thus the structure G ·x0 is stretched (resp. compressed) if a > a^∗ (resp. a < a^∗). Now we investigate its stability numerically with Al-gorithm 4.56.

(i) By Corollary 4.17 the functionχ_Gx0 is a critical point ofE for all a >0.

(ii) We have

∂_g∂_hV(y₀) =

















 6a⁻⁸

−2a⁻⁶+ 1 0 0 26a⁻⁶−7

ifg=h=t 2⁻⁴3a⁻⁸

−1 0

0 7

ifg=h=t²

02,2 else.

We have ({id} ∪suppV)⁻¹({id} ∪suppV) ={t⁻², . . . , t²} and

(iii) Since the set {id, t} has Property 1 and {t} generates G, the set R={id, t, t²}has Property 2. We define the functions U_iso(R) andU_iso,0,0(R), respectively. We define the bijectionϕ:R

→ {0,1,2}bytⁿ 7→nfor alln∈ {0,1,2}. Let ψbe the embedding Uiso(R),→R⁶, u7→(u(ϕ⁻¹(0)), . . . , u(ϕ⁻¹(2))).

A computation shows that the orthogonal projection matrices ofR⁶ with kernelsψ(U_iso(R)) andψ(U_iso,0,0(R)) are

respectively. Thus the functions g_R and g_R,0,0 of Definition 4.43 set of T F/∼d by Lemma 2.36(i). Recall Definition 2.28. We have S =h(I1, a)i, L_S =haiand L^∗_S =ha⁻¹i. By Proposition 2.39 we have {k ∈ R|(I₁, k) ∈ Gid} = ha⁻¹i and thus Gid = h(I1, a⁻¹)i.

The intervalK_id = [0, a⁻¹) is a representation set ofR/G_id. (v) For allk∈Kid we have Ind^G_{T F}χk =χk. We have

{k∈Kid|^Vg_R(χk) has full rank}=Kid\ {0}

and

{k∈K_id|^Vg_R,0,0(χ_k) has full rank}=K_id\ {0}.

For all k ∈ Kid \ {0} we can compute λmin(fV

V

(χk), g^V_R(χk)) and λ_min(f_V

V

(χ_k), g^V_R,0,0(χ_k)). In particular we can computeλ_a=λ_a(a) andλa,0,0=λa,0,0(a) numerically, see Figure 4.1.

(vi) In the compressed case a∈(0, a^∗) we haveλ_a =−∞andλ_a,0,0 ∈ (−∞,0) and thus (G, x₀, V) is not stable with respect to bothk · k_R and k · k_R,0,0. Now we investigate the stretched case, i. e. a > a^∗. Leta^∗∗=p⁶

26/7≈1.244455. For alla∈(a^∗, a^∗∗) we haveλa>0 and λa,0,0 > 0 and thus (G, x0, V) is stable with respect to both k · kR and k · kR,0,0. For alla > a^∗∗ we have E⁰⁰(χGx0)(u, u)<0, whereu=e2χ_{tⁿ_|n∈2Z}. In particular we haveλa<0 andλa,0,0<

0 and thus (G, x0, V) is not stable with respect to bothk · kR and k · kR,0,0 for alla > a^∗∗.

Notice that in the stretched casea∈(a^∗, a^∗∗), the appropriate seminorm for the stability isk · k_R,0,0. For the equilibrium casea≈a^∗, the weaker seminormk · k_Ris appropriate since lim_a→a^∗λa= 0 and lim_a&a^∗λa,0,0>

0.

Example 4.58. There exists a huge literature on the stability of (n, m) nanotubes as zigzag or armchair nanotubes, see, e. g., [30]. Each (n, m) nanotube is the orbit of some point inR³ under the action of a discrete subgroup of E(3). Thus its stability can be checked with Algorithm 4.56.

In this example we investigate the stability of a (5,1) nanotube, see Fig-ure 4.2.

For all scale factorsa >0 and angles α∈(0, π) we define: Let R(α)∈ O(2) be the rotation matrix as in (2.1), t = ta,α = (R(α)⊕I1, ae3) ∈ E(3), p= (I1⊕(−I2),03) ∈ E(3) and G = Ga,α be the discrete group ht, pi<E(3), i. e.G={t^mp^q|m∈Z, q∈ {0,1}}. For allx∈R³ we have G ·x⊂C_x, whereC_xis the cylinder{y∈R³|y²₁+y₂²=x²₁+x²₂}.

