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Preprint submitted on 20 May 2021
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SELF-ADJOINTNESS OF MAGNETIC LAPLACIANS
ON TRIANGULATIONS
Colette Anné, Hela Ayadi, Yassin Chebbi, Nabila Torki-Hamza
To cite this version:
Colette Anné, Hela Ayadi, Yassin Chebbi, Nabila Torki-Hamza. SELF-ADJOINTNESS OF MAG-NETIC LAPLACIANS ON TRIANGULATIONS. 2021. �hal-03230953�
SELF-ADJOINTNESS OF MAGNETIC LAPLACIANS ON TRIANGULATIONS
COLETTE ANN ´E, HELA AYADI, YASSIN CHEBBI, AND NABILA TORKI-HAMZA
ABSTRACT. The notions of magnetic difference operator defined on weighted graphs or magnetic exterior derivative are discrete analogues of the notion of covariant derivative on sections of a fibre bundle and its extension on differential forms. In this paper, we extend this notion to certain 2-simplicial complexes called triangulations, in a manner compatible with changes of gauge. Then we study the magnetic Gauß-Bonnet operator naturally defined in this context and introduce the geometric hypothesis of χ−completeness which ensures the essential self-adjointness of this operator. This gives also the essential self-adjointness of the magnetic Laplacian on triangulations. Finally we introduce an hypothesis of bounded curvature for the magnetic potential which permits to caracterize the domain of the self-adjoint extension.
1. INTRODUCTION
Considering weighted graphs, many works are interested recently in the character of essential self-adjointness of the magnetic Laplace operator such as [11], [6], [10], [8] and [2]... Furthermore, in [1], the authors gave a new geometric criterion of χ−completeness which assures essential self-adjointness of the Laplace operator, so without discrete magnetic potential. After that, still without magnetic poten-tial, [5] generalized this notion of χ−completeness on weighted triangulations, that is a 2-simplicial complex such as the faces are all triangles. On the other hand the authors of [2], generalized this notion of χ−completeness in the case of weighted magnetic graphs, they introduced the notion of χα−completeness related to the magnetic potential α, which is a mixing of discrete magnetic geometric
properties with the behaviour of the magnetic potential. In the present work, using the analogy with the smooth case as presented in [4], we give a generalization of [1], [2] and [5] to study χ−completeness on magnetic graphs in two ways.
Working on weighted triangulations, as in [5], we concider a magnetic potential α on it, ie. a 1-form with real values which has to be understood as defining a parallel transport along the edges by the quantity exp(iα). This gives a magnetic triangulation, we start with definitions for the change of gauge and the magnetic diffentials dα and its formal adjoint δα on functions on vertices (0-forms), on
skewsymmetric functions on edges (1-forms) and on skewsymmetric functions on faces (2-forms) and we study the action of a change of gauge.
This magnetic differential defines naturally a Gauß-Bonnet operator Tα = dα + δα and a magnetic
Laplace operator ∆α = Tα2.
We show (Theorem 1) that if the triangulation is χ−complete, in the sense of [5], then the operator Tα
is essentially self-adjoint, and the magnetic Laplace operator also (Corollary 7.1). We remark that this result is valid for any magnetic potential α.
But the notion of magnetic potential has a geometric meaning. Using this analogy we introduce a notion of magnetic potential with bounded curvature and apply it to caracterize the domain of the self-adjoint extension of the magnetic Gauß-Bonnet operator (Theorem 2).
Date: Version of May 20, 2021.
2010 Mathematics Subject Classification. 39A12, 05C63, 47B25, 05C12, 05C50.
Key words and phrases. Graph, 2-Simplicial complex, Discrete magnetic operators, Essential self-adjointness, χ-completeness.
2. PRELIMINARIES
2.1. The basic concepts. A graph K is considered as a simplicial complex of dimension one. We denote by V the set at most countable of vertices and E the set of edges, considered as a subset of V × V. We assume that E is symmetric and without loops:
(x, y) ∈ E ⇐⇒ (y, x) ∈ E ; x ∈ V =⇒ (x, x) /∈ E.
In this setting, we choose an orientation of the graph which consists of defining a partition of E as follows:
E = E−∪ E+; (x, y) ∈ E+⇐⇒ (y, x) ∈ E−.
Given an edge e = (x, y), let us set e− = x, e+ = y and − e = (y, x) where e− and e+ are called
the boundary points of e. We write y ∼ x when (x, y) or (y, x) is an edge.
A path from x to y is a finite sequence γ = (x0, ...., xn) of vertices in V, such that x0 = x, xn = y
and xi−1∼ xifor each i ∈ {1, ...., n}. The length of the path γ is the number n. If x0 = xn, we say that
the path is closed or that it is a cycle. If no cycles appear more than once in a path, the path is called a simple path. A graph is connected if for any two vertices x and y, there exists a path connecting x to y. For x ∈ V we denote V(x) = {y ∈ V; (x, y) ∈ E} the set of its neighbours. A graph is locally finite if any vertex has a finite number of neighbours.
As we are dealing with magnetic fields, we consider a magnetic potential on the graph K: α : V × V −→ R such that α(x, y) = −α(y, x).
To simplify the notations, we denote α(x, y) by αxy. The triplet (V, E, α) is a magnetic graph, denoted
by Kα.
In the sequel, we shall consider all magnetic graphs as connected, locally finite, and without loops.
We return now to talk about the notion of triangulation. If the graph contains cycles of length 3, we can consider it as a two dimensional simplicial complex, see [5]. This structure is called triangulation. It gives a general framework for Laplacians defined in terms of the combinatorial structure of a simplicial complex.
Definition 1. We say that Tα := (Kα, F ) is a magnetic triangulation, where Kα is its magnetic graph
andF is the set of triangular faces which is a symmetric subset of the set of simple cycles of length 3 (e.g. (x, y, z) ∈ F ⇒ (y, z, x) and (x, z, y) ∈ F ).
Remark 1. A triangulation is a 2-simplicial complex such that all faces are triangles and we consider (x, y, z) = (y, z, x) 6= (y, x, z).
To define weighted magnetic triangulations we need weights, let us give • c : V → R∗
+the weight on the vertices.
• r : E → R∗+the even weight on the oriented edges, i.e, r(x, y) = r(y, x).
• s : F → R∗
+the even weight on oriented faces, i.e, s(x, y, z) = s(z, y, x) = s(z, x, y).
The weighted triangulation (Tα, c, r, s) is given by the triangulation Tα = (Kα, F ). We say that Tαis
simple, if, the weights of vertices, edges and faces are all equal to 1. The set of vertices connected by a face to the edge (x, y) is given by
Fxy = Fyx:= {z ∈ V; (x, y, z) ∈ F } ⊆ V(x) ∩ V(y).
The weighted degrees of vertices degVand of edges degE are given respectively by:
degV(x) := 1 c(x) X y∼x r(x, y) and degE(x, y) := 1 r(x, y) X z∈Fxy s(x, y, z).
When Tα is simple, degV(x) := degcomb(x) the combinatorial degree, and degE(x, y) = |Fxy|,
where |A| = ]A if A is a set. In [6] and [8], the authors introduce the definition of the flux of a magnetic potential:
Definition 2. The space of cycles of Kα, denoted by Z1(Kα) is the Z-module with a basis of geometric
cyclesγ = (x0, ...., xn= x0). The holonomy map is the map
Holα: Z1(Kα) −→ R
given by
Holα(γ) := αx0x1+ αx1x2+ ... + αxn−1xn.
