Topological Crystals: Independence of Spectral Properties with Respect to Reference Systems

Abstract

It is a common postulate that spectral properties of operators describing physical systems are independent of the underlying reference frames. For the Laplace operator on topological crystals, we prove such a statement from a deeper analysis of the behavior of spectral properties with respect to arbitrary choices. In particular, we investigate the impact of the choice of a unit cell, and of the choice of a family of generators for the transformation group.

1. Introduction

In the framework of quantum mechanics, physical systems are usually described by self-adjoint operators on Hilbert spaces. As a result, spectral properties of these operators play an important role for our understanding of the systems. However, it is often tacitly assumed that these spectral properties are independent of the underlying reference system used for the definition of the operators. One can then naturally wonder to what extend this assumption is correct, and if it can be proved.

In this paper, we provide a thorough answer to this question in the framework of topological crystals. More precisely, we study the Laplace operator defined on them, and provide a detailed analysis of the spectral properties of this operator. In particular, we perform very explicit computations, and keep track of all implicit choices made during the investigations. This allows us to show the independence of spectral properties with respect to these choices, and to thoroughly understand the impact of these choices at different stages in the computations. This careful study is new, and shows that the mentioned independence can take place at different levels.

The notion of topological crystals, that is an infinite-fold Abelian covering graph over a finite graph, has been carefully introduced and explained in [1]. This subject is naturally linked to the notion of discrete geometric analysis. In particular, the discrete Laplace operator is one of the central objects in this field, and its study plays an important role for the study of the graph itself. An introduction to the discrete Laplace operator on locally finite graphs is provided, for example, in [2]. Spectral and scattering theory for operators acting on topological crystals and on their perturbations have then been investigated in [3,4]; see also [5,6,7,8,9] and references therein. For our investigations, we mainly follow the paper [4]. We also refer to [10,11] for recent papers in which spectral properties of graphs play an essential role.

Let us now describe the content of this paper and the main results obtained. Section 2 contains the introduction to topological crystals. We start with the definition of general graphs and quickly move to the special instance of topological crystals. These graphs can be described by quadruplet $(X,\mathfrak{X},\omega ,\Gamma )$, where X is a covering graph over the base graph $\mathfrak{X}$, $\omega :X\to \mathfrak{X}$ is the morphism which defines the local structure, and where $\Gamma$ is the transformation group of X leaving $\omega$ invariant. In addition, $\Gamma$ is assumed to be isomorphic to ${\mathbb{Z}}^d$. This invariance of $\omega$ with respect to $\Gamma$ provides the periodicity of the topological crystal. Note that one often refers to X as a topological crystal, but it is really the quadruplet $(X,\mathfrak{X},\omega ,\Gamma )$ which contains all the information. Note also that it is useful to choose a set of generators for the group $\Gamma$. This choice is highly non-unique, and will play a role subsequently.

Still in Section 2, the concept of unit cell U is introduced, and related maps are studied. Let us emphasize that a unit cell corresponds to a minimum subset of vertices of X in bijection with the vertices of the base graph $\mathfrak{X}$. In particular, it means that if $\mathfrak{X}$ contains n vertices, then any unit cell will also consist of n distinct vertices. However, the exact choice of these n vertices is highly non-unique. On the other hand, several maps, which will be important for the subsequent analysis on topological crystals, are uniquely defined once the choice of a unit cell is made. These maps determine the relations between the vertices and edges in the covering graph and those in the base graph. They also provide, to each vertex and edge, distinct coordinates. Together, the generators of $\Gamma$ and the unit cell correspond to the reference systems mentioned earlier.

In Section 3, we introduce the necessary material for performing the analysis on topological crystals. First of all, a periodic measure (or weight) function m is attached to each vertex and edge of the graph. A resulting degree function ${deg}_m$ is also defined. The Laplace operator on topological crystals $\Delta (X,m)$ is then introduced in the Hilbert space ${\ell }^2(X,m)$. This operator encodes the structure of the graph, and our next aim is to describe its spectral properties. As emphasized by the notation, the Laplace operator depends on the topological crystal and on the measure m, but not on other arbitrary choices. In our setting, it is also known that $\Delta (X,m)$ defines a bounded and self-adjoint operator on the Hilbert space ${\ell }^2(X,m)$.

For further investigations, some identifications on topological crystals are necessary. For example, each vertex in the covering graph is identified with the pair of an element of ${\mathbb{Z}}^d$ and a vertex in the base graph. Similarly, the dual group of ${\mathbb{Z}}^d$ is identified with the d-torus ${\mathbb{T}}^d$. Based on these identifications, three important unitary transforms ${\mathcal{U}}_U,\mathcal{F}$ and $\mathcal{I}$ are defined. The map ${\mathcal{U}}_U$ transforms a function from ${\ell }^2(X,m)$ to a function in ${\ell }^2\left({\mathbb{Z}}^d;{\ell }^2\left(\mathfrak{X}\right)\right)$. The map $\mathcal{F}$ transforms a function from ${\ell }^2\left({\mathbb{Z}}^d;{\ell }^2\left(\mathfrak{X}\right)\right)$ to a function in $L^2\left({\mathbb{T}}^d;{\ell }^2\left(\mathfrak{X}\right)\right)$. Finally, the map $\mathcal{I}$ transforms a function from $L^2\left({\mathbb{T}}^d;{\ell }^2\left(\mathfrak{X}\right)\right)$ to a function in $L^2({\mathbb{T}}^d;{\mathbb{C}}^n)$. Thus, the composition of these transforms $\mathcal{I}\mathcal{F}{\mathcal{U}}_U$ is a unitary transform from ${\ell }^2(X,m)$ to $L^2({\mathbb{T}}^d;{\mathbb{C}}^n)$. Among these unitary transforms, let us emphasize that only ${\mathcal{U}}_U$ depends upon the choice of the unit cell.

The next task is to look at the image of the Laplace operator through these unitary maps. More precisely, we show the equality

$$\left[\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\right]\Delta (X,m){\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\right]}^{*}g\right]\left(\xi \right)=h_U\left(\xi \right)g\left(\xi \right)$$

(1)

for $g\in L^2({\mathbb{T}}^d;{\mathbb{C}}^n)$, $\xi \in {\mathbb{T}}^d$, and where $h_U\left(\xi \right)$ is a $n\times n$ matrix explicitly provided in (2). In the literature, these unitary transforms are usually only sketched, while we exhibit them very explicitly, and provide the details of all computations. Let us still emphasize the meaning of the above equality; through these unitary transforms, the Laplace operator becomes a multiplication operator by a function taking values in the set of $n\times n$ Hermitian matrices.

In Section 4, we study the dependence of the spectral properties of the multiplication operator defined by the function $h_U$, with respect to the arbitrary choices mentioned before. First of all, as recalled in Proposition 1, it is well known that the following equality holds

$$\sigma \left(\Delta (X,m)\right)=\bigcup _{\xi \in {\mathbb{T}}^d}\sigma \left(h_U\left(\xi \right)\right).$$

The dependence of this equality with respect to the choice of the unit cell is investigated in Section 4.1. In particular, if $h_U\left(\xi \right)$ and $h_{U^{\prime }}\left(\xi \right)$ denote the matrices computed with respect to two different unit cells U and $U^{\prime }$, we show in Theorem 1 that the following equality

$$\sigma \left(h_U\left(\xi \right)\right)=\sigma \left(h_{U^{\prime }}\left(\xi \right)\right)$$

holds for any $\xi \in {\mathbb{T}}^d$. We observe that this equality corresponds to the strongest version of the invariance of the spectrum with respect to a change in reference system; it is an equality for each fixed $\xi$, and not only for the continuous union over all $\xi$.

In Section 4.2, we investigate the change in generators for the transformation group $\Gamma$, and observe that the outcome is quite different. More precisely, we show in Theorem 2 that the spectrum of the matrix $h_U\left(\xi \right)$ does depend on the choice of the generators for $\Gamma$. Nevertheless, when taking the union over all $\xi \in {\mathbb{T}}^d$, the same set of values is obtained, and corresponds to the spectrum of the Laplace operator.

In summary, the definition of the Laplace operator $\Delta (X,m)$ does not depend on any arbitrary choice, but the computation of its spectral properties involves the choice of a unit cell and the choice of a set of generators for the transformation group $\Gamma$. Both choices are highly non-unique, and these choices are reflected in the expression obtained for the matrix-valued function $h_U$. The choice of the unit cell has only a weak impact on $h_U$, since the spectrum of $h_U\left(\xi \right)$ is preserved for each individual $\xi$. On the other hand, this stability is not preserved for a change in the set of generators for the transformation groups, but the invariance is restored when the union of all $\xi \in {\mathbb{T}}^d$ is computed. As a consequence of these investigations, a refined version of the spectral invariance was obtained.

