The Magnetic Laplacian Acting on Discrete Cusps

Sylvain Golénia and Françoise Truc

Received: July 17, 2015 Revised: June 8, 2017

Communicated by Heinz Siedentop

Abstract. We introduce the notion of discrete cusp for a weighted graph. In this context, we prove that the form-domain of the magnetic Laplacian and that of the non-magnetic Laplacian can be different. We establish the emptiness of the essential spectrum and compute the asymptotic of eigenvalues for the magnetic Laplacian.

2010 Mathematics Subject Classification: 34L20, 47A10, 05C63, 47B25, 47A63, 81Q10 Keywords and Phrases: discrete magnetic Laplacian, locally finite graphs, eigenvalues, asymptotic, form-domain

1 Introduction

The spectral theory of discrete Laplacians on graphs has drawn a lot of attention for decades. The spectral analysis of the Laplacian associated to a graph is strongly related to the geometry of the graph. Moreover, graphs are discretized versions of manifolds. In [MoT, GM], it is shown that for a manifold with cusps, adding a magnetic field can drastically destroy the essential spectrum of the Laplacian. The aim of this article is to go along this line in a discrete setting. We recall some standard definitions of graph theory. A graph is a triple G := (E, V, m), where V is a countable set (the vertices), E : V × V → ℝ_{+} is symmetric, and m : V → (0, ∞) is a weight. We say that G is simple if m = 1 and E : V × V → {0, 1}. Given x, y ∈ V, we say that (x, y) is an edge (or x and y are neighbors) if E(x, y) > 0. We denote this relationship by x ∼ y and the set of neighbors of x by N_{G}(x). We say that there is a loop at x ∈ V if E(x, x) > 0. A graph is connected if for all x, y ∈ V, there exists a path γ joining x and y. Here, γ is a sequence x_{0}, x_{1}, ..., x_{n} ∈ V such that x = x_{0}, y = x_{n}, and x_{j} ∼ x_{j+1} for all 0 ≤ j ≤ n − 1. In this case, we set |γ| := n. A graph G is locally finite if |N_{G}(x)| is finite for all x ∈ V. In the sequel, we assume that:

All graphs are locally finite, connected with no loops.

We endow a graph G := (E, V, m) with the metric ρ_{G} defined by

ρ_{G}(x, y) := inf{|γ|, γ is a path joining x and y}.

The space of complex-valued functions acting on the set of vertices V is denoted by C(V) := {f : V → ℂ}. Moreover, C_{c}(V) is the subspace of C(V) of functions with finite support. We consider the Hilbert space

ℓ^{2}(V, m) := {f ∈ C(V), ∑_{x∈V} m(x)|f(x)|^{2} < ∞}

with the scalar product ⟨f, g⟩ := ∑_{x∈V} m(x)\overline{f(x)}g(x). We equip G with a magnetic potential θ : V × V → ℝ/2πℤ such that we have θ_{x,y} := θ(x, y) = −θ_{y,x} and θ(x, y) := 0 if E(x, y) = 0. We define the Hermitian form

Q_{G,θ}(f) := \frac{1}{2} ∑_{x,y∈V} E(x, y) |f(x) − e^{iθ_{x,y}}f(y)|^{2} ,

for all f ∈ C_{c}(V). The associated magnetic Laplacian is the unique non-negative self-adjoint operator Δ_{G,θ} satisfying ⟨f, Δ_{G,θ}f⟩_{ℓ^{2}(V,m)} = Q_{G,θ}(f), for all f ∈ C_{c}(V). It is the Friedrichs extension of Δ_{G,θ}|_{C_{c}(V)}, e.g., [CTT3, RS], where

(Δ_{G,θ}f)(x) = \frac{1}{m(x)} ∑_{y∈V} E(x, y) (f(x) − e^{iθ_{x,y}}f(y)) ,

for all f ∈ C_{c}(V). We set

deg_{G}(x) := \frac{1}{m(x)} ∑_{y∈V} E(x, y),

the degree of x ∈ V. We see easily that Δ_{G,θ} ≤ 2 deg_{G}(·) in the form sense, i.e.,

0 ≤ ⟨f, Δ_{G,θ}f⟩ ≤ ⟨f, 2 deg_{G}(·)f⟩, for all f ∈ C_{c}(V).  (1)