Let N = Na,α ={tp, t⁶p, t⁷p}. Let Ua,α ⊂R³ be the set of all points x∈R³ for which the map G →R³, g7→ g·xis injective and the three nearest neighbors ofxin G ·xare the pointsN ·x, i. e.

supn

kg·x−xk

g∈ No

<infn

kg·x−xk

g∈ G \(N ∪ {id})o . Let

W :=

(a, α, x)

a >0, α∈(0, π), x∈U_a,α .

0.0 0.2 0.4 0.6 0.8 0.2

0.4 0.6 0.8 1.0 1.2

1.12 1.14 1.16 1.18 1.20 1.22 1.24

−0.2 0.0 0.2 0.4 0.6 0.8

Figure 4.1.: For the toy model as described in Example 4.57, the graphs ofλmin(cfV(χk),gc_R(χk)) (blue) andλmin(cfV(χk),g\_R,0,0(χk)) (orange) dependent on k∈Kid\ {0} are plotted on the top plot for the choice a= 1.22. The points (a^∗,0) and (a^∗∗,0) and the graphs ofλa(blue) andλa,0,0(orange) dependent on the scale factor aare plotted on the bottom plot.

t⁰

Figure 4.2.: As described in Example 4.57, the orbit of the point xa₀

under the action of the groupGa₀,α₀ is a (5,1) nanotube. We have a natural bijection between the group elements and the atoms.

Analogously to [30] we define the interaction potential V = Va,α, see Definition 4.1 and Remark 4.2(iv), by

V(y) =1 2

X

g∈N

v₁(ky(g)k) +1 2

X

g,h∈N

v₂(y(g), y(h)), where

v1: (0,∞)→R, r7→(r−1)² is atwo-body potential and

v2:{(x, y)|x, y∈R³\ {0}} →R,(x, y)7→

hx, yi kxkkyk +1

2 ²

is athree-body potential. Thus the bonded points of G ·x tend to have distance 1 and thebond anglestend to form 2π/3 angles. By Lemma 4.6 for all (a, α, x) ∈ W we have E(χ_Gx) = V(y0), where E =Ea,α is the configurational energy andy0=y0,a,α,x= (g·x−x)_g∈Ga,α.

First we consider the (5,1) nanotube. We define α₀:= 11π/31≈1.115

and

x_a:=a(rcos(β), rsin(β),7/3)∈R³ for alla >0, wherer= 31/(π√

3) andβ= 5π/31. With the formulas in [22] it follows that for all (a, α, x) ∈ W the set G ·xis a so called (5,1) nanotube if and only ifα=α0 andx=xa. The bond length of the unrolled (5,1) nanotubeGa,α₀·xa, i. e. the distance of two neighbored points ofGa,α₀·xa

with respect to the induced metric of the manifold Cx, is equal to 1 if and only ifa=a0, where

a₀:= 3/(2√

31)≈0.269.

Now we investigate numerically with Algorithm 4.56 the stability of the (5,1) nanotube, more precisely of (G_a,α0, x_a, V_a,α0).

(i) For alla >0 we haveeV_a,α0 6= 0, see Figure 4.3, and thusχ_Ga,α

0xa

is not a critical point ofEa,α₀. Thus we can proceed with (vi).

(vi) By (i) for all a > 0 the triple (Ga,α0, x_a, V_a,α0) is not stable with respect to bothk · kR and k · kR,0,0.

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.0

0.5 1.0 1.5 2.0

0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25

Figure 4.3.: For the (5,1) nanotube as described in Example 4.58, the graphs of the energyE(χ_Ga,α

0x_a) and the norm ofe_Va,α

0 de-pendent on the scale factoraare plotted in blue and orange, respectively. For alla, we havee_Va,α

0 6= 0 and thus the (5,1) nanotube is not stable.