2. Introduction to Topological Crystals

We start by recalling some general notions related to graphs. A graph $X=\left(V\left(X\right),E\left(X\right)\right)$ consists of a set $V\left(X\right)$ of vertices and a set $E\left(X\right)$ of unoriented edges, with possibly multiple edges and loops. We use the notations $x,y$ for vertices and e for edges. If both $V\left(X\right)$ and $E\left(X\right)$ are finite, then X is said to be a finite graph. From the set $E\left(X\right)$ of unoriented edges, the set $A\left(X\right)$ of oriented edges is defined such that, for any unoriented edge between x and y, one creates a pair of oriented edges, one from x to y and one from y to x. Elements of $A\left(X\right)$ are also denoted by e. The origin of an oriented edge e is denoted by $o\left(e\right)$, and its terminal by $t\left(e\right)$. For an edge e we denote by $\overline{e}$ the second edge in the pair with the origin and terminal interchanged, namely $o\left(\overline{e}\right)=t\left(e\right)$ and $t\left(\overline{e}\right)=o\left(e\right)$. For each $x\in V\left(X\right)$, we define a subset of $E\left(X\right)$ by

$$E{\left(X\right)}_x:=\{e\in E\left(X\right)\mid x\mathrm{is}\mathrm{an}\mathrm{endpoint}\mathrm{of}e\}.$$

If $E{\left(X\right)}_x$ is finite for every $x\in V\left(X\right)$, we say that X is locally finite. Similarly, we define a subset of $A\left(X\right)$ for each $x\in V\left(X\right)$ by

$$A{\left(X\right)}_x:=\{e\in A\left(X\right)\mid o\left(e\right)=x\}.$$

We now introduce various notions for defining topological crystals. For that purpose, we consider a second graph $\mathfrak{X}=\left(V\left(\mathfrak{X}\right),E\left(\mathfrak{X}\right)\right)$ with its vertices and edges denoted by $\mathfrak{x}$ and $\mathfrak{e}$, respectively. Then, given two graphs X and $\mathfrak{X}$, a graph morphism $\omega :X\to \mathfrak{X}$ consists of two maps $\omega :V\left(X\right)\to V\left(\mathfrak{X}\right)$ and $\omega :E\left(X\right)\to E\left(\mathfrak{X}\right)$, preserving the adjacency relations between vertices and edges; namely, if e is an edge between x and y in X, then $\omega \left(e\right)$ is an edge between $\omega \left(x\right)$ and $\omega \left(y\right)$ in $\mathfrak{X}$.

Definition 1.

A graph morphism $\omega :X\to \mathfrak{X}$ is said to be a covering map and X a covering graph over the base graph $\mathfrak{X}$, if

1. 

$\omega :V\left(X\right)\to V\left(\mathfrak{X}\right)$ is surjective;

2. 

For all $x\in V\left(X\right)$, the restriction ${\omega |}_{E{\left(X\right)}_x}:E{\left(X\right)}_x\to E{\left(\mathfrak{X}\right)}_{\omega \left(x\right)}$ is a bijection.

Given a covering map $\omega :X\to \mathfrak{X}$, we also define the transformation group $\Gamma$ acting upon X as the subgroup of automorphisms of X such that $\omega \circ \mu =\omega$ for any $\mu \in \Gamma$. With these notions at hand, we can introduce a family of periodic graphs.

Definition 2 

([1], Section 6.2). A d-dimensional topological crystal is a quadruplet $(X,\mathfrak{X},\omega ,\Gamma )$ such that:

1. 

X and $\mathfrak{X}$ are graphs, with $\mathfrak{X}$ finite;

2. 

$\omega :X\to \mathfrak{X}$ is a covering map;

3. 

The transformation group Γ of ω is isomorphic to ${\mathbb{Z}}^d$;

4. 

ω is regular, i.e., for every $x,y\in V\left(X\right)$ satisfying $\omega \left(x\right)=\omega \left(y\right)$, there exists $\mu \in \Gamma$ such that $x=\mu y$.

For simplicity, we exclude topological crystals with multiple edges or loops, which means that X has no multiple edges or loops. However, $\mathfrak{X}$ often possesses multiple edges and loops. We gather in the following lemma the relations between $\omega$ and $\Gamma$.

Lemma 1.

1. 

For every $\mu \in \Gamma$, the equality $\omega \circ \mu =\omega$ holds. That is, $\omega \left(\mu x\right)=\omega \left(x\right)$ for all $x\in V\left(X\right)$, and in particular, $\omega \left(t\right(e\left)\right)=\omega \left(\mu t\right(e\left)\right)$ and $\omega \left(o\right(e\left)\right)=\omega \left(\mu o\right(e\left)\right)$ for any $e\in E\left(X\right)$.

2. 

The equalities $o\left(\mu e\right)=\mu o\left(e\right)$ and $t\left(\mu e\right)=\mu t\left(e\right)$ hold, where $\mu e$ denotes the action of Γ on an edge, while $\mu t\left(e\right)$ or $\mu o\left(e\right)$ corresponds to the action of Γ on vertices.

We also observe that since $\Gamma$ is isomorphic to ${\mathbb{Z}}^d$, there exists a set of generators ${\Gamma }_1,\dots ,{\Gamma }_d$ of $\Gamma$ such that for any $\mu \in \Gamma$, one has

$$\mu :=\prod _{j=1}^d{\left({\Gamma }_j\right)}^{{\mu }_j}$$

with ${\mu }_j\in \mathbb{Z}$. For the abstract group $\Gamma$, we use the multiplicative notation, but for the concrete group ${\mathbb{Z}}^d$, we shall subsequently use the additive notation. Clearly, the choice of generators is not unique, and this will be discussed in Section 4.2.

Let us now move to additional structures on topological crystals. These new notions are based on the choice of a unit cell in the covering graph X, in bijection with the set $V\left(\mathfrak{X}\right)$. The subsequent constructions and analysis are dependent on this choice, which is highly non-unique, and will be carefully discussed in Section 4.1.

For any $\mathfrak{x}\in V\left(\mathfrak{X}\right)$, we fix a representative $x\in V\left(X\right)$ satisfying $\omega \left(x\right)=\mathfrak{x}$. More precisely, for any ${\mathfrak{x}}_j\in V\left(\mathfrak{X}\right)$, we look for a representative $x_j\in V\left(X\right)$ satisfying $\omega \left(x_j\right)={\mathfrak{x}}_j$, and call a complete set of representative vertices a unit cell U. In other words, if $V\left(\mathfrak{X}\right)$ contains n elements, we have $U=\{x_1,x_2,\dots ,x_n\}$ with $\omega \left(x_j\right)={\mathfrak{x}}_j$. We observe that if we set $\Gamma \left(x_j\right):=\{\mu x_j\mid \mu \in \Gamma \}$, then $\Gamma \left(x_j\right)$ corresponds to the orbit of $x_j$ under the action of the transformation group $\Gamma$, and the family $\{\Gamma \left(x_j\right)\mid x_j\in U\}$ defines a partition of X for any unit cell U.

Given a topological crystal structure $(X,\mathfrak{X},\omega ,\Gamma )$ with $V\left(\mathfrak{X}\right)=\{{\mathfrak{x}}_1,\dots ,{\mathfrak{x}}_n\}$ and a unit cell $U=\{x_1,\dots ,x_n\}$, we define the lift $\widehat{·}:V\left(\mathfrak{X}\right)\to U$ given by $\widehat{{\mathfrak{x}}_j}=x_j\in U$ for any ${\mathfrak{x}}_j\in \mathfrak{X}$ and $j=1,\dots ,n$. The composition of the two maps, $\omega :V\left(X\right)\to V\left(\mathfrak{X}\right)$ and $\widehat{·}:V\left(\mathfrak{X}\right)\to U$, namely

$$\widehat{\omega }\equiv \widehat{\omega (·)}:V\left(X\right)\to U$$

gives a unique $x_j\in U$ for any $x\in V\left(X\right)$, such that $x_j=\widehat{\omega }\left(x\right)$. Clearly, the map $\widehat{\omega }$ depends on the choice of a unit cell U. Then, the following statement can easily be proved.

Lemma 2.

For any $\mu \in \Gamma$, any $x\in V\left(X\right)$ and any $e\in A\left(X\right)$, one has $\widehat{\omega }\left(\mu x\right)=\widehat{\omega }\left(x\right)$, $\widehat{\omega }\left(\mu t\left(e\right)\right)=\widehat{\omega }\left(t\left(e\right)\right)$, and $\widehat{\omega }\left(\mu o\left(e\right)\right)=\widehat{\omega }\left(o\left(e\right)\right)$.