We can easily see that if the magnetic field has no holonomy, then its integration on paths does not depend on the path joining two points. Because the graph is connected, this defines a function whose differential is α, we say that the magnetic field is trivial.
In the case of triangulations, any face (x, y, z) defines a cycle (x, y, z, x) so the holonomy defines a skewsymmetric functionα on F :b
(2.1) ∀(x, y, z) ∈ F , αbxyz = Holα(x, y, z, x) = αxy+ αyz+ αzx.
2.2. Functions spaces. Let (Tα, c, r, s) be a weighted triangulation. In this section, we endow Hilbert
structures on the spaces of functions (or cochains) on vertices, edges and faces. 2.2.1. Hilbert structure on 0-cochains. The set of 0-cochains is given by:
C(V) = {f : V → C} , and its subset of functions of finite support by Cc(V).
We turn to the Hilbert space: l2(V) := ( f ∈ C(V); X x∈V c(x)|f (x)|2 < ∞ ) , endowed with the scalar product given by
hf1, f2il2(V):=
X
x∈V
c(x)f1(x)f2(x),
for f1, f2∈ l2(V).
2.2.2. Hilbert structure on 1-cochains. The set of complex skewsymmetric 1-cochains is given by: Cskew(E ) = {ϕ : E −→ C; ϕ(x, y) = −ϕ(y, x)} .
Its subset of functions with finite support is denoted by Cskewc (E ). Let us define the Hilbert space
l2(E ) := ϕ ∈ Cskew(E ); X (x,y)∈E r(x, y)|ϕ(x, y)|2 < ∞
endowed with the scalar product hϕ1, ϕ2il2(E)= 1 2 X (x,y)∈E r(x, y)ϕ1(x, y)ϕ2(x, y)
2.2.3. Hilbert structure on 2-cochains. The set of complex skewsymmetric 2-cochains is given by: Cskew(F ) = {ψ : F −→ C; ψ(x, y, z) = ψ(z, x, y) = −ψ(z, y, x)} .
The set of functions with finite support is denoted by Ccskew(F ). Let us define the Hilbert space
l2(F ) := ψ ∈ Cskew(F ); X (x,y,z)∈F s(x, y, z) |ψ(x, y, z)|2 < ∞
endowed with the scalar product given by hψ1, ψ2il2(F ) = 1 6 X (x,y,z)∈F s(x, y, z)ψ1(x, y, z)ψ2(x, y, z)
when ψ1and ψ2are in l2(F ).
The direct sum of the spaces l2(V), l2(E ) and l2(F ) can be considered as a Hilbert space denoted by l2(T
α), indeed independent on α, that is
l2(Tα) = l2(V) ⊕ l2(E ) ⊕ l2(F ),
endowed with the scalar product given by h(f1, ϕ1, ψ1), (f2, ϕ2, ψ2)il2(T
α)= hf1, f2il2(V)+ hϕ1, ϕ2il2(E)+ hψ1, ψ2il2(F )
2.3. A change of gauge. We consider functions as sections of a C-line bundle on which is defined a connection α. We say that these functions are sections of the α-bundle. The gauge group consists of functions with values in O(1) = S1. A change of gauge is then given by any function f ∈ C(V, R), which is considered as acting on sections by multiplication by eif. We have to define its action on 1-forms, and on 2-1-forms, with values in the α-bundle.
• If F ∈ Cskew(E ), we define
(2.2) eif.F := ei ˜fF,
for the operator of symmetrization
e: C (V ) −→ Csym(E ) f 7−→ ef , ef (e) := f (e −) + f (e+) 2 . • If F ∈ Cskew(F ), we define (2.3) eif.F := eif˜˜F,
where the operator ≈: C(V) −→ Csym(F ) is defined by the formula ≈ f (x, y, z) := f (x, y) + ee f (y, z) + ef (z, x) 3 = 1 3 f (x) + f (y) + f (z). We remark that, on locally finite graphs, these formulae are defined for any function f ∈ C(V).
3. THE DIFFERENCE MAGNETIC OPERATOR
We recall the definition of the difference operator d0f (e) = f (e+) − f (e−) which can be defined for any function f ∈ C(V).
If there exists a function f ∈ C(V) such that α = d0f the magnetic field is trivial (it can be resolved by a change of gauge as we will see below, Corollary 3.1). So we look for covariant derivatives, with α as form of connection, which are equivariant by change of gauge. It is d0α, an analog of the covariant derivative with α as form of connection.
It is defined as follows :
(3.4) ∀g ∈ C(V) d0αg(x, y) := eiα(y,x)2 g(y) − e iα(x,y)
2 g(x)
It satisfies the following gauge invariance: ∀f ∈ C(V, R)
∀g ∈ C(V) d0α+d0feif.g = eif.d0α(g). Proof. d0α+d0f(eif.g)(x, y) = e i(α(y,x)+f (x)−f (y)) 2 eif (y)g(y) − e i(α(x,y)+f (y)−f (x)) 2 eif (x)g(x) = ei ef (x,y) eiα(y,x)2 g(y) − e iα(x,y) 2 g(x) = eif.d0α(g)(x, y) Corollary 3.1. Let α = d0f be a trivial magnetic field, then the covariant derivative d0α is conjugated to the flat oned0:
d0d0f = eif. ◦ d0◦ e−if.
The co-boundary magnetic operator is the formal adjoint of d0α, denoted δ0α, which satisfies
(3.5) ∀f ∈ Cc(V), ϕ ∈ Cskewc (E ) hd0αf, ϕil2(E)= hf, δ0αϕil2(V).
Lemma 3.1. The co-boundary magnetic operator δα0 : Cskewc (E ) −→ Cc(V), acts as
(δα0ϕ)(x) = 1 c(x)
X
e,e+=x
r(e) eiαe2 ϕ(e).
Let (f, ϕ) ∈ Cc(V) × Cskewc (E ), we have that hd0αf, ϕil2(E)= 1 2 X e∈E r(e)d0αf (e)ϕ(e) = 1 2 X e∈E
r(e)e−iαe2 f (e+) − eiαe2 f (e−)
ϕ(e) = 1 2 X x∈V f (x) X e,e+=x
r(e)e−iαe2 ϕ(e) − 1
2 X
x∈V
f (x) X
e,e−=x
r(e)eiαe2 ϕ(e)
= 1 2 X x∈V f (x) X e,e+=x
r(e)e−iαe2 ϕ(e) − 1
2 X
x∈V
f (x) X
−e,(−e)+=x
r(−e)e−iα−e2 ϕ(−e)
= 1 2 X x∈V f (x) X e,e+=x
r(e)e−iαe2 ϕ(e) +
X
x∈V
f (x) X
e,e+=x
r(e)e−iαe2 ϕ(e)
=X x∈V c(x)f (x) 1 c(x) X e,e+=x
r(e)eiαe2 ϕ(e)
. The equation (3.5) gives that
(δ0αϕ)(x) = 1 c(x)
X
e,e+=x
r(e)eiαe2 ϕ(e).
Remark 2. The operators (d0
α, Cc(V)) and (δ0α, Cskewc (E )) are closable, as done in [1].