Moreover, setting δ̃_{x}(y) := m^{−1/2}(x)δ_{x,y} for any x, y ∈ V, ⟨δ̃_{x}, Δ_{G,θ}δ̃_{x}⟩ = deg_{G}(x), so Δ_{G,θ} is bounded if and only if sup_{x∈V} deg_{G}(x) is finite, e.g. [KL, Go]. Another consequence of (1) is

D (deg_{G}^{1/2}(·)) ⊂ D (Δ_{G,θ}^{1/2}) ,  (2)

where D (deg_{G}^{1/2}(·)) := {f ∈ ℓ^{2}(V, m), deg_{G}(·)f ∈ ℓ^{2}(V, m)}. However, the equality of the form-domains

D (deg_{G}^{1/2}(·)) = D (Δ_{G,θ}^{1/2})  (3)

The Magnetic Laplacian Acting on Discrete Cusps 1711

is wrong in general for a simple graph, see [Go, BGK]. In fact if θ = 0, (2) is equivalent to a sparseness condition and holds true for planar simple graphs, see [BGK]. We refer to [BGKLM] for a magnetic sparseness condition. On a general weighted graph, if (3) holds true,

σ_{ess}(Δ_{G,θ}) = ∅ ⇔ (Δ_{G,θ} + 1)^{−1} is compact ⇔ lim_{|x|→∞} deg_{G}(x) = ∞,

where |x| := ρ_{G}(x_{0}, x) for a given x_{0} ∈ V. Note that the limit is independent of the choice of x_{0}. Besides if the latter is true and if the graph is sparse (simple and planar for instance), [BGK] ensures the following asymptotic of eigenvalues,

lim_{n→∞} \frac{λ_{n} (Δ_{G,θ})}{λ_{n} (deg_{G}(·))} = 1,  (4)

where λ_{n}(H) denotes the n-th eigenvalue, counted with multiplicity, of a selfadjoint operator H, which is bounded from below. The technique used in [BGK] does not apply when the graph is a discrete cusp (thin at infinity), see Definition 2.5. The aim of this article is to establish new behaviors for the asymptotic of eigenvalues for the magnetic Laplacian in that case, and also to prove that the form-domain of the non-magnetic Laplacian can be different from that of the magnetic Laplacian, see Theorem 2.14. We found the inspiration by mimicking the continuous case, which was studied in [MoT, GM]. Let us present a flavour of our results (in particular of Theorem 2.14) by introducing the following specific example of discrete cusp :

Example 1.1 Let n ≥ 3 be an integer and consider G_{1} := (E_{1}, V_{1}, m_{1}), where

V_{1} := ℕ, m_{1}(n) := exp(−n), and E_{1}(n, n + 1) := exp(−(2n + 1)/2),

for all n ∈ ℕ and G_{2} := (E_{2}, V_{2}, 1) a simple connected finite graph such that |V_{2}| = n. Set θ_{1} := 0 and θ_{2} such that Hol_{θ_{2}} ≠ 0. Let G := (E, V, m) be the twisted Cartesian product G_{1} ×_{V_{2}} G_{2}, given by:

\{ \begin{matrix}m(x, y) := & m_{1}(x), \\ E ((x, y), (x′, y′)) := & E_{1}(x, x′) × δ_{y,y′} + δ_{x,x′} × E_{2}(y, y′), \\ θ ((x, y), (x′, y′)) := & δ_{x,x′} × θ_{2}(y, y′), \end{matrix}

for all x, x′ ∈ V_{1} and y, y′ ∈ V_{2}. Then there exists a constant ν > 0 such that for all κ ∈ ℝ/νℤ

σ_{ess}(Δ_{G,κθ}) = ∅ ⇔ D (Δ_{G,κθ}^{1/2}) = D (deg_{G}^{1/2}(·)) ⇔ κ ≠ 0 in ℝ/νℤ

Moreover:

1) When κ ≠ 0 in ℝ/νℤ, we have:

lim_{λ→∞} \frac{N_{λ} (Δ_{G,κθ})}{N_{λ} (deg_{G}(·))} = 1,

where N_{λ}(H) := dim ran1_{]−∞,λ]}(H) for a self-adjoint operator H.