Based on the previous definition, we now define the floor function $\lfloor ·\rfloor :V\left(X\right)\to \Gamma$ by the equality

$$\lfloor x\rfloor \widehat{\omega }\left(x\right)=x$$

for all $x\in V\left(X\right)$. Note that this floor function depends on the choice of the unit cell U, and its existence follows from the regularity condition of the topological crystal. In other words, the floor function $\lfloor ·\rfloor$ identifies each vertex of the orbit $\Gamma \left(x_j\right)$ with an element of the transformation group $\Gamma$ by identifying $x_j\in \Gamma \left(x_j\right)\cap U$ with the identity element of $\Gamma$.

For any $x\in V\left(X\right)$, we denote by ${\lfloor x\rfloor }^{-1}\in \Gamma$ the inverse of the element $\lfloor x\rfloor \in \Gamma$. It then follows that

$${\lfloor x\rfloor }^{-1}\lfloor x\rfloor \widehat{\omega }\left(x\right)=\lfloor x\rfloor {\lfloor x\rfloor }^{-1}\widehat{\omega }\left(x\right)=\widehat{\omega }\left(x\right).$$

It is also easily observed that the following equalities hold for any $\mu \in \Gamma$ and $x\in V\left(X\right)$: $\lfloor \mu x\rfloor =\mu \lfloor x\rfloor =\lfloor x\rfloor \mu$ and ${\lfloor \mu x\rfloor }^{-1}={\lfloor x\rfloor }^{-1}{\mu }^{-1}={\mu }^{-1}{\lfloor x\rfloor }^{-1}$.

We can now introduce a key concept of the subsequent analysis, namely the index map $\eta :A\left(X\right)\to \Gamma$ given for any $e\in A\left(X\right)$ by

$$\eta \left(e\right):=\lfloor t\left(e\right)\rfloor {\lfloor o\left(e\right)\rfloor }^{-1}.$$

A few properties of this map are proven below.

Lemma 3.

The index map η is Γ-periodic, namely for any $\mu \in \Gamma$ and $e\in A\left(X\right)$ the following equality holds:

$$\eta \left(\mu e\right)=\eta \left(e\right).$$

Proof. 

For any $\mu \in \Gamma$ and $e\in A\left(X\right)$, one has

$$\eta \left(\mu e\right)=\lfloor t\left(\mu e\right)\rfloor {\lfloor o\left(\mu e\right)\rfloor }^{-1}=\lfloor \mu t\left(e\right)\rfloor {\lfloor \mu o\left(e\right)\rfloor }^{-1}=\mu \lfloor t\left(e\right)\rfloor {\lfloor o\left(e\right)\rfloor }^{-1}{\mu }^{-1}.$$

Since $\Gamma$ is an Abelian group, it follows that

$$\mu \lfloor t\left(e\right)\rfloor {\lfloor o\left(e\right)\rfloor }^{-1}{\mu }^{-1}=\lfloor t\left(e\right)\rfloor {\lfloor o\left(e\right)\rfloor }^{-1}=\eta \left(e\right),$$

leading directly to the statement. □

Because of this periodicity, an index can be attached to all edges of $A\left(\mathfrak{X}\right)$. However, note that this index cannot be computed by looking at the base graph only; it depends on the choice of U. In the next lemma, we state one more property of the index map, with respect to the change in orientation of an edge.

Lemma 4.

For any $e\in A\left(X\right)$, the equality $\eta \left(\overline{e}\right)=\eta {\left(e\right)}^{-1}$ holds.

Proof. 

For any $e\in A\left(X\right)$, one has

$$\eta \left(\overline{e}\right)=\lfloor t\left(\overline{e}\right)\rfloor {\lfloor o\left(\overline{e}\right)\rfloor }^{-1}=\lfloor o\left(e\right)\rfloor {\lfloor t\left(e\right)\rfloor }^{-1}={\left(\lfloor t\left(e\right)\rfloor {\lfloor o\left(e\right)\rfloor }^{-1}\right)}^{-1}=\eta {\left(e\right)}^{-1},$$

as stated. □

3. Analysis on Topological Crystals

In this section, we introduce the Laplace operator acting on topological crystals. Before this, we endow the graph with an additional structure. Let m be a measure on a locally finite graph X, namely two maps $m:V\left(X\right)\to (0,\infty )$ and $m:E\left(X\right)\to (0,\infty )$. A measure on oriented edges is defined by the measure on the corresponding unoriented edges, and consequently $m\left(e\right)=m\left(\overline{e}\right)$ for $e\in A\left(X\right)$. We also define the degree function on the set of vertices:

$${deg}_m:V\left(X\right)\to (0,\infty ),{deg}_m\left(x\right):=\sum _{e\in A{\left(X\right)}_x}{\displaystyle \frac{m\left(e\right)}{m\left(x\right)}}.$$

For any topological crystal $(X,\mathfrak{X},\omega ,\Gamma )$, we shall consider that the measure is $\Gamma$-periodic. The periodicity means that for every $\mu \in \Gamma$, $x\in V\left(X\right)$, and $e\in E\left(X\right)$, we have $m\left(\mu x\right)=m\left(x\right)$ and $m\left(\mu e\right)=m\left(e\right)$. As a consequence, the finite graph $\mathfrak{X}$ can be endowed with a measure which reflects the periodicity of m, namely $m\left({\mathfrak{x}}_j\right):=m\left(x_j\right)$, and $m\left(\mathfrak{e}\right):=m\left(e\right)$ if $\omega \left(e\right)=\mathfrak{e}$. In this framework, the degree function ${deg}_m$ is clearly bounded, and one has

$${deg}_m\left(x\right)=\sum _{e\in A{\left(X\right)}_x}{\displaystyle \frac{m\left(e\right)}{m\left(x\right)}}=\sum _{\mathfrak{e}\in A{\left(\mathfrak{X}\right)}_{\mathfrak{x}}}{\displaystyle \frac{m\left(\mathfrak{e}\right)}{m\left(\mathfrak{x}\right)}}={deg}_m\left(\mathfrak{x}\right)$$

where the last expression defines a function from $V\left(\mathfrak{X}\right)$ to $(0,\infty )$.

In the framework introduced in the previous paragraph, we also consider the set of compactly supported functions on $V\left(X\right)$ defined as:

$$C_c\left(X\right):=\left\{f:V\left(X\right)\to \mathbb{C}\mid f\left(x\right)=0\mathrm{except}\mathrm{for}\mathrm{a}\mathrm{finite}\mathrm{number}\mathrm{of}x\in V\left(X\right)\right\}.$$

Similarly, we define the Hilbert space of functions on $V\left(X\right)$

$${\ell }^2(X,m):=\left\{f:V\left(X\right)\to \mathbb{C}\mid {\parallel f\parallel }^2:=\sum _{x\in V\left(X\right)}m\left(x\right){\left|f\left(x\right)\right|}^2<\infty \right\}$$

endowed with the scalar product defined for any $f,g\in {\ell }^2(X,m)$ by

$$⟨f,g⟩:=\sum _{x\in V\left(X\right)}m\left(x\right)\overline{f\left(x\right)}g\left(x\right).$$

Clearly, the set $C_c\left(X\right)$ is dense in ${\ell }^2(X,m)$.

Definition 3.

Let $(X,\mathfrak{X},\omega ,\Gamma )$ be a topological crystal endowed with a periodic measure m. For any function $f\in C_c\left(X\right)$, the Laplace operator $\Delta (X,m)$ is defined on f and for any $x\in V\left(X\right)$ by

$$\begin{array}{cc}\hfill [\Delta (X,m\left)f\right]\left(x\right)& :=\sum _{e\in A{\left(X\right)}_x}{\displaystyle \frac{m\left(e\right)}{m\left(x\right)}}\left(f\left(t\left(e\right)\right)-f\left(x\right)\right)\hfill \\ \hfill & =\sum _{e\in A{\left(X\right)}_x}{\displaystyle \frac{m\left(e\right)}{m\left(x\right)}}f\left(t\left(e\right)\right)-{deg}_m\left(x\right)f\left(x\right).\hfill \end{array}$$

Since the degree function is bounded, it is well established that this operator $\Delta (X,m)$ extends continuously to a bounded and self-adjoint operator on the Hilbert space ${\ell }^2(X,m)$. We refer, for example, to ([12], Theorem 2.4) for the boundedness of $\Delta (X,m)$, and to ([13], Lemma 4.7) for its self-adjointness.