Proposition 3.1. Derivation properties Forf, g ∈ Cc(V) and ϕ ∈ Cc
skew(E ) it holds
d0α(f g)(x, y) = f (y)d0α(g)(x, y) + eiα(x,y)2 d0(f )(x, y)g(x) or
(3.6)
∀e ∈ E, d0α(f g)(e) = ef (e)d0αg(e) + e
iα(e)2 g(e−) + e−iα(e)
2 g(e+) 2 d 0(f )(e) δ0α e f ϕ (x) = f (x)δα0(ϕ)(x) − 1 2c(x) X y∼x
r(x, y)ei−α(x,y)2 d0f (x, y)ϕ(x, y)
(3.7) Proof.
Let f, g ∈ Cc(V), we have that
d0α(f g)(x, y) = eiα(y,x)2 (f g)(y) − ei α(x,y) 2 (f g)(x) = f (y) eiα(y,x)2 g(y) − ei α(x,y) 2 g(x) + eiα(x,y)2 (f (y)g(x) − f (x)g(x))
= f (y)d0α(g)(x, y) + eiα(x,y)2 d0(f )(x, y)g(x).
Furthermore, using the definition ef (x, y) = f (x) + f (y)
2 , we obtain d0α(f g)(x, y) = ef (x, y)d0αg(x, y) +e
iα(x,y)2 g(x) + e−iα(x,y)
2 g(y)
2 d
We return to the formula of δ0α. Given f ∈ Cc(V) and ϕ ∈ Cc
skew(E ), as ef (e) = f (e+) − 12d0f (e) we
obtain δα0 e f ϕ (x) = 1 c(x) X e,e+=x r(e)eiαe2 e f ϕ (e) = 1 c(x)f (x) X y∼x
r(x, y)eiα(y,x)2 ϕ(y, x) + −1
2c(x) X
y∼x
r(x, y)d0f (y, x)eiα(y,x)2 ϕ(y, x)
= f (x)δ0α(ϕ)(x) − 1 2c(x)
X
y∼x
r(x, y)(d0f )(x, y)e−iα(x,y)2 ϕ(x, y).
4. THE EXTERIOR MAGNETIC DERIVATIVE OPERATOR
The point here is to extend the definition of the covariant derivative with form of connection α on 1-forms.We define the exterior magnetic derivative operator d1αas follows on Cskew(E )
(4.8) d1α(ϕ)(x, y, z) := e6i(α(x,z)+α(y,z))ϕ(x, y) + e i
6(α(y,x)+α(z,x))ϕ(y, z) + e i
6(α(z,y)+α(x,y))ϕ(z, x)
It satisfies the following gauge invariance. Let f ∈ C(V, R)
d1α+d0feif.ϕ = eif.d1α(ϕ), ∀ϕ ∈ Cskew(E ). Proof. d1α+d0f(eif.ϕ)(x, y, z) = e i 6(α(x,z)+α(y,z)+2f (z)−(f (x)+f (y)))e i 2(f (x)+f (y))ϕ(x, y) + ei6(α(y,x)+α(z,x)+2f (x)−(f (z)+f (y)))e i 2(f (z)+f (y))ϕ(y, z) + ei6(α(z,y)+α(x,y)+2f (y)−(f (x)+f (z)))e i 2(f (x)+f (z))ϕ(z, x) = e3i(f (x)+f (y)+f (z))d1 α(ϕ)(x, y, z) The co-exterior magnetic derivative is the formal adjoint of d1
α, denoted by δα1. It satisfies
hd1αϕ, ψil2(F )= hϕ, δα1ψil2(E),
(4.9)
for all (ϕ, ψ) ∈ Cskewc (E ) × Ccskew(F ).
Lemma 4.1. The formal adjoint δ1α: Cskewc (F ) −→ Cskewc (E ), is given by
δα1(ψ)(x, y) = 1 r(x, y) X t∈Fxy s(x, y, t)ei6(α(t,x)+α(t,y))ψ(x, y, t). Proof.
To justify it, note that the expression of d1α contributing to the first sum is divided into three similar parts. Let ϕ ∈ Cskewc (E ) and ψ ∈ Cskewc (F ). The equation (4.9) gives
hd1αϕ, ψil2(F ) = 1 6 X (x,y,z)∈F s(x, y, z)d1α(ϕ)(x, y, z)ψ(x, y, z) = 1 2 X (x,y,z)∈F
s(x, y, z)e6i(α(x,z)+α(y,z))ϕ(x, y)ψ(x, y, z)
= 1 2 X (x,y)∈E r(x, y)ϕ(x, y) 1 r(x, y) X t∈Fxy s(x, y, t)e6i(α(t,x)+α(t,y))ψ(x, y, t) . Remark 3. The operators (d1α, Cskewc (E )) and (δ1α, Cskewc (F )) are closable as done in [5]. It is a simple consequence of the fact that on locally finite graphs convergence in norm implies punctual convergence and the operators arelocal.
To give a derivative property in the same way as Proposition 3.1 we need to define the wedge product of a scalar 1-form with a section 1-form.
Definition 3. Let (ξ, ϕ) be two 1-forms, we consider ξ as scalar values and ϕ as a 1-form with values in theα-bundle, so we can consider ξ ∧αϕ as a 2-form with values in the α-bundle with the formula
(ξ ∧αϕ) (x, y, z) = e−i(α(z,x)+α(z,y))/6(ξ(z, x) + ξ(z, y)) ϕ(x, y)
+ e−i(α(x,y)+α(x,z))/6(ξ(x, y) + ξ(x, z)) ϕ(y, z) + e−i(α(y,z)+α(y,x))/6(ξ(y, z) + ξ(y, x)) ϕ(z, x)
We remark that if we make α = 0, this definition coincides with wedge product ξ ∧discϕ given in [5]
for 1-forms with scalar values.
Proposition 4.1. Derivation properties
Let(f, ϕ, ψ) ∈ Cc(V) × Cskewc (E ) × Cskewc (F ). Then we have
(4.10) d1α( ef ϕ)(x, y, z) = ≈ f d1αϕ (x, y, z) +1 6 d 0f ∧ αϕ (x, y, z). δ1α( ≈
f ψ)(e) = ef (e)(δα1ψ)(e)
+ 1
6r(e) X
x∈Fe
s(e, x)e6i(α(x,e
−)+α(x,e+))
d0f (e−, x) + d0f (e+, x) ψ(e, x) (4.11)
(1) Let (f, ϕ) ∈ Cc(V) × Cskewc (E ), we have d1α( ef ϕ)(x, y, z) − ≈ f (x, y, z)d1α(ϕ)(x, y, z) = = e−i(α(z,x)+α(z,y))/6 f (x) + f (y) 2 − f (x) + f (y) + f (z) 3 ϕ(x, y) + e−i(α(x,y)+α(x,z))/6 f (y) + f (z) 2 − f (x) + f (y) + f (z) 3 ϕ(y, z) + e−i(α(y,z)+α(y,x))/6 f (z) + f (x) 2 − f (x) + f (y) + f (z) 3 ϕ(z, x) = e−i(α(z,x)+α(z,y))/6f (x) + f (y) − 2f (z) 6 ϕ(x, y) + e−i(α(x,y)+α(x,z))/6f (y) + f (z) − 2f (x) 6 ϕ(y, z) + e −i(α(y,z)+α(y,x))/6f (z) + f (x) − 2f (y) 6 ϕ(z, x) = e−i(α(z,x)+α(z,y))/6d 0f (z, x) + d0f (z, y) 6 ϕ(x, y) + e−i(α(x,y)+α(x,z))/6d 0f (x, y) + d0f (x, z) 6 ϕ(y, z) + e
−i(α(y,z)+α(y,x))/6d0f (y, z) + d0f (y, x)
6 ϕ(z, x)
(2) Let (f, ψ) ∈ Cc(V) × Cskewc (F ). In the same way as before, using Lemma 4.1 one has
δα1(
≈
f ψ)(e) − ef (e)δ1α(ψ)(e) = 1 r(e)
X
x∈Fe
s(e, x)e6i(α(x,e
−)+α(x,e+))≈ f (e, x) − ef (e) ψ(e, x) = 1 6r(e) X x∈Fe
s(e, x)ei6(α(x,e
−)+α(x,e+))
d0f (e−, x) + d0f (e+, x) ψ(e, x)
5. THE MAGNETIC OPERATORS
5.1. The magnetic Gauß-Bonnet operator. It was originally defined as a square root of the Laplacian, it is the operator Tαdefined on Cc(V) ⊕ Cskewc (E ) ⊕ Cskewc (F ) to itself by
0 δα0 0 d0α 0 δ1α 0 d1α 0
5.2. The magnetic Laplacian operator. The magnetic Gauß-Bonnet operator Tα is of Dirac type and
induces the magnetic Hodge Laplacian on Cc(V) ⊕ Cskewc (E ) ⊕ Cskewc (F ) to itself as follows ∆α:= Tα2.