We define

(a^∗, α^∗, x^∗) := arg min

(a,α,x)∈W

E_a,α(χ_Ga,αx)

≈(0.263,1.117,(1.388,0.776,0.626)) and

x^∗_a := arg min

x∈U_a,α∗

E(χ_Gx) for alla≈a^∗.

In particular we havex^∗=x^∗_a∗. We have (a^∗, α^∗, x^∗)≈(a0, α0, xa₀) and thus the nanotubeGa^∗,α^∗·x^∗is approximately equal to the (5,1) nanotube Ga0,α0·x_a0. Now for alla≈a^∗we check the stability of (Ga,α^∗, x^∗_a, V_a,α^∗) numerically with Algorithm 4.56.

(i) For all a ≈ a^∗ the function χ_Gx^∗_a is a critical point of E by Re-mark 4.15(ii) and Corollary 4.16.

(ii) We have

suppV ={tp, t⁶p, t⁷p}

and

suppfV ={t⁻⁶, t⁻⁵, t⁻¹, id, t, t⁵, t⁶, tp, t⁶p, t⁷p}

by Remark 4.15(ii). The first and second derivative of V can be computed, e. g., with the Python library SymPy and fV can be computed numerically by Definition 4.11.

(iii) Since{t⁻¹, id, t, p}has Property 1 and {t, p} generatesG, by Defi-nition 3.5 the set

R=Ra :={t⁻¹, id, t, t², t⁻¹p, p, tp}

has Property 2. We define the bijectionϕbetweenRand{0, . . . ,6}

byϕ(t^m) =m+ 1 for allm∈ {−1,0,1,2} andϕ(t^mp) =m+ 5 for allm∈ {−1,0,1}. For all a≈a^∗we define the functions

bi=bi,a:R →R³, g7→L(g)^Tei for alli∈ {1,2,3}

and

b_i=b_i,a:R →R³, g7→L(g)^TA_i(g·x^∗_a−x^∗_a) for alli∈ {4,5,6}, where

A4=

0−1 0 1 0 0 0 0 0

, A5=

0 0−1 0 0 0 1 0 0

andA6= 0 0 0

0 0−1 0 1 0

.

By Proposition 3.28 the sets{b₁, . . . , b₆}and{b₁, . . . , b₄}are bases of Uiso(R) and Uiso,0,0(R), respectively. With, e. g., the Gram-Schmidt process we can determine functions b⁰₁, . . . , b⁰₆:R → R³ such that {b⁰₁, . . . , b⁰₆} and {b⁰₁, . . . , b⁰₄} are orthonormal bases of Uiso(R) andUiso,0,0(R), respectively. A bijection between{u:R → C³} and C²¹ is given by u 7→ (u(ϕ⁻¹(0)), . . . , u(ϕ⁻¹(6))). Let B = (b⁰₁, . . . , b⁰₆) ∈ R^21×6 and B0 = (b⁰₁, . . . , b⁰₄) ∈ R^21×4. The matricesP=I21−BB^TandP0=I21−B0B₀^Tare orthogonal pro-jection matrices with kernels Uiso(R) andUiso,0,0(R), respectively.

Letp0, . . . , p6, p0,0, . . . , p0,6∈R^21×3such thatP = (p0, . . . , p6) and P₀ = (p_0,0, . . . , p_0,6). For the functions g_R and g_R,0,0 of Defini-tion 4.43 we have

suppg_R= suppg_R,0,0=R,

g_R(g) =p_ϕ(g) for allg∈ R

and

g_R,0,0(g) =p_0,ϕ(g) for allg∈ R.

(iv) We haveT F=T =hti,M0=Nand{id}is a representation set of T F/∼d by Lemma 2.36(i). Recall Definition 2.28. We haveL_S =hai and L^∗_S = ha⁻¹i. By Proposition 2.39 we have {k ∈ R|(I1, k) ∈ Gid} =ha⁻¹iand thus Gid ={((−I1)^q, ma⁻¹)|m∈Z, q ∈ {0,1}}.

The intervalKid = [0,1/(2a)) is a representation set ofR/Gid. (v) The set{id, p} is a complete set of representatives of the cosets of

T F in G. For all k∈Kid andg∈ G we have

Ind^G_{T F}χk(g) =











χ_k(g) 0 0 χk(p⁻¹gp)

ifg∈ T F 0 χ_k(gp)

χk(p⁻¹g) 0

else.