2) When κ = 0 in ℝ/νℤ, the absolutely continuous part of the Δ_{G,κθ} is

σ_{ac} (Δ_{G,κθ}) = [e^{1/2} + e^{−1/2} − 2, e^{1/2} + e^{−1/2} + 2] ,

with multiplicity 1 and

lim_{λ→∞} \frac{N_{λ} (Δ_{G,κθ}P_{ac,κ}^{⊥})}{N_{λ} (deg_{G}(·))} = \frac{n − 1}{n},

where P_{ac,κ} denotes the projection onto the a.c. part of Δ_{G,κθ}.

We now describe heuristically the phenomenon. Compared with the first case, the constant (n − 1)/n that appears in the second case encodes the fact that a part of the wave packet diffuses. Moreover, switching on the magnetic field is not a gentle perturbation because the form domain of the operator is changed. By Riemann-Lebesgue Theorem, the particle, which is localized in the a.c. part of the operator, escapes from every compact set. More precisely, for a finite subset X ⊂ V and all f ∈ D(Δ_{G,0})

∥1_{X}(·)e^{itΔ_{G,0}}P_{ac,0}f∥ → 0, as t → ∞.

In the first case, when the magnetic potential is active, the spectrum of Δ_{G,κθ} is purely discrete. The particle cannot diffuse anymore. More precisely, for a finite subset X ⊂ V and an eigenvalue f of Δ_{G,κθ} such that f|_{X} ≠ 0, there is c > 0 such that:

\frac{1}{T} ∫_{0}^{T} ∥1_{X}(·)e^{itΔ_{G,κθ}}f∥^{2} dt → c, as T → ∞.

The particle is trapped by the magnetic field.

⋯ Diffusion Magnetic effect Representation of a discrete cusp:

The magnetic field traps the particle by spinning it, whereas its absence lets the particle diffuse.

We now describe the structure of the paper. In Section 2.1, we recall some properties of the holonomy of a magnetic potential. In Section 2.2 we present our main hypotheses and several notions of (weighted) product for graphs. We introduce the notion of discrete cusp and analyze it under the light of the radius of injectivity. Then in Section 2.3 we give a criteria concerning the absence of essential spectrum. Next, in Section 2.4, we refine the analysis and give our central theorem, a general statement for discrete cusps, computing the form domain and the asymptotic of eigenvalues. We finish the section by proving Theorem 1.1.

Notation: ℕ denotes the set of non negative integers and ℕ^{∗} that of the positive integers. We denote by D(H) the domain of an operator H. Its (essential) spectrum is denoted by σ(H) (by σ_{ess}(H)). We set δ_{x,y} equals 1 if and only if x = y and 0 otherwise and given a set X, 1_{X}(x) equals 1 if x ∈ X and 0 otherwise.

Acknowledgments: We would like to thank Colette Anné, Michel Bonnefont, Yves Colin de Verdière, Matthias Keller, and Sergiu Moroianu for useful discussions. SG and FT were partially supported by the ANR project GeRaSic (ANR-13-BS01-0007-01) and by SQFT (ANR-12-JS01-0008-01).

2 Main results

2.1 Holonomy of a magnetic potential

We recall some facts about the gauge theory of magnetic fields, see [CTT3, HS] for more details and also [LLPP] for a different point of view. We recall that a gauge transform U is the unitary map on ℓ^{2}(V, m) defined by

(Uf)(x) = u_{x}f(x),

where (u_{x})_{x∈V} is a sequence of complex numbers with |u_{x}| ≡ 1 (we write u_{x} = e^{iσ_{x}}). The map U acts on the quadratic forms Q_{G,θ} by U^{⋆}(Q_{G,θ})(f) = Q_{G,θ}(Uf), for all f ∈ C_{c}(V). The magnetic potential U^{⋆}(θ) is defined by:

U^{⋆}(Q_{G,θ}) = Q_{G,U^{⋆}(θ)}.

More explicitly, we get:

U^{⋆}(θ)_{xy} = θ_{x,y} + σ_{y} − σ_{x}.

We turn to the definition of the flux of a magnetic potential, the Holonomy.