In general, it does not preserve the degree of a form. This default is measured by the magnetic field. 5.3. The magnetic field. It is indeed a curvature term which measures how the connection is not flat, it could be defined as the operator d1α◦ d0α(in the smooth case, a connection form A defines a covariant
see [4] Prop. 1.15. We calculate d1α[d0αf ](x, y, z) = e6i(α(x,z)+α(y,z))d0 αf (x, y) + e i 6(α(y,x)+α(z,x))d0 αf (y, z) + e i 6(α(z,y)+α(x,y))d0 αf (z, x) = e6i(α(x,z)+α(y,z)) eiα(y,x)2 f (y) − e iα(x,y) 2 f (x) + e6i(α(y,x)+α(z,x)) eiα(z,y)2 f (z) − e iα(y,z) 2 f (y) + e6i(α(z,y)+α(x,y)) eiα(x,z)2 f (x) − e iα(z,x) 2 f (z) = − sin b α(x, y, z) 6 ei3(α(x,z)+α(x,y))f (x) + e i 3(α(y,z)+α(y,x))f (y) + e i 3(α(z,x)+α(z,y))f (z)
where we use, to obtain the last line, that 12 = 16 + 13. So when the holomomy of α is zero, in particular b
α = 0 and d1α◦ d0
α = 0. Reciprocally if d1α ◦ d0α = 0, by testing the formula of f a Dirac mass at
point x (ie. f (x) = 1, and f (y) = 0 for y 6= x) we conclude that sin b α 6 = 0 on F , a kind of Bohr-Sommerfeld condition.
In the same way we can calculate δα0[δ1αψ](x) = 1 c(x) X e,e+=x r(e)eiαe2 δ1 αψ(e) = 1 c(x) X y∼x r(x, y)eiα(y,x)2 1 r(x, y) X t∈Fxy
s(x, y, t)ei6(α(t,x)+α(t,y))ψ(y, x, t)
.
We remark that it is less clear (but true) that this term is zero when α has no holonomy. To see this it is better to remember that δ0α◦ δα1 is the formal adjoint of d1α◦ d0α. It gives an other formula, namely
δα0◦ δ1α(ψ)(x) = −3 c(x) X (x,y,z)∈F s(x, y, z) sin b α(x, y, z) 6 e−i3 (α(x,z)+α(x,y))ψ(x, y, z).
Remark 4. We have proved indeed that on the magnetic triangulation Tα, if the magnetic potential has
no holonomy, then it is trivial: there exists a function f ∈ C(V) such that α = df ; as a consequence by a change of gauge the magnetic operators are those of a triangulation without magnetic potential. Moreover the magnetic field is nul (d10◦ d0
0 = 0). We can see also that the reciproque is not evident. If
the magnetic field is zero, or more strongly ifα = 0 we are not sure that the holonomy of α is trivial, web need the triangulation to be simply connected: all the cycles are combinations of boundaries of faces.
6. GEOMETRICHYPOTHESIS
6.1. Completeness for the magnetic graphs. We first recall the notion of χ−completeness of a trian-guation as defined in [5]. Let T = (K, F ) be a weighted triangulation. T is χ−complete if
(C1) K is χ−complete following the definition of [1]: there exists an increasing sequence of finite
sets (Bn)n∈Nsuch that V = ∪n∈NBnand a sequence of functions (χn)nsuch that
i) χn∈ Cc(V, R), 0 ≤ χn≤ 1 and χn(x) = 1 for all x ∈ Bn,
ii) there exists a positive constant C such that for all x ∈ V and n ∈ N we have 1
c(x) X
e∈E,e+=x
(C2) there exists a positive constant M such that for all (x, y) ∈ E and n ∈ N we have 1 r(x, y) X t∈Fxy s(x, y, t) d0χn(t, x) + d0χn(t, y) 2 ≤ M (in the following we will take M = C).
Remark 5. In [2] is defined the notion of χα−completeness of a magnetic weighted graph, a notion
mixing geometric properties of the graph with the behavior of the magnetic potential.
The magnetic weighted graphKα= (V, E , α) is considered χα−complete if there exists an increasing
sequence of finite sets(Bn)n∈Nsuch thatV = ∪n∈NBnand there exist(ηn)nand(φn)nwith
i) ηn∈ Cc(V), 0 ≤ ηn≤ 1 and ηn(x) = 1 for all x ∈ Bn.
ii) φn∈ C(V), such that (φn)nconverges to0.
iii) there exists a non-negative C such that for all x ∈ V and n ∈ N, the cut-off function χn :=
ηneiφnsatisfies 1 c(x) X e∈E,e±=x r(e)|d0αχn(e)|2 ≤ C.
We remark that under these hypothesis one has that there exists a non-negativeC such that for all x ∈ V (6.12) 1 c(x) X e∈E,e±=x r(e)| sin(α(e)/2)|2 ≤ C
Indeed, letx ∈ V, for n big enough, the vertex x and all its neighbours are in Bnand one has, because
of hypothesisii), for any e ∈ E , e±= x: lim n→∞d 0 αχn(e) = e −iα(e) 2 − e iα(e)
2 = −2i sin α(e)
2
As the graph is locally finite, the conclusion follows. But now, by the general formula on functions (6.13) d0αf (x, y) = e−iα(x,y)2 f (y) − e iα(x,y) 2 f (x) = e −iα(x,y) 2 d0f (x, y) − 2i sin α(x, y) 2 f (x) one has thatχα-completeness is stronger thanχ-completeness in the sense of the following definition,
indeed it impliesχ-completeness and Equation (6.12). We will take the following definition
Definition 4. Let Tα = (Kα, F ) be a weighted triangulation. Tα isχ-complete if the underlying
trian-gulation isχ-complete: there exists a positive constant C and
(C1) there exists an increasing sequence of finite sets (Bn)n∈Nsuch thatV = ∪n∈NBnand a sequence
of functions(χn)nsuch that
i) χn∈ Cc(V, C), 0 ≤ |χn| ≤ 1, |χn(x)| = 1 for all x ∈ Bnand∀x ∈ V, limn→∞χn(x) =
1 ;
ii) for all x ∈ V and n ∈ N we have 1
c(x) X
e∈E,e+=x
r(e)|d0χn(e)|2 ≤ C;
(C2) for all (x, y) ∈ E and n ∈ N we have
1 r(x, y) X t∈Fxy s(x, y, t) d0χn(t, x) + d0χn(t, y) 2 ≤ C. Notice that we consider in this definition cut-off functions with complex values.