Now for all k ∈ Kid, it is easy to compute the complex 6-by-6 matricesf_V

V

(Indχ_k),^Vg_R(Indχ_k) and^Vg_R,0,0(Indχ_k). We have {k∈Kid|^Vg_R(Indχk) has full rank}=Kid\ {0, α^∗/(2πa)}

and

{k∈Kid|^Vg_R,0,0(Indχk) has full rank}=Kid\ {0, α^∗/(2πa)}.

For all k ∈K_id\ {0, α^∗/(2πa)} we can compute λ_min(f_V

V

(Indχ_k), gR

V

(Indχk)) andλmin(fV

V

(Indχk), gR,0,0

V

(Indχk)). In particular we can computeλ_a(a, α^∗) andλ_a,0,0(a, α^∗) numerically, see Figure 4.4.

(vi) In the stretched case a > a^∗, we have both λ_a(a, α^∗) > 0 and λ_a,0,0(a, α^∗) > 0 and thus (G_a,α^∗, x_a,α^∗, V_a,α^∗) is stable with re-spect to both k · k_R and k · k_R,0,0. In the compressed case a ∈ (0, a^∗) we have λ_a(a, α^∗) = −∞ and λ_a,0,0(a, α^∗) < 0 and thus (Ga,α^∗, xa,α^∗, Va,α^∗) is not stable with respect to both k · k_R and k · k_R,0,0.

Notice that in the stretched case a > a^∗, the appropriate seminorm for the stability is k · k_R,0,0. For the equilibrium case a ≈ a^∗, the weaker seminorm k · k_R is appropriate since lim_a&a^∗λa,0,0(a, α^∗) = 0 and lima&a^∗λa(a, α^∗)>0.

For all a ≈ a^∗ and α ≈ α^∗ we can compute λa(a, α) and λa,0,0(a, α) analogously. For α ≈ α^∗ the graphs of λa(·, α) and λa,0,0(·, α) are similar to the graphs ofλa(·, α^∗) and λa,0,0(·, α^∗). As an example, we consider

α_a:= arg min

α∈(0,π)

E(χ_Gx_a,α) for alla≈α^∗, see Figure 4.5. In Figure 4.5 the graphs of the functions

a7→Relative difference λ_a(a, α^∗), λ_a(a, α_a) and

a7→Relative difference λ_a,0,0(a, α^∗), λ_a,0,0(a, α_a) are plotted, where

Relative difference(x, y) :=|x−y|/max{|x|,|y|} for allx, y∈R.

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 0.0

0.1 0.2 0.3 0.4 0.5

0.25 0.26 0.27 0.28 0.29 0.30

−0.01 0.00 0.01 0.02 0.03 0.04

Figure 4.4.: For the nanotube as described in Example 4.58, the point (α^∗/(2πa^∗),0) and the graphs ofλ_min(fc_V(χ_k),gc_R(χ_k)) (blue) and λmin(cfV(χk),g\_R,0,0(χk)) (orange) dependent on k ∈ Kid\{0, α^∗/(2πa^∗)}are plotted on the top plot for the choice a =a^∗. The point (a^∗,0) and and the graphs of λa (blue) andλa,0,0 (orange) dependent on the scale factor are plotted on the bottom plot.

0.25 0.26 0.27 0.28 0.29 0.30 0.0000

0.0002 0.0004 0.0006 0.0008 0.0010

+1.116

0.25 0.26 0.27 0.28 0.29 0.30

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Figure 4.5.: For the nanotube as described in Example 4.57, the point (a^∗, α(a^∗)) and the graph of the angleα(a) dependent on the scale factor aare plotted on the top plot. The point (a^∗,0) and the graphs of Relative difference λa(a, α^∗), λa(a, αa) (blue) and Relative difference λ_a,0,0(a, α^∗), λ_a,0,0(a, α_a)

(or-ange) dependent on the scale factor a are plotted on the bottom plot.

A. The configurational energy restricted to U _iso,0,0 ∩ U _per

In the following we prove Remark 4.30(iii), see Proposition A.3. Propo-sition A.3 is similar to PropoPropo-sition 4.29.