Our next aim will be to study this operator, particularly its spectrum. For that purpose, it is necessary to introduce three unitary transforms, and look at the representation of the Laplace operator through these transforms. The first of these unitary transforms is based on the following identification: $V\left(X\right)\cong {\mathbb{Z}}^d\times V\left(\mathfrak{X}\right)$. Indeed, for every $x\in V\left(X\right)$, we give a coordinate with respect to the unit cell U, namely

$${\phi }_U:V\left(X\right)\to {\mathbb{Z}}^d\times V\left(\mathfrak{X}\right)$$

with ${\phi }_U\left(x\right)=(\mu ,{\mathfrak{x}}_j)$ where, for $x\in V\left(X\right)$, we set $\mu :=\lfloor x\rfloor \in {\mathbb{Z}}^d$ and ${\mathfrak{x}}_j$ given by $\omega \left(x\right)={\mathfrak{x}}_j$. We observe that ${\phi }_U$ is bijective. For every $(\mu ,{\mathfrak{x}}_j)\in {\mathbb{Z}}^d\times V\left(\mathfrak{X}\right)$, we can set ${\phi }_U^{-1}(\mu ,{\mathfrak{x}}_j):=\mu x_j\in V\left(X\right)$, and it defines the inverse of ${\phi }_U$. Thus, the set $V\left(X\right)$ and the set ${\mathbb{Z}}^d\times V\left(\mathfrak{X}\right)$ are in bijection. We can now introduce these unitary transforms one by one.

• The unitary transform ${\mathcal{U}}_U:{\ell }^2(X,m)\to {\ell }^2\left({\mathbb{Z}}^d;{\ell }^2\left(\mathfrak{X}\right)\right)$.

For any function $h:{\mathbb{Z}}^d\to {\ell }^2\left(\mathfrak{X}\right)$, we use the notation

$$\left[h\right(\mu \left)\right]\left(\mathfrak{x}\right)=:h(\mu ,\mathfrak{x})$$

for any $\mu \in {\mathbb{Z}}^d$ and $\mathfrak{x}\in \mathfrak{X}$. We then set

$$\begin{array}{cc}\hfill {\ell }^2\left({\mathbb{Z}}^d;{\ell }^2\left(\mathfrak{X}\right)\right)& :=\left\{h:{\mathbb{Z}}^d\to {\ell }^2\left(\mathfrak{X}\right)\mid \sum _{\mu \in {\mathbb{Z}}^d}{\left|\right|h\left(\mu \right)\left|\right|}_{{\ell }^2\left(\mathfrak{X}\right)}^2<\infty \right\}\hfill \\ \hfill & =\left\{h:{\mathbb{Z}}^d\to {\ell }^2\left(\mathfrak{X}\right)\mid \sum _{\mu \in {\mathbb{Z}}^d}\sum _{\mathfrak{x}\in \mathfrak{X}}m\left(\mathfrak{x}\right){\left|h(\mu ,\mathfrak{x})\right|}^2<\infty \right\}.\hfill \end{array}$$

and define

$${\mathcal{U}}_U:{\ell }^2(X,m)\to {\ell }^2\left({\mathbb{Z}}^d;{\ell }^2\left(\mathfrak{X}\right)\right)$$

by

$$\left[{\mathcal{U}}_{U}f\right](\mu ,{\mathfrak{x}}_j):=f\left({\phi }_U^{-1}(\mu ,{\mathfrak{x}}_j)\right)=f\left(\mu x_j\right)$$

for $f\in {\ell }^2(X,m)$, $\mu \in {\mathbb{Z}}^d$ and ${\mathfrak{x}}_j\in \mathfrak{X}$. Since $m\left({\mathfrak{x}}_j\right):=m\left(x_j\right)$, it is easily observed that this transform defines a unitary map.

• The Fourier transform $\mathcal{F}:{\ell }^2\left({\mathbb{Z}}^d;{\ell }^2\left(\mathfrak{X}\right)\right)$ to $L^2\left({\mathbb{T}}^d;{\ell }^2\left(\mathfrak{X}\right)\right)$.

Let $\mathbb{T}$ denote the set of equivalence classes $\mathbb{R}/\mathbb{Z}$, namely the set $[0,1)$ with the addition of modulo 1. We also let ${\mathbb{T}}^d\equiv {(\mathbb{R}/\mathbb{Z})}^d$ denote the d-torus. For any $\mu \in {\mathbb{Z}}^d$ and $\xi \in {\mathbb{T}}^d$, we also set $\xi ·\mu$ by $\xi ·\mu :={\sum }_{j=1}^d{\xi }_j{\mu }_j\in \mathbb{R}$. We then define the Fourier transform for arbitrary $h\in {\ell }^1\left({\mathbb{Z}}^d;{\ell }^2\left(\mathfrak{X}\right)\right)\cap {\ell }^2\left({\mathbb{Z}}^d;{\ell }^2\left(\mathfrak{X}\right)\right)$ and for all $\xi \in {\mathbb{T}}^d$ by

$$\left[\mathcal{F}h\right]\left(\xi \right)=\sum _{\mu \in {\mathbb{Z}}^d}{e}^{-2\pi i\xi ·\mu }h\left(\mu \right)$$

Its inverse is given for $u\in C\left({\mathbb{T}}^d;{\ell }^2\left(\mathfrak{X}\right)\right)$ and $\mu \in {\mathbb{Z}}^d$ by

$$\left[{\mathcal{F}}^{*}u\right]\left(\mu \right)={\int }_{{\mathbb{T}}^d}{e}^{2\pi i\xi ·\mu }u\left(\xi \right)\mathrm{d}\xi$$

with $\mathrm{d}\xi$ the normalized measure on ${\mathbb{T}}^d$. It is known that the Fourier transform extends to a unitary map from ${\ell }^2\left({\mathbb{Z}}^d\right)$ to $L^2\left({\mathbb{T}}^d\right)$; see, for example, ([14], Theorem 4.26). In our setting, it means that $\mathcal{F}$ extends to a unitary transform from ${\ell }^2\left({\mathbb{Z}}^d;{\ell }^2\left(\mathfrak{X}\right)\right)$ to $L^2\left({\mathbb{T}}^d;{\ell }^2\left(\mathfrak{X}\right)\right)$.

• The identification operator $\mathcal{I}:L^2\left({\mathbb{T}}^d;{\ell }^2\left(\mathfrak{X}\right)\right)\to L^2({\mathbb{T}}^d;{\mathbb{C}}^n)$.

We finally define the map

$$\mathcal{I}:L^2\left({\mathbb{T}}^d;{\ell }^2\left(\mathfrak{X}\right)\right)\to L^2({\mathbb{T}}^d;{\mathbb{C}}^n)$$

by

$$\left[\mathcal{I}g\right]\left(\xi \right):={\left(m_0{\left({\mathfrak{x}}_1\right)}^{\frac{1}{2}}g(\xi ,{\mathfrak{x}}_1),m_0{\left({\mathfrak{x}}_2\right)}^{\frac{1}{2}}g(\xi ,{\mathfrak{x}}_2),\dots ,m_0{\left({\mathfrak{x}}_n\right)}^{\frac{1}{2}}g(\xi ,{\mathfrak{x}}_n)\right)}^T.$$

for all $g\in L^2\left({\mathbb{T}}^d;{\ell }^2\left(\mathfrak{X}\right)\right)$ and $\xi \in {\mathbb{T}}^d$. Here and in the sequel, the superscript T denotes the transpose of a vector or of a matrix. This transform is again unitary, as it can be easily checked, and allows us to use the standard scalar product in ${\mathbb{C}}^n$.

The product of the three unitary transforms which we defined above, namely

$$\mathcal{I}\mathcal{F}{\mathcal{U}}_U:{\ell }^2(X,m)\to L^2({\mathbb{T}}^d;{\mathbb{C}}^n),$$

is a unitary transform acting on the Hilbert space ${\ell }^2(X,m)$. This space is the natural ${\ell }^2$-space related to topological crystals, but $L^2({\mathbb{T}}^d;{\mathbb{C}}^n)$ is a more convenient space for further investigations. Thus, we shall now look at the expression of the Laplace operator through these unitary transforms. For this, we consider suitable $g\in L^2({\mathbb{T}}^d;{\mathbb{C}}^n)$, and let $j\in \{1,\dots ,n\}$ and $\xi \in {\mathbb{T}}^d$. In the following computation, we shall also set $f:={\mathcal{U}}_U^{*}{\mathcal{F}}^{*}{\mathcal{I}}^{*}g$. Then, one has