6.2. Magnetic graph with bounded curvature.
Definition 5. Let Tα = (Kα, F ) be a weighted magnetic triangulation. We say that this magnetic
triangulation has abounded curvature if there exists a positive constant C such that
(6.14) ∀x ∈ V, 1 c(x) X e∈Fx s(x, e) sin2 b α(x, e) 6 ≤ C
where Fxdenotes the set of edges connected by a face to the vertex x. For the reason of this
terminol-ogy, see Remark 6 below.
7. ESSENTIALSELF-ADJOINTNESS
In [1] and [5], the authors use the χ−completeness hypothesis on a graph to ensure essential self-adjointness for the Gauß-Bonnet operator and the Laplacian. In this section, with the same idea we will give a theorem which assures for the Gauß-Bonnet operator Tαto be essentially selfadjoint. Let us begin
with this result
Proposition 7.1. Let Tα = (Kα, F ) be a magnetic χ−complete triangulation, then the operator d0α+ δα0
is essentially self-adjoint onCc(V) ⊕ Cc(E ).
Proof.
It suffices to show that d0α,min = d0
α,maxand δα,min0 = δ0α,max. Indeed, suppose it is proved, d0α+ δα0is
a direct sum and if F = (f, ϕ) ∈ Dom((d0α+ δ0
α)max) then f ∈ Dom(d0α,max) and ϕ ∈ Dom(δ0α,max).
By hypothesis, we have then f ∈ Dom(d0α,min) and ϕ ∈ Dom(δα,min0 ), thus F ∈ Dom((d0α+ δ0α),min).
1) Let f ∈ Dom(d0α,max), we will show that
kf − χnf kl2(V)+ kd0α(f − χnf ) kl2(E)→ 0 when n → ∞.
By using dominated convergence theorem, we have lim n→∞(f − χnf ) (x) = limn→∞((1 − χn)f ) (x) = 0 kf − χnf k2l2(V)≤ 4 X x∈V c(x)|f (x)|2, f ∈ l2(V) so lim n−→∞kf − χnf kl2(V)= 0.
From the derivation formula (3.6) in Proposition 3.1, we have d0α((1 − χn)f )(x, y) = (1 − χn)(y)d0α(f )(x, y) + f (x)e iα(x,y) 2 d0((1 − χn))(x, y) = (1 − χn)(y)d0α(f )(x, y) − f (x)e iα(x,y) 2 d0(χn)(x, y).
We remark that d0(1 − χn)(e) = −d0χn(e) has finite support. Since d0αf ∈ l2(E ), one has by
dominated convergence theorem, lim
n→∞k (1 − χn) d 0
Now, for the second term, using the triangle inequality, the completeness hypothesis and that the graph is locally finite, we obtain
kf (.)d0χ nk2l2(E) = 1 2 X e r(e)|f (e+)|2|d0χ n(e)|2 = 1 2 X x c(x)|f (x)|2 1 c(x) X e,e+=x r(e) |d0χn(e)|2 ≤ CX x∈V c(x)|f (x)|2. But ∀x ∈ V, lim n→∞|f (x)| 2 X e,e−=x r(e)|d0χn(e)|2= 0.
We conclude then, by the dominated convergence theorem, that lim
n kf (.)d 0
α(1 − χn)kl2(E)= 0.
2) Let ϕ ∈ Dom(δα,max0 ), we will show that
kϕ −χfnϕkl2(E)+ kδα0(ϕ −χfnϕ) kl2(V)→ 0 when n → ∞.
By using dominated convergence theorem, we have lim
n→∞(ϕ −χfnϕ) (e) = limn→∞((1 −χfn)ϕ) (e) = 0 kϕ −χfnϕk2l2(E)≤ 4 X e∈E r(e)|ϕ(e)|2, ϕ ∈ l2(E ) so lim n−→∞kϕ −χfnϕkl2(E)= 0.
From the derivation formula (3.7) in Proposition 3.1, we have δ0α((1 −χfn)ϕ) (x) = (1 − χn)(x)(δα0ϕ)(x) −
1 2c(x)
X
y∼x
r(x, y)e−iα(x,y)2 d0(1 − χn)(x, y)ϕ(x, y)
= (1 − χn)(x)(δα0ϕ)(x) +
1 2c(x)
X
y∼x
r(x, y)e−iα(x,y)2 d0(χn)(x, y)ϕ(x, y)
On the first hand, we have δα0ϕ ∈ l2(V), then by dominated convergence theorem, lim
n→∞k (1 − χn) δ 0
αϕkl2(V)= 0.
On the other hand, fixing x ∈ V, we have An(x) =
X
e,e+=x
r(e)d0(χn)(e)ϕ(e) → 0 when n → ∞
|An(x)|2 ≤ X e,e+=x r(e)|d0(χn)(e)|2 . X e,e+=x r(e)|ϕ(e)|2 ≤ Cc(x) X e,e+=x r(e)|ϕ(e)|2. Then, we obtain, c(x) 1 2c(x) 2 |An(x)|2 ≤ C 4 X e+=x r(e)|ϕ(e)|2
which is sommable on V, as ϕ ∈ l2(E ).
We conclude then by the dominated convergence theorem that the second term 1
2c(x)An(x) converges
to 0 in l2(V).
Proposition 7.2. Let Tα = (Kα, F ) be a χ-complete magnetic triangulation then the operator d1α+ δα1
is essentially self-adjoint onCc(E ) ⊕ Cc(F ).
Proof.
Again, as d1α+ δα1 is a direct sum, it suffices to show that d1α,min= d1α,maxand δα,min1 = δα,max1 . 1) Let ϕ ∈ Dom(d1α,max), we will show that
kϕ −χfnϕkl2(E)+ kd1α(ϕ −
f
χnϕ) kl2(F )→ 0 when n → ∞.
By using dominated convergence theorem, we have ∀e ∈ E, lim
n→∞(ϕ −χfnϕ) (e) = limn→∞((1 −χfn)ϕ) (e) = 0 kϕ−χfnϕk2l2(E)≤ 4 X e∈E r(e)|ϕ(e)|2, ϕ ∈ l2(E ) so lim n−→∞kϕ −χfnϕkl2(E)= 0.
From the derivation formula (4.10) in Proposition 4.1, we have d1α(ϕ −χfnϕ) (x, y, z) = d1α((1 −χfn) ϕ) (x, y, z) = 1 −χ≈n d1αϕ (x, y, z) + 1 6 d 0(1 − χ n) ∧αϕ (x, y, z) = 1 −χ≈n d1αϕ (x, y, z) − 1 6 d 0χ n∧αϕ (x, y, z)
using d0(1 − χn)(e) = −d0χn(e).
Since d1αϕ ∈ l2(F ), one has by dominated convergence theorem, lim n→∞ 1 −χ≈n d1αϕ l2(F ) = 0.