Lemma A.1. Suppose thatV is weakly* sequentially continuous. Then for all functions y:G \ {id} →R^d and constantsC, c >0 there exists a finite setA ⊂ G \ {id}such that

|V(y+z)−V(y)|< c

for allz∈L^∞(G \ {id},R^d)with kzk_∞≤C andz(g) = 0for all g∈ A.

Proof. This is clear since V is weakly* sequentially continuous and by Exercise 2.51b) in [37].

Remark A.2. A sequence (y_n)_n∈N in L^∞(G \ {id},R^d) converges to y ∈ L^∞(G \ {id},R^d) with respect to the weak* topology if and only if the sequence (y_n)_n∈N is bounded and (y_n)_n∈N converges componentwise to y, i. e. lim_n→∞yn(g) =y(g) for allg∈ G \ {id}, see Exercise 2.51 in [37].

Proposition A.3. Suppose that V is weakly* sequentially continuous, E⁰(χGx0) = 0and letu∈Uiso,0,0∩Uper. Then it holdsE⁰⁰(χGx0)(u, u) = 0and _dτ^d3₃E(χ_Gx₀+τ u)

τ=0= 0.

Proof. Suppose thatVis weakly* sequentially continuous andE⁰(χ_Gx0) = 0. Thus for the monotonically increasing function

r: [0,∞)→[0,∞) t7→sup

|E(χ_Gx0+u)−E(χ_Gx0)|

u∈Bt(0) it holds

limt&0

r(t)

t² = sup{E⁰⁰(χ_Gx₀)(u, u)|u∈U_per}<∞, (A.1) whereB_t(0) ={u∈U_per| kuk∞< t} for allt >0.

Letu∈Uiso,0,0∩U_per. There exist somea∈R^d and S∈ ⊕(Skew(d₁)× {0d₂,d₂}) such that

L(g)u(g) =a+S(g·x0−x0) for allg∈ G.

Since differentiability implies locally boundedness, there exist someδ >0 andC1>0 such that

|V(y₀+w)| ≤C₁ for allw∈B_δ(0), whereB_δ(0) ={w∈L^∞(G \ {id},R^d)| kwk_∞< δ}. Let

C2= 2kx0ksup

e^{−τ S}−Id+τ Se^{−τ S} /τ²

τ ∈(−1,1) ≥0.

By Taylor’s theorem we haveC2<∞. Let t0= min{1,p

δ/(2C2)}>0, wherea/0 :=∞for alla >0.

Now we show that

|E(χ_Gx₀+tu)−E(χ_Gx₀)| ≤r(C₂t²) +t⁴ for allt∈(−t₀, t₀). (A.2) Lett∈(−t0, t0)\ {0}. We define the functionv:G →R^d by

g·v(g) =x0+ e^−tS(Id+tS)(g·x0−x0) for allg∈ G, see also Figure A.1. We have

kv−χ_Gx₀k∞= sup{kv(g)−x₀k |g∈ G}

= sup{kg·v(g)−g·x0k |g∈ G}

= sup

e^−tS−Id+tSe^−tS

(g·x0−x0) g∈ G

= sup

e^−tS−I_d+tSe^−tS

(L(g)x₀−x₀) g∈ G

≤2

e^−tS−Id+tSe^−tS kx0k

≤C₂t², (A.3)

where in the forth step we used thatSτ(g) = 0 for allg∈ G. In particular, we havev∈L^∞(G,R^d) and

kv−χGx0k∞<δ

2. (A.4)

For allg∈ G we define the map

ϕ_g:U_per→ {w:G \ {id} →R^d}

w7→ G \ {id} →R^d, h7→(gh)·w(gh)−g·w(g) .

x0

S₁ S2

S3

g·x₀ g·v(g)

g·(x₀+tu(g))−ta

Figure A.1.: In this figure foru,v,a,Sandtas in the proof of Lemma A.3 and g ∈ G, the points x₀, g·x₀, g·(x₀+u(g))−a and g·v(g) and the setsS₁={x0+A(g·x₀−x₀)|A∈O(d)}, S₂={x₀+ (I_d+ ˜S)(g·x₀−x₀)|S˜∈ ⊕(Skew(d₁)× {0_d2,d2})}

and S₃ = {x₀+A(I_d +tS)(g·x₀−x₀)|A ∈ O(d)} are displaced.