Proposition 2.1 Let us denote by Z_{1}(G) the space of cycles of G. It is is a free ℤ−module with a basis of geometric cycles γ = (x_{0}, x_{1}) + (x_{1}, x_{2}) + … + (x_{N−1}, x_{N}) with, for i = 0, ⋯, N − 1, E(x_{i}, x_{i+1}) ≠ 0, and x_{N} = x_{0}. We define the holonomy map Hol_{θ} : Z_{1}(G) → ℝ/2πℤ, by

Hol_{θ} ((x_{0}, x_{1}) + (x_{1}, x_{2}) + ⋯ + (x_{N}, x_{0})) := θ_{x_{0},x_{1}} + ⋯ + θ_{x_{N} ,x_{0}}.

Then

1) The map θ ↦ Hol_{θ} is surjective onto Hom_{ℤ}(Z_{1}(G), ℝ/2πℤ).

2) Hol_{θ_{1}} = Hol_{θ_{2}} if and only if there exists a gauge transform U so that U^{⋆}(θ_{2}) = θ_{1}.

In consequence Hol_{θ_{1}} = Hol_{θ_{2}} if and only if the magnetic Laplacians Δ_{G,θ_{1}} and Δ_{G,θ_{2}} are unitarily equivalent.

Lemma 2.2 Let G := (E, V, m) be a connected graph such that 1 ∈ ker Δ_{G,0}. Let θ be magnetic potential. Then ker Δ_{G,θ} ≠ {0} if and only if Hol_{θ} = 0.

Remark 2.3 By construction of the Friedrichs extension, the domain of Δ_{G,0} is given by

D(Δ_{G,0}) = \{f ∈ ℓ^{2}(V, m), x ↦ \frac{1}{m(x)} ∑_{y∈V} E(x, y)(f(x) − f(y)) ∈ ℓ^{2}(V, m)\}

⋂ \overline{C_{c}(V)}^{(∥·∥^{2}+Q_{G,0}(·))^{1/2}}.

The hypothesis 1 ∈ ker Δ_{G,0} is trivially satisfied if G is a finite graph. In general, it is satisfied if and only if:

(∗) 1 belongs to the closure of C_{c}(V) with respect to the norm (∥ · ∥^{2} + Q_{G,0}(·))^{1/2}.

A sufficient condition to guarantee (∗) is that the following two conditions hold true:

1) G is of finite volume, i.e., such that ∑_{x∈V} m(x) < ∞,

2) Δ_{G,0} is essentially self-adjoint on C_{c}(V).

Proof: If Hol_{θ} = 0 then Δ_{G,θ} is unitarily equivalent to Δ_{G,0} by Proposition 2.1 and 1 ∈ ker(Δ_{G,0}) ≠ {0} by hypothesis. Conversely, let f ≠ 0 with Δ_{G,θ}f = 0 and hence Q_{G,θ}(f) = 0. This implies that all terms in the expression of Q_{G,θ}(f) vanish. In particular, if E(x, y) ≠ 0 we have

f(x) = e^{iθ_{x,y}}f(y).  (5)

Assume that there is a cycle γ = (x_{0}, x_{1}, …, x_{N} = x_{0}), such that Hol_{θ}(γ) ≠ 0. Using (5), we obtain that

f(x_{i}) = e^{−iHol_{θ}(γ)}f(x_{i}) .

for all i = 0, …, N − 1. Therefore f|_{γ} = 0. Then, since f ≠ 0, there is x ∈ V such that f(x) ≠ 0. Using again (5) and by connectedness between x and γ, it yields that f(x) = 0. Contradiction. Therefore if there exists f ∈ ker (Δ_{G,θ}) \ {0} then Hol_{θ} = 0. ■

We exhibit the following coupling constant effect.

Corollary 2.4 Let G := (E, V, m) be a connected graph of finite volume, i.e., such that ∑_{x∈V} m(x) < ∞ and let θ be a magnetic potential such that Hol_{θ} ≠ 0. Assume that the function 1 is in ker Δ_{G,θ}. Then there is ν ∈ ℝ such that

ker Δ_{G,λθ} ≠ {0} ⇔ λ = 0 in ℝ/νℤ.