For the second term, we recall the definition of ∧α
d0χn∧αϕ (x, y, z) = e−i/6(α(z,x)+α(z,y)) d0χn(z, x) + d0χn(z, y) ϕ(x, y)
+ e−i/6(α(x,y)+α(x,z)) d0χn(x, y) + d0χn(x, z) ϕ(y, z)
+ e−i/6(α(y,z)+α(y,x)) d0χn(y, z) + d0χn(y, x) ϕ(z, x).
We have by property (C2) of the completeness hypothesis for the triangulation
X (e,x)∈F s(e, x)|ϕ(e)|2 d0χn(x, e−) + d0χn(x, e+) 2 =X e∈E r(e)|ϕ(e)|2 1 r(e) X x∈Fe s(e, x) d0χn(x, e−) + d0χn(x, e+) 2 ≤ CX e∈E r(e)|ϕ(e)|2.
But for any e ∈ E, limn→∞Px∈Fes(e, x)
d0χn(x, e−) + d0χn(x, e+)
= 0, we conclude then again by the dominated convergence theorem that this second term converges to 0.
2) Let ψ ∈ Dom(δ1α,max), we will show that kψ −χ≈nψkl2(F )+ kδ1α(ψ −
≈
χnψ)kl2(E)→ 0 when n → ∞.
First, by dominated convergence theorem, we have lim n→∞ ψ −χ≈nψ (x, y, z) = lim n→∞ (1 −χ≈n)ψ (x, y, z) = 0. kψ −χ≈nψk2l2(F )= 1 6 X (x,y,z)∈F s(x, y, z)|1 −χ≈n(x, y, z)|2|ψ(x, y, z)|2 ≤ 4 6 X (x,y,z)∈F s(x, y, z)|ψ(x, y, z)|2= 4kψk2l2(F ). Then, lim n→∞kψ − ≈ χnψkl2(F ) = 0.
Secondly, by the derivation formula (4.11) in Proposition 4.1, we have Bn(e) = δα1
(1 −χ≈n)ψ
(e) − (1 −χfn)(e)(δ1αψ)(e)
= 1
6r(e) X
x∈Fe
s(e, x)ei6(α(x,e
−)+α(x,e+))
d0(1 − χn)(e−, x) + d0(1 − χn)(e+, x) ψ(e, x)
= 1
6r(e) X
x∈Fe
s(e, x)ei6(α(x,e
−)+α(x,e+))
d0χn(x, e−) + d0χn(x, e+) ψ(e, x)
Then, by the hypothesis of completeness,
(7.15) ∀e ∈ E, lim
n→∞Bn(e) = 0.
On the other hand, using the Cauchy-Schwarz inequality and again the property (C2) of χ−completeness,
we have for any e ∈ E,
|Bn(e)|2= 1 6r(e) 2 X x∈Fe
s(e, x)e6i(α(x,e
−)+α(x,e+)) d0(χn)(x, e−) + d0(χn)(x, e+) ψ(e, x) 2 ≤ 1 6r(e) 2 X x∈Fe s(e, x) d0(χn)(x, e−) + d0(χn)(x, e+) 2 X x∈Fe s(e, x)|ψ(e, x)|2 ≤ C r(e) X x∈Fe s(e, x)|ψ(e, x)|2. (7.16)
Therefore, by the dominated convergence theorem ∀n ∈ N, X
e∈E
r(e)|Bn(e)|2 ≤ Ckψk2 ⇒ lim n→∞
X
e∈E
r(e)|Bn(e)|2= 0
Finally, we have by dominated convergence theorem as δ1αψ ∈ l2(E )
lim
n→∞k (1 −χfn) δ
1
α(ψ)k = 0.
This completes the proof.
Theorem 1. Let Tα = (Kα, F ) be a χ−complete magnetic triangulation then the operator Tαis
essen-tially self-adjoint onCc(V) ⊕ Cc(E ) ⊕ Cc(F ).
Proof. To show that Tαis essentially self-adjoint, we will prove that Dom(Tα,max) ⊂ Dom(Tα,min).
Let us take the sequence of cut-off functions (χn)n⊆ Cc(V) assured by the hypothesis of χ−completness
of Tα. Let F = (f, ϕ, ψ) ∈ Dom(Tα,max), then F and TαF are in l2(Tα). This implies that δα0ϕ ∈
l2(V), d0αf + δα1ψ ∈ l2(E ) and d1αϕ ∈ l2(F ). Consequently, by the definition of δ0α,maxand d1α,maxwe have ϕ ∈ Dom(δα,max0 ) ∩ Dom(d1
α,max). But the proofs of Proposition 7.1 and Proposition 7.2 show
that ϕn=χenϕ satisfies
∀n ∈ N ϕn∈ C(E) and lim
n→∞kϕ − ϕnk 2+ kd1
α(ϕ − ϕn)k2+ kδα0(ϕ − ϕn)k2 = 0,
so ϕ ∈ Dom(δα,min0 ) ∩ Dom(d1α,min).
Now, it remains to prove that d0α(χnf ) + δα1( ≈
χnψ) converges in l2(E ) to d0αf + δ1αψ. Indeed, we need
some derivation formula taken in Proposition 3.1 and Equation (7.16) in the proof of Proposition (7.2). It gives: d0α((1 − χn)f )(e) = (1 −χfn)(e)d 0 α(f )(e) − eiα(e)2 f (e−) + e−i α(e) 2 f (e+) 2 ! d0χn(e) | {z } Sn(e) δα1(1 −χ≈n)ψ
(e) = (1 −χfn)(e)(δ1αψ)(e)
− 1
6r(e) X
x∈Fe
s(e, x)e6i(α(x,e
−)+α(x,e+))
d0(χn)(e−, x) + d0(χn)(e+, x) ψ(e, x)
| {z }
In(e)
.
Therefore, we have by the triangle inequality kd0α(f − χnf ) + δα1(ψ −
≈
χnψ)k2l2(E)
= k(1 −χfn)(d0αf + δα1ψ) − Sn− Ink2l2(E)
≤ 3k(1 −χfn)(d0αf + δα1ψ)k2l2(E)+ kSnk2l2(E)+ kInk2l2(E)
Because d0αf + δα1ψ ∈ l2(E ), we have lim n→∞k(1 −χfn)(d 0 αf + δ1αψ)k2l2(E)= 0.
Because ψ ∈ l2(F ) we have, as in the proof of Proposition 7.2 lim
n→∞kInk 2
l2(E)= 0.
Because f ∈ l2(V) we have, as in the proof of Proposition 7.1, lim
n→∞kSnk 2
l2(E)= 0.
Corollary 7.1. Let Tα = (Kα, F ) be a χ−complete magnetic triangulation then the magnetic Laplace
operator∆αis essentially self-adjoint onCc(V) ⊕ Cc(E ) ⊕ Cc(F ).
Indeed, we know that Tαis essentially self-adjoint, we conclude then that ∆α= Tα2is also essentially
Theorem 2. Let Tα= (Kα, F ) be a χ−complete triangulation with bounded curvature (ie. the property
(6.14) is satisfied) then the operatorTαsatisfies
Dom(Tα,min) = Dom(d0α,min) ⊕ Dom(δ0α,min) ∩ Dom(d1α,min) ⊕ Dom(δ1α,min).