$$\begin{array}{cc}\hfill & {[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\Delta (X,m){\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\right]}^{*}g]}_j\left(\xi \right)\hfill \\ \hfill & =m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}[\mathcal{F}{\mathcal{U}}_U\Delta (X,m)f](\xi ,{\mathfrak{x}}_j)\hfill \\ \hfill & =m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}\sum _{\mu \in {\mathbb{Z}}^d}{e}^{-2\pi i\xi ·\mu }[{\mathcal{U}}_U\Delta (X,m)f](\mu ,{\mathfrak{x}}_j)\hfill \\ \hfill & =m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}\sum _{\mu \in {\mathbb{Z}}^d}{e}^{-2\pi i\xi ·\mu }[\Delta (X,m)f]\left(\mu x_j\right)\hfill \\ \hfill & =m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}\sum _{\mu \in {\mathbb{Z}}^d}{e}^{-2\pi i\xi ·\mu }\sum _{e\in A{\left(X\right)}_{\mu x_j}}{\displaystyle \frac{m\left(e\right)}{m\left(\mu x_j\right)}}\left(f\left(t\left(e\right)\right)-f\left(\mu x_j\right)\right)\hfill \\ \hfill & =m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}\sum _{\mu \in {\mathbb{Z}}^d}{e}^{-2\pi i\xi ·\mu }\sum _{e\in A{\left(X\right)}_{\mu x_j}}{\displaystyle \frac{m\left(e\right)}{m\left(x_j\right)}}\left(f\left(t\left(e\right)\right)-f\left(\mu x_j\right)\right)\hfill \\ \hfill & =\sum _{\mu \in {\mathbb{Z}}^d}{e}^{-2\pi i\xi ·\mu }\sum _{\mathfrak{e}\in A{\left(\mathfrak{X}\right)}_{{\mathfrak{x}}_j}}{\displaystyle \frac{m\left(\mathfrak{e}\right)}{m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}}}f\left(\left(\mu +\eta \left(\mathfrak{e}\right)\right)\widehat{t\left(\mathfrak{e}\right)}\right)\hfill \\ \hfill & -m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}\sum _{\mu \in {\mathbb{Z}}^d}{e}^{-2\pi i\xi ·\mu }\sum _{\mathfrak{e}\in A{\left(\mathfrak{X}\right)}_{{\mathfrak{x}}_j}}{\displaystyle \frac{m\left(\mathfrak{e}\right)}{m\left({\mathfrak{x}}_j\right)}}f\left(\mu x_j\right)\hfill \\ \hfill & =\sum _{\mathfrak{e}\in A{\left(\mathfrak{X}\right)}_{{\mathfrak{x}}_j}}{\displaystyle \frac{m\left(\mathfrak{e}\right)}{m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}}}\sum _{\mu \in {\mathbb{Z}}^d}{e}^{-2\pi i\xi ·\mu }f\left(\left(\mu +\eta \left(\mathfrak{e}\right)\right)\widehat{t\left(\mathfrak{e}\right)}\right)-deg\left({\mathfrak{x}}_j\right){\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_{U}f\right]}_j\left(\xi \right)\hfill \\ \hfill & =\sum _{\mathfrak{e}\in A{\left(\left(X\right)\right)}_{{\mathfrak{x}}_j}}{\displaystyle \frac{m\left(\mathfrak{e}\right)}{m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}}}{e}^{2\pi i\xi ·\eta \left(\mathfrak{e}\right)}\sum _{\mu \in {\mathbb{Z}}^d}{e}^{-2\pi i\xi ·\left(\mu +\eta \left(\mathfrak{e}\right)\right)}f\left(\left(\mu +\eta \left(\mathfrak{e}\right)\right)\widehat{t\left(\mathfrak{e}\right)}\right)-deg\left({\mathfrak{x}}_j\right)g_j\left(\xi \right)\hfill \\ \hfill & =\sum _{\mathfrak{e}\in A{\left(\mathfrak{X}\right)}_{{\mathfrak{x}}_j}}{\displaystyle \frac{m\left(\mathfrak{e}\right)}{m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}m{\left(t\left(\mathfrak{e}\right)\right)}^{\frac{1}{2}}}}{e}^{2\pi i\xi ·\eta \left(\mathfrak{e}\right)}m{\left(t\left(\mathfrak{e}\right)\right)}^{\frac{1}{2}}\sum _{{\mu }^{\prime }\in {\mathbb{Z}}^d}{e}^{-2\pi i\xi ·{\mu }^{\prime }}f\left({\mu }^{\prime }\widehat{t\left(\mathfrak{e}\right)}\right)-deg\left({\mathfrak{x}}_j\right)g_j\left(\xi \right)\hfill \\ \hfill & =\sum _{k=1}^n\sum _{\mathfrak{e}\in A\left(\mathfrak{X}\right),o\left(\mathfrak{e}\right)={\mathfrak{x}}_j,t\left(\mathfrak{e}\right)={\mathfrak{x}}_k}{\displaystyle \frac{m\left(\mathfrak{e}\right)}{m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}m{\left({\mathfrak{x}}_k\right)}^{\frac{1}{2}}}}{e}^{2\pi i\xi ·\eta \left(\mathfrak{e}\right)}m{\left({\mathfrak{x}}_k\right)}^{\frac{1}{2}}\sum _{{\mu }^{\prime }\in {\mathbb{Z}}^d}{e}^{-2\pi i\xi ·{\mu }^{\prime }}f\left({\mu }^{\prime }{x}_k\right)\hfill \\ \hfill & -deg\left({\mathfrak{x}}_j\right)g_j\left(\xi \right)\hfill \\ \hfill & =\sum _{k=1}^n\sum _{\mathfrak{e}\in A\left(\mathfrak{X}\right),o\left(\mathfrak{e}\right)={\mathfrak{x}}_j,t\left(\mathfrak{e}\right)={\mathfrak{x}}_k}{\displaystyle \frac{m\left(\mathfrak{e}\right)}{m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}m{\left({\mathfrak{x}}_k\right)}^{\frac{1}{2}}}}{e}^{2\pi i\xi ·\eta \left(\mathfrak{e}\right)}{g}_k\left(\xi \right)-deg\left({\mathfrak{x}}_j\right)g_j\left(\xi \right).\hfill \end{array}$$

By looking carefully at this expression, we observe that

$$[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\Delta (X,m){\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\right]}^{*}g]\left(\xi \right)=h_U\left(\xi \right)g\left(\xi \right),$$

where $h_U\left(\xi \right)$ is $n\times n$ matrix with the entries given by

$${\left(h_U\right)}_{jk}\left(\xi \right):=\sum _{\mathfrak{e}\in A\left(\mathfrak{X}\right),o\left(\mathfrak{e}\right)={\mathfrak{x}}_j,t\left(\mathfrak{e}\right)={\mathfrak{x}}_k}{\displaystyle \frac{m\left(\mathfrak{e}\right)}{m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}m{\left({\mathfrak{x}}_k\right)}^{\frac{1}{2}}}}{e}^{2\pi i\xi ·\eta \left(\mathfrak{e}\right)}-deg\left({\mathfrak{x}}_j\right){\delta }_{jk}.$$

(2)

Note that this expression already appeared in ([3], Proposition 4.7) and in ([4], Proposition 3.2), but the details of the computation were not shown there. We also emphasize in the notation that the matrix $h_U$ is computed through a transform which does depend on U.

4. Independence of Spectral Properties with Respect to Reference Systems

In this section, we discuss firstly the change in reference systems of topological crystals, then provide the spectral analysis of the Laplace operator on topological crystals, and conclude that the spectral properties are independent from the choice of reference systems.

In Section 2, we introduce the notion of a unit cell and various maps in topological crystals. The choice of the unit cell was arbitrary, but all the subsequent constructions depended on this initial choice. In other words, the unit cell worked as a reference system for many subsequent concepts. Up to now, in [1,4], the authors dealt with the case of a single unit cell. We investigate here what happens when we choose another unit cell. The topological crystal structure $(X,\mathfrak{X},\omega ,\Gamma )$ remains unchanged.

4.1. Independence with Respect to the Unit Cell

Assume that we choose two unit cells $U=\{x_1,\dots ,x_n\}$ and $U^{\prime }=\{x_1^{\prime },\dots ,x_n^{\prime }\}$ in a graph X, where $x_j$ and $x_j^{\prime }$ belong to the same $\Gamma$-orbit, that is

$$\omega \left(x_j\right)=\omega \left(x_j^{\prime }\right)={\mathfrak{x}}_{j}j=1,\dots ,n.$$

We consider two lifts and two floor functions based on U and $U^{\prime }$, namely

$$\widehat{·}:\mathfrak{X}\to U\mathrm{and}{\widehat{·}}^{\prime }:\mathfrak{X}\to U^{\prime },$$

together with

$$\lfloor ·\rfloor :V\left(X\right)\to \Gamma \mathrm{and}{\lfloor ·\rfloor }^{\prime }:V\left(X\right)\to \Gamma$$

such that $\lfloor x\rfloor \widehat{\omega }\left(x\right)=x$ and ${\lfloor x\rfloor }^{\prime }{\widehat{\omega }}^{\prime }\left(x\right)=x$. The following statement provides the link between these maps.

Lemma 5.

In the framework introduced above, one has

$${\lfloor x\rfloor }^{\prime }=\lfloor x\rfloor {\lfloor {\widehat{\omega }}^{\prime }\left(x\right)\rfloor }^{-1}$$

for any $x\in V\left(X\right)$.

Proof. 