For allg∈ G we have

ϕ_g(χ_Gx₀+tu) = (gh)·x₀+tL(gh)u(gh)

−(g·x₀+tL(g)u(g))

h∈G\{id}

= (gh)·x0+ta+tS((gh)·x0−x0)

−(g·x0+ta+tS(g·x0−x0))

h∈G\{id}

= (Id+tS)((gh)·x0−g·x0)

h∈G\{id}

= e^tS((gh)·v(gh)−g·v(g))

h∈G\{id}

=e^tSϕg(v). (A.5)

For allA ⊂ G \ {id} we denote B_A:=

w∈L^∞(G \ {id},R^d)

kwk_∞≤Randw(g) = 0 for allg∈ A , whereR= 2(kx0k+t₀kuk∞). Let N ∈M₀ such that uis T^N-periodic.

SinceV is weakly* sequentially continuous, by Lemma A.1 for allg∈ CN

there exists a finite setAg⊂ G \ {id} such that V(ϕg(χ_Gx0+tu) +w)−V(ϕg(χ_Gx0+tu))

< t⁴

2 for allw∈B_Ag. (A.6) LetA =S

g∈CNAg. Equation (A.5), (H1) and (A.6) imply that for all g∈ G we have

sup

w∈BA

V(ϕ_g(v) +w)−V(ϕ_g(v))

= sup

w∈BA

V(e^−tSϕg(χ_Gx0+tu) +w)−V(e^−tSϕg(χ_Gx0+tu))

= sup

w∈BA

V(ϕ_g(χ_Gx₀+tu) +w)−V(ϕ_g(χ_Gx₀+tu))

= sup

w∈BA

V(ϕ_˜g(χ_Gx₀+tu) +w)−V(ϕ_˜g(χ_Gx₀+tu))

≤t⁴

2, (A.7)

where in the third line ˜g∈ Gis defined by the condition{˜g}=gT^N∩ C_N. Let m ∈ N such that M₀ = mN. Since T^m is isomorphic to Z^d2, there existt₁, . . . , t_d2 ∈ T^msuch that {t₁, . . . , t_d2}generatesT^m. With-out loss of generality we assume that Cn ={tⁿ₁¹. . . tⁿ_d^d2

2 g|n1, . . . , nd₂ ∈ {0, . . . , n/m−1}, g ∈ Cm} for all n ∈ M0, see Remark 2.51(ii). There exists somen⁰ ∈Nsuch that

CmA ⊂ {tⁿ₁¹. . . tⁿ_d^d2

2 |n1, . . . , nd₂ ∈ {−n⁰, . . . , n⁰}}Cm. Thus there exists someN⁰∈M₀ such thatN divides N⁰ and

|CN⁰\ D|

|C_N⁰| < t⁴

4C₁, (A.8)

where D={g ∈ C_N⁰|gA ⊂ C_N⁰}. We define the T^N0-periodic function

˜

v∈U_per by

˜

v(g) :=v(g) for allg∈ C_N⁰. It holds

|E(˜v)−E(χ_Gx0)| ≤r(k˜v−χ_Gx0k_∞)≤r(kv−χ_Gx0k_∞)≤r(C2t²), (A.9) where we used (A.3) in the last step. Moreover, we have

|E(χ_Gx₀+tu)−E(˜v)| ≤ 1

|CN⁰| X

g∈C_N0

|V(ϕ_g(χ_Gx₀+tu))−V(ϕ_g(˜v))|

= 1

|CN⁰| X

g∈C_N0

|V(e^tSϕg(v))−V(ϕg(˜v))|

= 1

|CN⁰| X

g∈C_N0

|V(ϕg(v))−V(ϕg(˜v))|

≤ 1

|CN⁰| X

g∈D

sup

w∈BA

|V(ϕ_g(v))−V(ϕ_g(v) +w)|

+ 2

|CN⁰| X

g∈C_N0\D

sup

w∈Bδ(0)

|V(ϕ_g(χ_Gx₀) +w)|

≤ t⁴ 2 +t⁴

2 =t⁴, (A.10)

where we used (A.5) in the second step, (H1) in the third step, (A.4) in the forth step and (A.7) and (A.8) in the fifth step. Equation (A.9) and (A.10) imply (A.2).