Remark 6. This theorem is the reason why we call our hypothesis (6.14) bounded curvature. Indeed it sounds like, in the smooth case, the theorem which says that on a complete manifold with bounded geometry (ie. with injectivity radius positive and Ricci curvature bounded from below) the Laplace Beltrami operator is essentially self adjoint and the domain of its selfadjoint extension is the second Sobolev space (the closure of the smooth function with compact support for the Sobolev norm of degree 2), see[9] Prop. 2.10.
Proof.
First inclusion.Let F = (f, ϕ, ψ) ∈ Dom(d0α,min)⊕
Dom(δα,min0 ) ∩ Dom(d1α,min)
⊕Dom(δ1 α,min).
Then by Proposition 7.1 and Proposition 7.2 we have (fn)n⊆ Cc(V) and (ψn)n⊆ Cc(F ) such that :
- fn= χnf → f in l2(V) and dα0fn→ d0α,minf in l2(E ).
- ψn= ≈
χnψ → ψ in l2(F ) and δ1αψn→ δα,min1 ψ in l2(E )
We have also ϕ ∈ Dom(δα,min0 ) ∩ Dom(d1α,min) so by these two propositions (ϕn =χfnϕ)n ⊆ C
c(E )
satisfies
kϕ −χfnϕkl2(E)+ kδα0(ϕ −χfnϕ)kl2(V)+ kd1α(ϕ −χfnϕ)kl2(F )→ 0, when n → ∞.
Hence, Fn= (fn, ϕn, ψn) satisfies
Fn→ F in l2(Tα), TαFn→ Tα,minF in l2(Tα),
where Tα,minF = (δ0α,minϕ, d0α,minf + δ1α,minψ, d1α,minϕ). Then, F ∈ Dom(Tα,min).
Second inclusion. Let F = (f, ϕ, ψ) ∈ Dom(Tα,min). It means that there exists a sequence
((fn, ϕn, ψn))nin Cc(V) × Cc(E ) × Cc(F ) with lim n→∞kfn− f kl2(V)+ kϕn− ϕkl2(E)+ kψn− ψkl2(F )= 0 and lim n→∞kδ 0 α(ϕn− ϕ)kl2(V)= 0, lim n→∞kd 1 α(ϕn− ϕ)kl2(F )= 0, lim n→∞kd 0 α(fn− f ) + δ1α(ψn− ψ)kl2(E)= 0.
We obtain directly that ϕ ∈ Dom(δ0α,min) ∩ Dom(d1α,min). We have to show now that d0αfnand δα1ψnare Cauchy sequences.
Lemma 7.1. In the situation of the theorem, there exists a positive constant C0 such that for anyg ∈ Cc(V) and η ∈ Cc(F ) (7.17) |d0 αg, δα1η l2(E)| ≤ C 0kgk l2(V)kηkl2(F ).
Proof.. Let g ∈ Cc(V) and η ∈ Cc(F ), then d0 αg, δ1αη l2(E)=d 1 α◦ d0αg, η l2(F ) = − X (x,y,z)∈F s(x, y, z)¯η(x, y, z) sin b α(x, y, z) 6 e3i(α(x,z)+α(x,y))g(x) + e i 3(α(y,z)+α(y,x))g(y) + e i 3(α(z,x)+α(z,y))g(z)
by the calculus of section 5.3. Now using the hypothesis of bounded curvature (6.14) one has X (x,y,z)∈F s(x, y, z) sin2 b α(x, y, z) 6 |g(x)|2+ |g(y)|2+ |g(z)|2 = 3X x∈V |g(x)|2 X e∈Fx s(x, e) sin2 b α(x, y, z) 6 ≤ 3Ckgk2l2(V).
The Cauchy-Schwarz inequality gives finally the result with C0 = 3√C. Now we return to the proof of the theorem; let m, n ∈ N
kd0 α(fn− fm)k2l2(E)+kδ1α(ψn− ψm)k2l2(E) = kd0α(fn− fm) + δ1α(ψn− ψm)k2l2(E)− 2d0α(fn− fm), δ1α(ψn− ψm) l2(E) ≤kd0 α(fn− fm) + δα1(ψn− ψm)k2l2(E)+ 2C0kfn− fmkl2(V)kψn− ψmkl2(F ).
We conclude that both (d0αfn)nand (δ1ψn)nconverge so
f ∈ Dom(d0α,min) and ψ ∈ Dom(δα,min1 ).
8. APPLICATION TO MAGNETIC TRIANGULATIONS
The aim of this section is to give a concrete way to prove that the notion of χ−completeness covers the χα-completeness that have been studied in [2]. Let α be an arbitrary magnetic potential. Given Tα
a triangulation, fix an origin vertex O. The combinatorial distance dcomb between two vertices x, y is
given by
dcomb(x, y) := min γ∈Γxy
L(γ)
where Γxy is the set of the paths from x to y and L(γ) denotes the length of a path γ. To simplify the
notation, we denote dcomb(O, x) by |x|. We denote by Bn the ball with center the origin vertex O and
radius n, i.e Bn:= {x ∈ V : |x| ≤ n} .
Theorem 3. Let Tαbe a weighted magnetic triangulation endowed with an origin. Assume that
sup
x∈Bn
degV(x) = O(n2) and sup (x,y)∈Bn×Bn±1
degE(x, y) = O(n2) when n −→ ∞.
ThenTαisχ−complete.
Proof.
Let n ∈ N, we define θn: V −→ R and ηn: V −→ R by
θn(x) := |x| n + 1 and ηn(x) := 2 − |x| n + 1 ∨ 0 ∧ 1.
We set the cut-off function χn(x) := ηn(x)eiθn(x). If x ∈ Bn+1, we have that ηn(x) = 1 and if
x ∈ B2(n+1)c , we have that ηn(x) = 0. Then, ηnhas a finite support.
We remark that if y ∼ x then, by the triangular inequality, ||x| − |y|| ≤ 1 so y ∼ x ⇒ |ηn(x) − ηn(y)| ≤
1
n + 1and |θn(x) − θn(y)| ≤ 1 n + 1
and we notice also the general fact concerning two complex numbers r1eiβ1 and r2eiβ2:
|r1eiβ1− r2eiβ2| ≤ |r1eiβ1 − r1eiβ2| + |r1eiβ2− r2eiβ2|
≤ |r1|.2| sin β1− β2 2
| + |r1− r2| ≤ |r1|.|β1− β2| + |r1− r2|
So if |r1| ≤ 1 then |r1eiβ1 − r2eiβ2| ≤ |β1− β2| + |r1− r2|.
Combining these two results conducts to
(8.18) y ∼ x ⇒ |d0χn(x, y)| = |χn(x) − χn(y)| ≤
2 n + 1.
We conclude that, thanks to the hypothesis on the degrees, there exists a constant C such that for any x ∈ V 1 c(x) X y∼x r(x, y) d0χn(x, y) 2 ≤ 4degV(x) (n + 1)2 ≤ C.
Now, let (x, y) ∈ E. Using again Equation (8.18), we have for some M > 0 independent from (x, y) and n 1 r(x, y) X t∈Fxy s(x, y, t) d0χn(t, x) − d0χn(t, y) 2 ≤ 2 r(x, y) X t∈Fxy s(x, y, t) d0χn(t, x) 2 + d0χn(t, x) 2 = 4 r(x, y) X t∈Fxy s(x, y, t) |χn(x) − χn(t)|2 ≤ 16degE(x, y) (n + 1)2 ≤ M.