Assume that $\omega \left(x\right)={\mathfrak{x}}_j$, that is, $\widehat{\omega }\left(x\right)=x_j$ for some fixed $x\in V\left(X\right)$. Then, one has

$$\begin{array}{cc}\hfill x& =\lfloor x\rfloor x_j\mathrm{with}\mathrm{respect}\mathrm{to}\mathrm{the}\mathrm{unit}\mathrm{cell}U=\{x_1,\dots ,x_n\},\hfill \\ \hfill x& ={\lfloor x\rfloor }^{\prime }{x}_j^{\prime }\mathrm{with}\mathrm{respect}\mathrm{to}\mathrm{the}\mathrm{unit}\mathrm{cell}{U}^{\prime }=\{x_1^{\prime },\dots ,x_n^{\prime }\}.\hfill \end{array}$$

On the other hand, since $x_j^{\prime }=\lfloor x_j^{\prime }\rfloor \widehat{\omega }\left(x_j^{\prime }\right)=\lfloor x_j^{\prime }\rfloor x_j$, it follows that

$$x={\lfloor x\rfloor }^{\prime }{x}_j^{\prime }={\lfloor x\rfloor }^{\prime }\left(\lfloor x_j^{\prime }\rfloor x_j\right)={\lfloor x\rfloor }^{\prime }\lfloor x_j^{\prime }\rfloor x_j.$$

Thus, from the equalities

$$x=\lfloor x\rfloor x_j={\lfloor x\rfloor }^{\prime }\lfloor x_j^{\prime }\rfloor x_j,$$

one infers that $\lfloor x\rfloor ={\lfloor x\rfloor }^{\prime }\lfloor x_j^{\prime }\rfloor$ or equivalently ${\lfloor x\rfloor }^{\prime }=\lfloor x\rfloor {\lfloor x_j^{\prime }\rfloor }^{-1}$. Without reference to j, it means

$${\lfloor x\rfloor }^{\prime }=\lfloor x\rfloor {\lfloor {\widehat{\omega }}^{\prime }\left(x\right)\rfloor }^{-1}\mathrm{for}\mathrm{any}x\in V\left(X\right),$$

as claimed. □

In the framework introduced above, we now consider two index maps based on U and $U^{\prime }$, respectively, namely for any $e\in A\left(X\right)$

$$\eta \left(e\right):=\lfloor t\left(e\right)\rfloor {\lfloor o\left(e\right)\rfloor }^{-1}\mathrm{and}{\eta }^{\prime }\left(e\right):={\lfloor t\left(e\right)\rfloor }^{\prime }{\lfloor o\left(e\right)\rfloor }^{\prime -1}.$$

The relation between these two index maps is provided in the following statement:

Lemma 6.

For any $e\in A\left(X\right)$, one has

$${\eta }^{\prime }\left(e\right)=\lfloor {\widehat{\omega }}^{\prime }\left(o\left(e\right)\right)\rfloor \eta \left(e\right){\lfloor {\widehat{\omega }}^{\prime }\left(t\left(e\right)\right)\rfloor }^{-1}.$$

Proof. 

By using Lemma 5, the Abelian property of $\Gamma$, and the definitions of the index maps, one has

$$\begin{array}{cc}\hfill {\eta }^{\prime }\left(e\right)& ={\lfloor t\left(e\right)\rfloor }^{\prime }{\lfloor o\left(e\right)\rfloor }^{\prime -1}\hfill \\ \hfill & =\left(\lfloor t\left(e\right)\rfloor {\lfloor {\widehat{\omega }}^{\prime }\left(t\left(e\right)\right)\rfloor }^{-1}\right){\left(\lfloor o\left(e\right)\rfloor {\lfloor {\widehat{\omega }}^{\prime }\left(o\left(e\right)\right)\rfloor }^{-1}\right)}^{-1}\hfill \\ \hfill & =\lfloor {\widehat{\omega }}^{\prime }\left(o\left(e\right)\right)\rfloor \lfloor t\left(e\right)\rfloor {\lfloor o\left(e\right)\rfloor }^{-1}{\lfloor {\widehat{\omega }}^{\prime }\left(t\left(e\right)\right)\rfloor }^{-1}\hfill \\ \hfill & =\lfloor {\widehat{\omega }}^{\prime }\left(o\left(e\right)\right)\rfloor \eta \left(e\right){\lfloor {\widehat{\omega }}^{\prime }\left(t\left(e\right)\right)\rfloor }^{-1},\hfill \end{array}$$

as stated. □

Let us now move to spectral theory, and recall a few definitions and properties related to the spectrum of a bounded operator.

Let A be a bounded operator in $\mathcal{H}$. The resolvent set of A is defined as the set $\rho \left(A\right)$ of all $z\in \mathbb{C}$ for which $A-z\equiv A-zI$ is invertible with ${(A-z)}^{-1}\in \mathcal{B}\left(\mathcal{H}\right)$. This means that $z\in \mathbb{C}$ belongs to $\rho \left(A\right)$ if and only if

• $\mathcal{N}(A-z)=\left\{0\right\}$ (The null space of A is empty);
• $\mathcal{R}(A-z)=\mathcal{H}$ (The range of A is $\mathcal{H}$);
• ${(A-z)}^{-1}$ is bounded.

The complementary set of $\rho \left(A\right)$ in $\mathbb{C}$ is the spectrum $\sigma \left(A\right)$ of A:

$$\sigma \left(A\right)=\mathbb{C}\setminus \rho \left(A\right).$$

The spectrum of an operator A contains all eigenvalues of A. Apart from the eigenvalues (if any), the spectrum of A contains $z\in \mathbb{C}$ for which $A-z$ is invertible but with unbounded or not densely defined as inverse. We finally recall that the spectrum of any self-adjoint operator is real, and that this set is invariant if the operator is conjugated by any unitary transform.

Let us look at the implication of the previous remark for our investigations. In our setting, since $\mathcal{I}\mathcal{F}{\mathcal{U}}_U$ is a unitary transform, it means that the equality

$$\sigma \left(\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\right]\Delta (X,m){\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\right]}^{*}\right)=\sigma \left(\Delta (X,m)\right)$$

(3)

holds. In particular, we can use the explicit formula obtained for

$$\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\right]\Delta (X,m){\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\right]}^{*}$$

in order to compute the spectrum of the Laplace operator $\Delta (X,m)$. In fact, (3) says even more that in the product $\mathcal{I}\mathcal{F}{\mathcal{U}}_U$, we observe that only ${\mathcal{U}}_U$ depends on the choice of a unit cell U, not $\mathcal{F}$ and $\mathcal{I}$. Accordingly, let us denote by ${\mathcal{U}}_{U^{\prime }}$ the unitary transform defined for another unit cell $U^{\prime }$. Then, one has

$$\sigma \left(\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\right]\Delta (X,m){\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\right]}^{*}\right)=\sigma \left(\Delta (X,m)\right)=\sigma \left(\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_{U^{\prime }}\right]\Delta (X,m){\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_{U^{\prime }}\right]}^{*}\right).$$

(4)

Hence, the spectrum of the operator $\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\right]\Delta (X,m){\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\right]}^{*}$ that we are going to compute is independent from the choice of a unit cell, even if the explicit expression of this operator depends on the choice of the unit cell U. We shall provide a stronger statement subsequently.

Let us now compute the spectrum of the operator $\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\right]\Delta (X,m){\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\right]}^{*}$. We firstly recall that

$$\left[\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\right]\Delta (X,m){\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\right]}^{*}g\right]\left(\xi \right)=h_U\left(\xi \right)g\left(\xi \right),$$

where the entries of the matrix $h_U\left(\xi \right)\in M_n\left(\mathbb{C}\right)$ are given in (2). This equality means that the image of $\Delta (X,m)$ in $L^2({\mathbb{T}}^d;{\mathbb{C}}^n)$ through the unitary transform defined by $\mathcal{I}\mathcal{F}{\mathcal{U}}_U$ is equal to a matrix-valued multiplication operator $H_U$. Note that in our case, the operator $H_U$ is defined by a continuous function $h_U:{\mathbb{T}}^d\to M_n\left(\mathbb{C}\right)$, which is sometimes called the symbol of $H_U$.

For the computation of the spectrum of a multiplication operator, we need the following statement adapted to our setting:

Lemma 7

([15], Proposition 2). Let $d,n\in \mathbb{N}$, $\mathcal{H}:=L^2({\mathbb{T}}^d;{\mathbb{C}}^n)$ and consider a function $h\in C\left({\mathbb{T}}^d;M_n\left(\mathbb{C}\right)\right)$. The spectrum of the corresponding multiplication operator H in $\mathcal{H}$ satisfies

$$\sigma \left(H\right)=\bigcup _{\xi \in {\mathbb{T}}^d}\sigma \left(h\left(\xi \right)\right).$$

This result means that the spectrum of a multiplication operator (with continuous symbol) can be obtained as a continuous union of eigenvalues of $n\times n$ matrices. Thus, if we collect the information obtained so far, we obtain:

Proposition 1.