By (A.2) and (A.1) we have lim sup

t→0

E(χ_Gx₀+tu)−E(χ_Gx₀) t³

≤lim sup

t→0

r(C₂t²)

t³ +t= 0 and thus,E⁰⁰(χ_Gx0)(u, u) = 0 and _dτ^d33E(χ_Gx0+τ u)

_τ=0= 0.

B. Representation theory

We need the following propositions in Chapter 2.

In general, the dual space of a locally compact group contains infinite-dimensional representations. In contrast to the rest of the thesis, in the following when we use the term representation, we mean a finite- or infinite-dimensional representation on a Hilbert space.

Proposition B.1 (Proposition 1.35 in [43]). Let ρbe a continuous uni-tary representation of a locally compact group G on the Hilbert space H(ρ). Thenρis irreducible if and only if

commutant ofρ(G) :={T ∈ B(H(ρ))|T ρ(g) =ρ(g)T for allg∈G}

=CI,

whereB(H(ρ))denotes the space of bounded linear operators from H(ρ) toH(ρ)andI is the identity operator on H(ρ).

Proposition B.2 (Proposition 1.71 in [43]). Let N be a closed normal subgroup of a locally compact group Gandq:G→G/N be the quotient homomorphism. The mapρ7→ρ◦q is a homeomorphism of G/N[ with the closed subset ofGb consisting of those elements ofGbwhich annihilate N.

C. Seminorms

We need the following definitions and lemma in Chapter 3.

Definition C.1. Given a vector space V over a field K ∈ {R,C}, a seminorm is a functionp:V →[0,∞) such that

p(u+v)≤p(u) +p(v) (subadditivty) and

p(αv) =|α|p(v) (absolute homogeneity) for allu, v∈V andα∈K.

Definition C.2. We say that two seminormsp1andp2on a vector space areequivalent if there exist two constantsc, C >0 such that

cp1≤p2≤Cp1.

RemarkC.3. It is clear that for a given vector space this definition induces an equivalence relation on the set of all seminorms on that vector space.

The following lemma is well-known, see, e. g., [35, Exercise 36, p.206].

Lemma C.4. Letp₁ andp₂two seminorms on a finite-dimensional vec-tor space. Thenp₁ andp₂are equivalent if and only ifker(p₁) = ker(p₂).

Proof. Ifp₁andp₂ are equivalent then it is clear that ker(p₁) = ker(p₂).

Let ker(p1) = ker(p2) and call the domain of p1 andp2 the vector space V. Thenp1andp2are norms on the quotient spaceV /ker(p1). Since all norms on a finite-dimensional vector space are equivalent the normsp1

andp2onV /ker(p1) are equivalent. This implies that also the seminorms p1 andp2 onV are equivalent.

D. Miscellaneous results

In [10, p. 440] the Kronecker product is defined.

Definition D.1 (Kronecker product). LetA= (aij)∈C^m×n andB = (bij)∈C^p×q. Then, theKronecker product A⊗B ∈C^(mp)×(nq)ofAand B is the partitioned matrix

A⊗B:=







a₁₁B · · · a_1nB ... . .. ... am1B · · · amnB





.

Remark D.2. If we sayv∈Cⁿ, thenv is a column vector, i. e.v∈C^n×1. Thus, the Kronecker productA⊗Bis also defined ifA∈Cⁿ orB∈Cⁿ. For the basic properties of the Kronecker product we refer to [10].

Lemma D.3. For all m, n ∈ N let P_m,n ∈ O(mn) be the Kronecker permutation matrix such that

Pp,m(A⊗B)Pn,q=B⊗A for all A∈C^m×n andB∈C^p×q, see [10, Fact 7.4.30]. For all natural numbers m, n₁, . . . , n_k ∈ N let Q_m,n1,...,nk ∈O(m(n₁+· · ·+n_k)) be the permutation matrix (P_m,n1⊕

· · · ⊕P_m,nk)P_{n1+···+nk,m}. Then the following statements hold:

(i) For allAi∈C^mi×ni,i∈ {1, . . . , k}, andB∈C^p×q we have (A1⊕ · · · ⊕Ak)⊗B= (A1⊗B)⊕ · · · ⊕(Ak⊗B).