Proof: Let Φ : (ℝ, +) → (Hom_{ℤ}(Z_{1}(G), ℝ/2πℤ), +) be defined by Φ(λ) := Hol_{λθ}. It is a homomorphism of group. Hence its kernel is a subgroup of (ℝ, +). In particular it is either dense with respect to the Euclidean norm or equal to νℤ for some ν ∈ ℝ, e.g., [Bou, Section V.1.1]. Suppose by contradiction that the kernel is dense. Since for any cycle γ of G, the map λ ↦ Hol_{λθ}(γ) is continuous from ℝ to ℝ/2πℤ, we infer that Hol_{λθ}(γ) = 0 for all λ ∈ ℝ. Hence, Φ(λ) = 0 for all λ ∈ ℝ. This is a contradiction with Hol_{θ} ≠ 0. We conclude that there is ν ∈ ℝ such that ker(Φ) = νℤ, i.e., using Proposition 2.1, that

{λ ∈ ℝ, ker Δ_{G,λθ} ≠ {0}} = {λ ∈ ℝ, Hol_{λθ} = 0} = νℤ.

This ends the proof. ■

2.2 The setting

Given G_{1} := (E_{1}, V_{1}, m_{1}) and G_{2} := (E_{2}, V_{2}, m_{2}), the Cartesian product of G_{1} by G_{2} is defined by G := (E, V, m), where V := V_{1} × V_{2}.

\{ \begin{matrix}m(x, y) := & m_{1}(x) × m_{2}(y), \\ E ((x, y), (x′, y′)) := & E_{1}(x, x′) × δ_{y,y′}m_{2}(y) + m_{1}(x)δ_{x,x′} × E_{2}(y, y′), \\ θ ((x, y), (x′, y′)) := & θ_{1}(x, x′) × δ_{y,y′} + δ_{x,x′} × θ_{2}(y, y′), \end{matrix}

We denote by G := G_{1}×G_{2}. This definition generalizes the unweighted Cartesian product, e.g., [Ha]. It is used in several places in the literature, e.g., [Ch][Section 2.6] and in [BGKLM] for a generalization.

⋯ ⋯ The graph of ℤ × ℤ/3ℤ

The terminology is motivated by the following decomposition:

Δ_{G,θ} = Δ_{G_{1},θ_{1}} ⊗ 1 + 1 ⊗ Δ_{G_{2},θ_{2}},

where ℓ^{2}(V, m) ≃ ℓ^{2}(V_{1}, m_{1}) ⊗ ℓ^{2}(V_{2}, m_{2}). The spectral theory of Δ_{G,θ} is wellunderstood since

e^{itΔ_{G,θ}} = e^{itΔ_{G_{1},θ_{1}}} ⊗ e^{itΔ_{G_{2},θ_{2}}} , for t ∈ ℝ.

We refer to [RS][Section VIII.10] for an introduction to the tensor product of self-adjoint operators. In this paper, we are motivated by a geometrical situation. A hyperbolic manifold of finite volume is the union of a compact part and of a cusp, e.g., [Th, Theorem 4.5.7]. The cusp part can be seen as the product of (1, ∞) × M, where (M, g_{M}) is a possibly disconnected Riemannian manifold, endowed with the metric,

y^{−1}(dy^{2} + g_{M}).

On the cusp part, the infimum of the radius of injectivity is 0. To analyze the Laplacian on this product one separates the variables and obtain a decomposition which is not of the type of a Cartesian product, e.g., [GM, Eq. (5.22)] for some details. We aim at mimicking this situation and introduce a modified Cartesian product. Given G_{1} := (E_{1}, V_{1}, m_{1}) and G_{2} := (E_{2}, V_{2}, m_{2}) and I ⊂ V_{2}, we define the product of G_{1} by G_{2} through I by G := (E, V, m), where V := V_{1} × V_{2} and

\{ \begin{matrix}m(x, y) := & m_{1}(x) × m_{2}(y), \\ E ((x, y), (x′, y′)) := & E_{1}(x, x′) × δ_{y,y′} (∑_{z∈I} δ_{y,z}) + δ_{x,x′} × E_{2}(y, y′), \\ θ ((x, y), (x′, y′)) := & θ_{1}(x, x′) × δ_{y,y′} + δ_{x,x′} × θ_{2}(y, y′), \end{matrix}

for all x, x′ ∈ V_{1} and y, y′ ∈ V_{2}. We denote G by G_{1} ×_{I} G_{2}. If I is empty, the graph is disconnected and of no interest for our purpose. If |I| = 1, G_{1} ×_{I} G_{2} is the graph G_{1} decorated by G_{2}, see [SA] for its spectral analysis in the unweighted case. If I = V_{2} and m = 1, we notice that G_{1} ×_{I} G_{2} = G_{1} × G_{2}.