The following equality holds:

$$\sigma \left(\Delta (X,m)\right)=\bigcup _{\xi \in {\mathbb{T}}^d}\sigma \left({\left(\sum _{\mathfrak{e}\in A\left(\mathfrak{X}\right),o\left(\mathfrak{e}\right)={\mathfrak{x}}_j,t\left(\mathfrak{e}\right)={\mathfrak{x}}_k}{\displaystyle \frac{m\left(\mathfrak{e}\right)}{m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}m{\left({\mathfrak{x}}_k\right)}^{\frac{1}{2}}}}{e}^{2\pi i\xi ·\eta \left(\mathfrak{e}\right)}-deg\left({\mathfrak{x}}_j\right){\delta }_{jk}\right)}_{jk}\right).$$

Proof. 

We see by the paragraph preceding Lemma 7, that

$$\sigma \left(\Delta (X,m)\right)=\sigma \left(\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\right]\Delta (X,m){\left[\mathcal{I}\mathcal{F}{\mathcal{U}}_U\right]}^{*}\right)=\sigma \left(H_U\right).$$

Then, by Lemma 7, one infers that

$$\sigma \left(H_U\right)=\bigcup _{\xi \in {\mathbb{T}}^d}\sigma \left(h_U\left(\xi \right)\right),$$

leading to the statement with the $jk$-entry of the matrix $h_U\left(\xi \right)$. □

Let us emphasize that the previous result was computed with respect to a unit cell U. In the expression for the matrix $h_U\left(\xi \right)$, this dependence appears only in the index map $\eta$. If we compute this matrix with respect to another unit cell, we obtain the following relation:

Theorem 1.

Let $h_U\left(\xi \right)$ and $h_{U^{\prime }}\left(\xi \right)$ denote the matrices computed with respect to the unit cells $U=\{x_1,\dots ,x_n\}$ and $U^{\prime }=\{x_1^{\prime },\dots ,x_n^{\prime }\}$, respectively. Then, the equality

$$\sigma \left(h_U\left(\xi \right)\right)=\sigma \left(h_{U^{\prime }}\left(\xi \right)\right)$$

(5)

holds for any $\xi \in {\mathbb{T}}^d$.

Proof. 

As shown in Lemma 6, one has

$$\begin{array}{cc}\hfill & {\left(h_{U^{\prime }}\right)}_{jk}\left(\xi \right)\hfill \\ \hfill & =\sum _{\mathfrak{e}\in A\left(\mathfrak{X}\right),o\left(\mathfrak{e}\right)={\mathfrak{x}}_j,t\left(\mathfrak{e}\right)={\mathfrak{x}}_k}{\displaystyle \frac{m\left(\mathfrak{e}\right)}{m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}m{\left({\mathfrak{x}}_k\right)}^{\frac{1}{2}}}}{e}^{2\pi i\xi ·{\eta }^{\prime }\left(\mathfrak{e}\right)}-deg\left({\mathfrak{x}}_j\right){\delta }_{jk}\hfill \\ \hfill & =\sum _{\mathfrak{e}\in A\left(\mathfrak{X}\right),o\left(\mathfrak{e}\right)={\mathfrak{x}}_j,t\left(\mathfrak{e}\right)={\mathfrak{x}}_k}{\displaystyle \frac{m\left(\mathfrak{e}\right)}{m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}m{\left({\mathfrak{x}}_k\right)}^{\frac{1}{2}}}}{e}^{2\pi i\xi ·\left(\lfloor {\widehat{o\left(\mathfrak{e}\right)}}^{\prime }\rfloor \eta \left(\mathfrak{e}\right){\lfloor {\widehat{t\left(\mathfrak{e}\right)}}^{\prime }\rfloor }^{-1}\right)}-deg\left({\mathfrak{x}}_j\right){\delta }_{jk}\hfill \\ \hfill & =e^{2\pi i\xi ·\lfloor x_j^{\prime }\rfloor }\left(\sum _{\mathfrak{e}\in A\left(\mathfrak{X}\right),o\left(\mathfrak{e}\right)={\mathfrak{x}}_j,t\left(\mathfrak{e}\right)={\mathfrak{x}}_k}{\displaystyle \frac{m\left(\mathfrak{e}\right)}{m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}m{\left({\mathfrak{x}}_k\right)}^{\frac{1}{2}}}}{e}^{2\pi i\xi ·\eta \left(\mathfrak{e}\right)}-deg\left({\mathfrak{x}}_j\right){\delta }_{jk}\right)e^{-2\pi i\xi ·\lfloor x_k^{\prime }\rfloor }\hfill \\ \hfill & =e^{2\pi i\xi ·\lfloor x_j^{\prime }\rfloor }{\left(h_U\right)}_{jk}\left(\xi \right)e^{-2\pi i\xi ·\lfloor x_k^{\prime }\rfloor }.\hfill \end{array}$$

(6)

Thus, if we define the unitary $n\times n$ matrices

$$V\left(\xi \right):=\left(\begin{array}{cccc}{e}^{2\pi i\xi ·\lfloor x_1^{\prime }\rfloor }& 0& \cdots & 0\\ 0& e^{2\pi i\xi ·\lfloor x_2^{\prime }\rfloor }& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & e^{2\pi i\xi ·\lfloor x_n^{\prime }\rfloor }\end{array}\right),$$

then we infer from (6)

$$h_{U^{\prime }}\left(\xi \right)=V\left(\xi \right)h_U\left(\xi \right)V{\left(\xi \right)}^{-1}.$$

Since $V\left(\xi \right)$ is a unitary matrix, we deduce from the invariance of the spectrum through unitary conjugation that

$$\sigma \left(h_{U^{\prime }}\left(\xi \right)\right)=\sigma \left(V\left(\xi \right)h_U\left(\xi \right)V{\left(\xi \right)}^{-1}\right)=\sigma \left(h_U\left(\xi \right)\right),$$

leading to the statement, for any $\xi \in {\mathbb{T}}^d$. □

Remark 1.

With the notation of this Theorem, let us emphasize that the content of Theorem 1 is stronger than the equality $\sigma \left(H_U\right)=\sigma \left(H_{U^{\prime }}\right)$, which directly follows from (4). Indeed, one has

$$\sigma \left(H_U\right)=\sigma \left(H_{U^{\prime }}\right)⟺\bigcup _{\xi \in {\mathbb{T}}^d}\sigma \left(h_U\left(\xi \right)\right)=\bigcup _{\xi \in {\mathbb{T}}^d}\sigma \left(h_{U^{\prime }}\left(\xi \right)\right)$$

which are equalities of sets. On the other hand, the equality (5) is an equality for each ξ, which clearly implies the equality for the continuous union over all ξ. Thus, we have proven the strongest version of the invariance of the spectrum with respect to a change in reference system; it is an equality for each fixed ξ, and not only for the continuous union over all ξ.

4.2. Independence with Respect to Generators

So far, we have constantly kept the same set of generators for $\Gamma$. We investigate here what happens when we choose other generators for $\Gamma$, letting the topological crystal structure $(X,\mathfrak{X},\omega ,\Gamma )$ and a given unit cell unchanged. Let $({\Gamma }_1,\dots ,{\Gamma }_d)$ and $({\Gamma }_1^{\prime },\dots ,{\Gamma }_d^{\prime })$ be two distinct sets of generators for $\Gamma$. Then, there are ${\alpha }_{jk}\in \mathbb{Z},j,k=1,\dots ,d$, such that

$${\Gamma }_j={\Pi }_{k=1}^d{\left({\Gamma }_k^{\prime }\right)}^{{\alpha }_{jk}}j=1,\dots ,d.$$

Note that we use the multiplicative notations for transformations. Then, a floor function at any $x\in V\left(X\right)$ is expressed with respect to each set of generators, respectively, as follows:

$$\begin{array}{cc}\hfill \lfloor x\rfloor & ={\Pi }_{j=1}^d{\left({\Gamma }_j\right)}^{{\mu }_j}\hfill \\ \hfill & ={\Pi }_{j=1}^d{\left({\Pi }_{k=1}^d{\left({\Gamma }_k^{\prime }\right)}^{{\alpha }_{jk}}\right)}^{{\mu }_j}\hfill \\ \hfill & ={\Pi }_{j=1}^d{\Pi }_{k=1}^d{\left({\Gamma }_k^{\prime }\right)}^{{\alpha }_{jk}{\mu }_j}\hfill \\ \hfill & ={\Pi }_{k=1}^d{\left({\Gamma }_k^{\prime }\right)}^{{\sum }_{j=1}^d{\alpha }_{jk}{\mu }_j}\hfill \\ \hfill & ={\Pi }_{k=1}^d{\left({\Gamma }_k^{\prime }\right)}^{{\left({\alpha }^T\mu \right)}_k}\hfill \\ \hfill & ={\Pi }_{k=1}^d{\left({\Gamma }_k^{\prime }\right)}^{{\mu }_k^{\prime }}\hfill \end{array}$$

where $\alpha ={\left({\alpha }_{jk}\right)}_{j,k=1}^d$ ${\mu }_k^{\prime }:={\sum }_{j=1}^d{\alpha }_{jk}{\mu }_j={\left({\alpha }^T\mu \right)}_k\in \mathbb{Z}$.