The definition of χ−completeness is satisfied.
8.1. Case of a book-like triangulation. We recall the definition of 1-dimensional decomposition given in [3] for the case of graphs. This notion generalizes the decomposition in spheres on a graph with origin, where the spheres are defined by
Sn:= {x ∈ V : |x| = n} .
Definition 6. A 1-dimensional decomposition of a graph (V, E) is a family of finite sets (Sn)n∈Nwhich
forms a partitions ofV and such that for all (x, y) ∈ Sn× Sm,
(x, y) ∈ E =⇒ |n − m| ≤ 1. The following definition is introduced in [5].
Definition 7. We say that a triangulation is a book-like triangulation endowed with an origin O (Fig.1), if there exists a 1-dimensional decomposition(Sn)n∈Nof its graph such that for alln ∈ N,
i) S0= {O} , |S2n+1| = 2 and S2 2n+1∩ E = 1. ii) ∀x, y ∈ S2n+2=⇒ (x, y) /∈ E. iii) F = S2n+12 × S2n S S2 2n+1× S2n+2 .
FIGURE1. A book-like triangulation
Example 8.1. Let Tαbe a magnetic book-like triangulation endowed with an originO and an arbitrary
magnetic potentialα. Take 0 < β ≤ 2. We set c(x) = 1 and val(x) := ] {y ∈ V, y ∼ x} , for all x ∈ V. If
val(x) = b(2n + 1)βc + 4, for all n ∈ N and for x ∈ S2n+1
r(x, y) = val(x)val(y)
|x||y| , for all (x, y) ∈ E and
s(x, y, z) = r(x, y)r(y, z)r(z, x), for all (x, y, z) ∈ F . Then using Theorem 3,Tαsatisfies the definition ofχ-completeness.
Indeed, at first, we remark that
(8.19) val(x) = |S2n| + |S2n+2| + 1, for all n ∈ N and for x ∈ S2n+1.
Letn ∈ N∗, the equation (8.19) gives:
First case. If x ∈ S2n+1, we have for some C > 0 independent from x and n
X y∼x r(x, y) c(x) = val(x) |x| val(y) |y| + val(x) |x| X y∼x, y∈S2n∪S2n+2 val(y) |y| = b(2n + 1) βc + 4 2n + 1 2 +b(2n + 1) βc + 4 2n + 1 2|S2n| n + 2|S2n+2| n + 1 ≤ b(2n + 1) βc + 4 2n + 1 2 + 2b(2n + 1) βc + 4 2n + 1 |S2n| + |S2n+2| n ≤ b(2n + 1) βc + 4 2n + 1 2 +2 b(2n + 1) βc + 42 n(2n + 1) ≤ Cn2β−2, (0 < β ≤ 2) ≤ Cn2.
Second case. If x ∈ S2n, we have for some C0 > 0 independent from x and n X y∼x r(x, y) c(x) = val(x) |x| X y∼x, y∈S2n±1 val(y) |y| = 2 n |S2n−1| b(2n − 1)βc + 4 2n − 1 + |S2n+1| b(2n + 1)βc + 4 2n + 1 ! = 4 n b(2n − 1)βc + 4 2n − 1 + b(2n + 1)βc + 4 2n + 1 ≤ C0nβ−2 ≤ C0, (0 < β ≤ 2).
On the other hand, let(x, y) ∈ S2n+12 . We have for some M > 0 independent from (x, y) and n X t∈Fxy s(x, y, t) r(x, y) = X t∈Fxy r(y, t)r(t, x) = val(x)val(y) |x||y| X t∈S2n∪S2n+2 val(t) |t| 2 = 16 b(2n + 1) βc + 4 2n + 1 2 |S2n| (2n)2 + |S2n+2| (2n + 2)2 ≤ 16 b(2n + 1) βc + 4 2n + 1 2 |S2n| + |S2n+2| (2n)2 ≤ 4 b(2n + 1) βc + 4 2n + 1 2 b(2n + 1)βc + 4 n2 ≤ M n3β−4, (0 < β ≤ 2) ≤ M n2.
If(x, y) ∈ S2n× S2n+1, we have for some M0 > 0 independent from (x, y) and n
X t∈Fxy s(x, y, t) r(x, y) = X t∈Fxy r(y, t)r(t, x) = val(x)val(y) |x||y| X t∈S2n+1 val(t) |t| 2 = 4 b(2n + 1) βc + 4 2n(2n + 1) 2 b(2n + 1)βc + 4 2n + 1 2 ≤ M0n3β−4, (0 < β ≤ 2) ≤ M0n2.
8.2. Case of a 1-dimensional decomposition triangulation. In this section under a specific choice of magnetic potential α, we construct an example of a triangulation that is χ-complete and not χα-complete.
The following definition is introduced in [5].
Definition 8. We say that a triangulation is a 1-dimensional decomposition triangulation if there exists a 1-dimensional decomposition(Sn)n∈Nof its graph (Fig.2).
FIGURE2. A 1-dimensional decomposition triangulation
We now divide the degree with respect to the simple 1-dimensional decomposition triangulation ηn±:= sup x∈Sn |V(x) ∩ Sn±1| , βn±:= sup e∈Sn×Sn±1 |Fe| , γn±:= sup e∈S2 n (Fe∩ Sn±1) × Sn2 ∩ F .
Example 8.2. Let Tαbe a simple 1-dimensional decomposition magnetic triangulation and
αxy := (|x| − |y|) π. Thus, X y∼x sin2 αxy 2 = val(x)
for all x ∈ V. Using (6.12), if val(.) is unbounded, then Tαis not χα-complete.
In contrast, in [5, Thm 6.2], the author proves that if X
n∈N
1
pξ(n, n + 1) = ∞
where ξ(n, n + 1) = η+n + ηn+1− + βn++ γn++ γn+1− , then Tαis χ−complete.
As a consequence, the operators Tα and ∆α are essentially self-adjoint on Cc(V) ⊕ Cc(E ) ⊕ Cc(F ).
But we remark also that the magnetic field is trivial, it is the differential of a function, then its holonomy is null and the curvature also. We can then apply to TαTheorem 2.
Acknowledgments: The authors would like to thank Luc Hillairet for fruitful discussions and helpful comments.
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UNIVERSIT´E DE NANTES, LABORATOIRE DE MATHEMATIQUE´ JEAN LERAY, CNRS, FACULTE DES´ SCIENCES, BP 92208, 44322 NANTES, (FRANCE).
Email address: colette.anne@univ-nantes.fr
UNIVERSIT´E DEMONASTIR, (LR/18ES15) & ACADMIENAVALE, 7050 MENZEL-BOURGUIBA(TUNISIE) Email address: halaayadi@yahoo.fr
UNIVERSIT´E DEMONASTIR, (LR/18ES15) (TUNISIE)., UNIVERSITE DE´ NANTES, LABORATOIRE DEMATHEMATIQUE´ JEANLERAY, CNRS, FACULTE DES´ SCIENCES, BP 92208, 44322 NANTES, (FRANCE).
Email address: chebbiyassin88@gmail.com
UNIVERSIT´E DEMONASTIR, LR/18ES15 & INSTITUTSUPERIEUR D’INFORMATIQUE DE´ MAHDIA(ISIMA) B.P 49, CAMPUSUNIVERSITAIRE DEMAHDIA; 5111-MAHDIA(TUNISIE).