⋯ ⋯ The graph of ℤ

The graph of ℤ/3ℤ

⋯ ⋯ The graph of ℤ ×_{I} ℤ/3ℤ, with |I| = 1

⋯

⋯

The graph of ℤ ×_{I} ℤ/3ℤ, with |I| = 2

⋯

⋯

The graph of ℤ ×_{I} ℤ/3ℤ, with |I| = 3

Under the representation ℓ^{2}(V, m) ≃ ℓ^{2}(V_{1}, m_{1}) ⊗ ℓ^{2}(V_{2}, m_{2}),

deg_{G}(·) = deg_{G_{1}}(·) ⊗ \frac{1_{I}(·)}{m_{2}(·)} + \frac{1}{m_{1}(·)} ⊗ deg_{G_{2}}(·)  (6)

and

Δ_{G,θ} = Δ_{G_{1},θ_{1}} ⊗ \frac{1_{I}(·)}{m_{2}(·)} + \frac{1}{m_{1}(·)} ⊗ Δ_{G_{2},θ_{2}}.  (7)

If m is non-trivial, we stress that the Laplacian obtained with our product is usually not unitarily equivalent to the Laplacian obtained with the Cartesian product. However, there is a potential V : V → ℝ such that Δ_{G_{1}×G_{2}} is unitarily equivalent to Δ_{G_{1}×_{V_{2}} G_{2}} + V (·), in ℓ^{2}(V, m).

Definition 2.5 Set G_{1} := (E_{1}, V_{1}, m_{1}), G_{2} := (E_{2}, V_{2}, m_{2}), and I ⊂ V_{2}. We say that G = G_{1}×_{I} G_{2} is a discrete cusp if the following hypotheses are satisfied:

(H1) m_{1}(x) tend to 0 as |x| → ∞,

(H2) G_{2} is finite,

(H3) Δ_{G_{1},θ_{1}} is bounded (or equivalently sup_{x∈V_{1}} deg_{G_{1}}(x) < ∞).

We now motivate the choice of the above hypotheses by discussing the radius of injectivity. We start by defining a different metric on V, this choice is motivated by the works of [CTT2] and [MiT] but it needs a small adaptation for our purpose.

Definition 2.6 Given G := (E, V, m), the weighted length of an edge (x, y) ∈ E defined by:

L_{G}((x, y)) := \sqrt{\frac{min (m(x), m(y))}{E(x, y)}}.

Documenta Mathematica 22 (2017) 1709–1727

Given x, y ∈ V, we define the weighted distance from x to y with respect to this length by:

ρ_{L_{G}}(x, y) := inf_{γ} ∑_{i=0}^{|γ|−1} L_{G}(γ(i), γ(i + 1)),

where γ is a path joining x to y and with the convention that ρ_{L_{G}}(x, x) := 0 for all x ∈ V.

Remark 2.7 Since G is assumed connected, ρ_{L_{G}} is a metric on V. Observe that, by [Ke, Section 3.2.5], ρ_{L_{G}} belongs to the class of intrinsic metrics if and only if the combinatorial vertex degree is bounded. We refer to [Ke] for a general definition, historical references, properties, and applications. However, since Propositions 2.9 and 2.10 do not hold in general with an arbitrary intrinsic metric, we stick to our specific choice of metric.

We turn to the definitions of the girth and of the weighted radius of injectivity. This is essentially a weighted version of the standard ones, e.g, [EGL].

Definition 2.8 Given G := (E, V, m), the girth at x ∈ V of G w.r.t. the weighted length L_{G} is

girth(x) := inf{L_{G}(γ),

γ simple cycle of unweighted length ≥ 3 and containing x},

where simple cycle means a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex. We use the convention that the girth is +∞ if there is no such cycle.

girth(G) := inf_{x∈V} girth(x).

The radius of injectivity (at x) of G with respect to L_{G} is half the girth (at x). We denote the radius of injectivity by rad(G) (at x by rad(x) respectively)

Note that with this definition, the radius of injectivity of a tree is +∞.

Proposition 2.9 Given G_{1} := (E_{1}, V_{1}, m_{1}) and G_{2} := (E_{2}, V_{2}, m_{2}) and I ⊂ V_{2} Assume that G := G_{1} ×_{I} G_{2} is a discrete cusp. We have:

1) rad(G_{1}) > 0.