We identify the transformation group $\Gamma$ with ${\mathbb{Z}}^d$ and use the abuse of notations in the sequel. The coordinates of $\lfloor x\rfloor$ with respect to the generators $({\Gamma }_1,\dots ,{\Gamma }_d)$ and $({\Gamma }_1^{\prime },\dots ,{\Gamma }_d^{\prime })$ are given, respectively, by

$$\begin{array}{cc}\hfill \mu & :={({\mu }_1,\dots ,{\mu }_d)}^T\in {\mathbb{Z}}^d\mathrm{with}\mathrm{respect}\mathrm{to}({\Gamma }_1,\dots ,{\Gamma }_d),\hfill \\ \hfill {\mu }^{\prime }& :={({\mu }_1^{\prime },\dots ,{\mu }_d^{\prime })}^T={\left({\left({\alpha }^T\mu \right)}_1,\dots ,{\left({\alpha }^T\mu \right)}_d\right)}^T\hfill \\ \hfill & ={\alpha }^T\mu \in {\mathbb{Z}}^d\mathrm{with}\mathrm{respect}\mathrm{to}({\Gamma }_1^{\prime },\dots ,{\Gamma }_d^{\prime }).\hfill \end{array}$$

Next, we examine the effect of the change in the generators for the index of any $e\in A\left(X\right)$ as well. Let $\eta \left(e\right),{\eta }^{\prime }\left(e\right)$ be the indices with respect to the generators $({\Gamma }_1,\dots ,{\Gamma }_d)$ and $({\Gamma }_1^{\prime },\dots ,{\Gamma }_d^{\prime })$ for $\Gamma$, respectively. Set

$$\eta \left(e\right)=\lfloor t\left(e\right)\rfloor {\lfloor o\left(e\right)\rfloor }^{-1}=\mu -\nu ,$$

where $\mu =\lfloor t\left(e\right)\rfloor$, $\nu =\lfloor o\left(e\right)\rfloor \in {\mathbb{Z}}^d$ are the coordinates of the vertices with respect to $({\Gamma }_1,\dots ,{\Gamma }_d)$. With respect to the generators $({\Gamma }_1^{\prime },\dots ,{\Gamma }_d^{\prime })$ for $\Gamma$, one has

$${\eta }^{\prime }\left(e\right)={\mu }^{\prime }-{\nu }^{\prime }={\alpha }^T\mu -{\alpha }^T\nu ={\alpha }^T(\mu -\nu )={\alpha }^T\eta \left(e\right).$$

From this, we find for $\xi \in {\mathbb{T}}^d$ and ${\eta }^{\prime }\left(e\right)\in {\mathbb{Z}}^d$,

$$\xi ·{\eta }^{\prime }\left(e\right)=\xi ·{\alpha }^T\eta \left(e\right)={\xi }^T{\alpha }^T\eta \left(e\right)={\left(\alpha \xi \right)}^T\eta \left(e\right)=\left(\alpha \xi \right)·\eta \left(e\right).$$

We substitute this result in the equation of Proposition 1,

$$\begin{array}{cc}\hfill & \sigma \left(\Delta (X,m)\right)\hfill \\ \hfill & =\bigcup _{\xi \in {\mathbb{T}}^d}\sigma \left({\left(\sum _{\mathfrak{e}\in A\left(\mathfrak{X}\right),o\left(\mathfrak{e}\right)={\mathfrak{x}}_j,t\left(\mathfrak{e}\right)={\mathfrak{x}}_k}{\displaystyle \frac{m\left(\mathfrak{e}\right)}{m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}m{\left({\mathfrak{x}}_k\right)}^{\frac{1}{2}}}}{e}^{2\pi i\xi ·{\eta }^{\prime }\left(\mathfrak{e}\right)}-deg\left({\mathfrak{x}}_j\right){\delta }_{jk}\right)}_{jk}\right)\hfill \\ \hfill & =\bigcup _{\xi \in {\mathbb{T}}^d}\sigma \left({\left(\sum _{\mathfrak{e}\in A\left(\mathfrak{X}\right),o\left(\mathfrak{e}\right)={\mathfrak{x}}_j,t\left(\mathfrak{e}\right)={\mathfrak{x}}_k}{\displaystyle \frac{m\left(\mathfrak{e}\right)}{m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}m{\left({\mathfrak{x}}_k\right)}^{\frac{1}{2}}}}{e}^{2\pi i\left(\alpha \xi \right)·\eta \left(\mathfrak{e}\right)}-deg\left({\mathfrak{x}}_j\right){\delta }_{jk}\right)}_{jk}\right)\hfill \\ \hfill & =\bigcup _{{\alpha }^{-1}{\xi }^{\prime }\in {\mathbb{T}}^d}\sigma \left({\left(\sum _{\mathfrak{e}\in A\left(\mathfrak{X}\right),o\left(\mathfrak{e}\right)={\mathfrak{x}}_j,t\left(\mathfrak{e}\right)={\mathfrak{x}}_k}{\displaystyle \frac{m\left(\mathfrak{e}\right)}{m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}m{\left({\mathfrak{x}}_k\right)}^{\frac{1}{2}}}}{e}^{2\pi i{\xi }^{\prime }·\eta \left(\mathfrak{e}\right)}-deg\left({\mathfrak{x}}_j\right){\delta }_{jk}\right)}_{jk}\right)\hfill \\ \hfill & =\bigcup _{\xi \in {\mathbb{T}}^d}\sigma \left({\left(\sum _{\mathfrak{e}\in A\left(\mathfrak{X}\right),o\left(\mathfrak{e}\right)={\mathfrak{x}}_j,t\left(\mathfrak{e}\right)={\mathfrak{x}}_k}{\displaystyle \frac{m\left(\mathfrak{e}\right)}{m{\left({\mathfrak{x}}_j\right)}^{\frac{1}{2}}m{\left({\mathfrak{x}}_k\right)}^{\frac{1}{2}}}}{e}^{2\pi i\xi ·\eta \left(\mathfrak{e}\right)}-deg\left({\mathfrak{x}}_j\right){\delta }_{jk}\right)}_{jk}\right).\hfill \end{array}$$

We can thus discuss the effect of the change in the generators for $\Gamma$. Let $h_U\left(\xi \right)$ denote the matrix (2) computed with the family of generators $({\Gamma }_1,\dots ,{\Gamma }_d)$, and therefore with the function $\eta$, and let $h_U^{\prime }\left(\xi \right)$ denote the same matrix but computed with the family of generators $({\Gamma }_1^{\prime },\dots ,{\Gamma }_d^{\prime })$, and therefore with the function ${\eta }^{\prime }$. Then, with the previous discussion, one has obtained the following:

Theorem 2.

In the above framework,

$$\sigma \left(\Delta (X,m)\right)=\bigcup _{\xi \in {\mathbb{T}}^d}\sigma \left(h_U\left(\xi \right)\right)=\bigcup _{\xi \in {\mathbb{T}}^d}\sigma \left(h_U^{\prime }\left(\xi \right)\right),$$

and for any $\xi \in {\mathbb{T}}^d$

$$\sigma \left(h_U^{\prime }\left(\xi \right)\right)=\sigma \left(h_U\left(\alpha \xi \right)\right)$$

but in general

$$\sigma \left(h_U^{\prime }\left(\xi \right)\right)\ne \sigma \left(h_U\left(\xi \right)\right).$$

Let us stress that this situation is quite different from the one of the previous sections. For a change in unit cell, the spectrum of each matrix $h_U\left(\xi \right)$ is independent of the choice of the unit cell, while for the change in generators, this is no more true. Only the entire spectrum of $\Delta (X,m)$ is invariant under the change in the set of generators.

5. Conclusions

Let us finally summarize the content of this paper. The definition of the Laplace operator $\Delta (X,m)$ on any topological crystal does not depend on any arbitrary choice, but the computation of its spectral properties involves the choice of a unit cell and the choice of a set of generators for the transformation group $\Gamma$. The unit cell and the generators of the transformation group can be understood as the necessary reference frame for the computation of the spectral properties of the operator $\Delta (X,m)$. These choices are highly non-unique, and are reflected in the expression obtained for the matrix-valued function $h_U$. The choice of the unit cell has only a weak impact on $h_U$, since the spectrum of $h_U\left(\xi \right)$ is preserved for each individual $\xi$. On the other hand, this stability is not preserved for a change in the set of generators for the transformation groups, but the invariance is restored when the union of all $\xi \in {\mathbb{T}}^d$ is computed. As a consequence of these investigations, a refined version of the spectral invariance has been obtained, and it particularly follows from our investigations that reference frames do not play any role for spectral properties.