(ii) For all A∈C^m×n andBi∈C^pi×qi,i∈ {1, . . . k}, we have A⊗(B1⊕· · ·⊕Bk) =Q^T_m,p

1,...,p_k((A⊗B1)⊕· · ·⊕(A⊗Bk))Q_n,q1,...,qk. (iii) For allA∈C^m×n andB1, . . . , Bk∈C^p×q we have

A⊗(B₁⊕· · ·⊕B_k) = (P_m,k⊗I_p)((A⊗B₁)⊕· · ·⊕(A⊗B_k))(P_k,n⊗I_q).

Proof. (i) This is easy to check.

(ii) For allA∈C^m×n andBi∈C^pi×qi,i∈ {1, . . . , k}, we have A⊗(B1⊕ · · · ⊕Bk)

=Pm,p₁+···+pk((B1⊕ · · · ⊕Bk)⊗A)Pq₁+···+qk,n

=P_{m,p1+···+pk}((B₁⊗A)⊕ · · · ⊕(B_k⊗A))P_{q1+···+qk,n}

=P_{m,p1+···+pk}((P_p1,m(A⊗B₁)P_n,q1)⊕ · · ·

⊕(Pp_k,m(A⊗Bk)Pn,q_k))Pq₁+···+q_k,n

=Q^T_m,p1,...,pk((A⊗B1)⊕ · · · ⊕(A⊗Bk))Qn,q1,...,q_k. (iii) By Fact 7.4.30viii) in [10] we have

Qn,q,...,q= (Ik⊗Pn,q)Pkq,n =Pk,n⊗Iq.

It is well-known that commuting orthogonal matrices are simultaneously quasidiagonalisable:

Theorem D.4. Let S ⊂O(n)be a nonempty commuting family of real orthogonal matrices. Then there exist a real orthogonal matrixQ and a nonnegative integerqsuch that, for eachA∈ S,Q^TAQis a real quasidi-agonal matrix of the form

Λ(A)⊕R(θ1(A))⊕ · · · ⊕R(θq(A))

in which each Λ(A) = diag(±1, . . . ,±1) ∈ R(n−2q)×(n−2q), R(θ) is the rotation matrix(^cos_sinθ^θ−_cos^sin_θ^θ)and eachθj(A)∈[0,2π).

Proof. This follows immediately by [39, Corollary 2.5.11.(c), Theorem 2.5.15].

We now state Kronecker’s approximation theorem, see, e. g., Corollary 2 on page 20 in [38].

Theorem D.5 (Kronecker’s approximation theorem). For each irra-tional number α the set of numbers {αnreduced modulo 1|n ∈ N} is dense in the whole interval[0,1).

We also need Turán’s third theorem, see Theorem 11.1 on page 126 in [57].

Theorem D.6 (Turán’s third theorem). Let b1, . . . , bn, z1, . . . , zn ∈C. Ifm is a nonnegative integer and thezj are restricted by

min_µ6=ν|z_µ−z_ν|

maxj|zj| ≥δ(>0), z_j6= 0

then the inequality

ν=m+1,...,m+nmax Pn

j=1b_jz_j^ν Pn

j=1|bj| |zj|^ν ≥ 1 n

δ 2

n−1

holds.

We also need Theorem A.1 of [4].

Theorem D.7. Let(X, d)be a metric space,(Y,F, µ)be a measure space andf:X×Y →R be such that

(i) f(x,·)isµ-integrable for allx∈X,

(ii) f(·, y)is continuous in X forµ-almost ally∈Y, (iii) there existsm∈L¹(Y, µ) satisfying

sup

x∈X

|f(x, y)| ≤m(y) forµ-a.e. y∈Y.

Then the map

X →R, x7→

Z

Y

f(x, y)dµ(y) is bounded and continuous.

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