2) If rad(G_{2}) < ∞, then rad(G) = 0.

Proof: (1) Assume that rad(G_{1}) = 0. Then for all ε > 0, there is x ∼ y in V_{1} such that L_{G_{1}}((x, y)) < ε. In particular, we have deg_{G_{1}}(x) > ε^{−2} or deg_{G_{1}}(y) > ε^{−2}. This is in contradiction with (H3).

(2) Since rad(G_{2}) < ∞, for all x ∈ V_{1}, there is a pure cycle contained in {x} × V_{2}. Moreover, for all x ∈ V_{1} and a ∼ b in V_{2}, since E(x, x) = 0, we have:

L_{G_{1}×_{I}G_{2}}(((x, a), (x, b))) = \sqrt{m_{1}(x)}L_{G_{2}}((a, b))

By (H1) we obtain that rad(G) = 0. ■

In contrast with this result we see that under the same hypotheses, the Cartesian product is not small at infinity. More precisely, we have:

Proposition 2.10 Set G_{1} := (E_{1}, V_{1}, m_{1}) and G_{2} := (E_{2}, V_{2}, m_{2}). Assume that (H1), (H2), and (H3) are satisfied. Then rad(G_{1} × G_{2}) > 0.

Proof: Assume that rad(G_{1} × G_{2}) = 0. For all ε > 0, there are x_{1} ∼ y_{1} in V_{1} and x_{2} ∼ y_{2} in V_{2} such that

ε > L_{G_{1}×G_{2}}(((x_{1}, x_{2}), (x_{1}, y_{2}))) = L_{G_{2}}((x_{2}, y_{2}))
or ε > L_{G_{1}×G_{2}}(((x_{1}, x_{2}), (y_{1}, x_{2}))) = L_{G_{1}}((x_{1}, y_{1})).

The first line is in contradiction with (H2) and the second line with (H3). ■

2.3 Absence of essential spectrum

We have a first result of absence of essential spectrum. We refer to [CTT3] for related results based on the non-triviality of Hol_{θ} in the context of non-complete graphs. See also [BGKLM] for similar ideas.

Proposition 2.11 Set G_{1} := (E_{1}, V_{1}, m_{1}), G_{2} := (E_{2}, V_{2}, m_{2}), and G := G_{1} ×_{I} G_{2}, with |I| > 0. Assume that (H1), (H2), and Hol_{θ_{2}} ≠ 0 hold true. Then Δ_{G,θ} has a compact resolvent, and

N_{λ} (m_{1}^{−1}(·) ⊗ Δ_{G_{2},θ_{2}}) ≥ N_{λ}(Δ_{G,θ}), for all λ ≥ 0.

Proof: Note that

Δ_{G,θ} ≥ \frac{1}{m_{1}(·)} ⊗ Δ_{G_{2},θ_{2}}

in the form sense on C_{c}(V). Since (H1) and (H2) hold, Lemma 2.2 ensures that 0 is not in the spectrum of (Δ_{G_{2},θ_{2}}). Hence the spectrum of the r.h.s. is purely discrete. By the min-max Principle, e.g., [Go, RS], Δ_{G,θ} has a compact resolvent. ■

2.4 The asymptotic of the eigenvalues

From now on, we focus on the case when the graph is a discrete cusp and aim at a more precise result. To start off, we give the key-stone of our approach:

Proposition 2.12 Set G_{1} := (E_{1}, V_{1}, m_{1}), G_{2} := (E_{2}, V_{2}, m_{2}), and I ⊂ V_{2} non-empty. Assume that G := G_{1} ×_{I} G_{2} is a discrete cusp. We set

M := sup_{x∈V_{1}} deg_{G_{1}}(x) × max_{y∈V_{2}}(1/m_{2}(y)) < ∞.  (8)

We have:

\frac{1}{m_{1}(·)} ⊗ deg_{G_{2}}(·) ≤ deg_{G}(·) ≤ \frac{1}{m_{1}(·)} ⊗ deg_{G_{2}}(·) + M,  (9)

\frac{1}{m_{1}(·)} ⊗ Δ_{G_{2},θ_{2}} ≤ Δ_{G,θ} ≤ 2M + \frac{1}{m_{1}(·)} ⊗ Δ_{G_{2},θ_{2}},  (10)

in the form sense on C_{c}(V).

Proof: Use (1), (6), and (7). ■

We work in the spirit of [Go, BGK, BGKLM] and compare the Laplacian directly with the degree.

Proposition 2.13 Set G_{1} := (E_{1}, V_{1}, m_{1}), G_{2} := (E_{2}, V_{2}, m_{2}), and I ⊂ V_{2} non-empty. Assume that G := G_{1} ×_{I} G_{2} is a discrete cusp. Set M as in (8). We have:

\frac{inf σ(Δ_{G_{2},θ_{2}})}{max_{y∈V_{2}} deg_{G_{2}}(y)} (deg_{G}(·) − M) ≤ Δ_{G,θ} ≤ 2M + 2 deg_{G}(·),  (11)

in the form sense on C_{c}(V).

Moreover, assuming that inf σ(Δ_{G_{2},θ_{2}}) > 0, then D(Δ_{G,θ}^{1/2}) = D (deg_{G}^{1/2}(·)). Furthermore, since lim_{|x|→∞} deg_{G}(x) = ∞, Δ_{G,θ} has a compact resolvent and

0 < \frac{inf σ(Δ_{G_{2},θ_{2}})}{max_{y∈V_{2}} deg_{G_{2}}(y)} ≤ lim inf_{n→∞} \frac{λ_{n}(Δ_{G,θ})}{λ_{n}(deg_{G}(·))} ≤ lim sup_{n→∞} \frac{λ_{n}(Δ_{G,θ})}{λ_{n}(deg_{G}(·))} ≤ 2.

Proof: Use (10) and (1) to get

\frac{inf σ(Δ_{G_{2},θ_{2}})}{max_{y∈V_{2}} deg_{G_{2}}(y)} \frac{1}{m_{1}(·)} ⊗ deg_{G_{2}}(·) ≤ Δ_{G,θ} ≤ 2M + \frac{2}{m_{1}(·)} ⊗ deg_{G_{2}}(·),

Then apply (9) to obtain (11). Concerning the statement about the eigenvalue this follows from the standard consequences of the min-max Principle, e.g., [Go]. ■

Here, trying to compare directly Δ_{G,θ} to deg_{G} to get sharp results about eigenvalues is too optimistic because it is unclear how to obtain constants arbitrarily close to 1 in front of deg_{G}, as in [Go, BGK]. To obtain some sharp asymptotics for the eigenvalues of Δ_{G,θ}, as in (15), we will use directly (10) and analyze very carefully the operator m_{1}^{−1}(·) ⊗ Δ_{G_{2},θ_{2}}.

Theorem 2.14 Set G_{1} := (E_{1}, V_{1}, m_{1}), G_{2} := (E_{2}, V_{2}, m_{2}), and I ⊂ V_{2} nonempty. Assume that G := G_{1} ×_{I} G_{2} is a discrete cusp. We obtain that

D(Δ_{G,θ}^{1/2}) = D (m_{1}^{−1/2}(·) ⊗ Δ_{G_{2},θ_{2}}^{1/2}) .  (12)

Moreover, we have:

1) Δ_{G,θ} has a compact resolvent if and only if Hol_{θ_{2}} ≠ 0.

2) If Hol_{θ_{2}} ≠ 0, then

D(Δ_{G,θ}^{1/2}) = D (deg_{G}^{1/2}(·))

and

lim_{n→∞} \frac{λ_{n} (Δ_{G,θ})}{λ_{n} (m_{1}^{−1}(·) ⊗ Δ_{G_{2},θ_{2}})} = 1.  (13)

Furthermore, setting M as in (8),

N_{λ−2M} (m_{1}^{−1}(·) ⊗ Δ_{G_{2},θ_{2}}) ≤ N_{λ}(Δ_{G,θ}) ≤ N_{λ} (m_{1}^{−1}(·) ⊗ Δ_{G_{2},θ_{2}}) , (14)

for all λ ≥ 0.

Proof: First note that (12) follows directly from (10). Denoting by {g_{i}}_{i=1,..,|V_{2}|} the eigenfunctions associated to the eigenvalues {λ_{i}}_{i=1,..,|V_{2}|} of Δ_{G_{2},θ_{2}}, where λ_{j} ≤ λ_{j+1}, we see that the eigenfunctions of m_{1}^{−1}(·) ⊗ Δ_{G_{2}} are given by {δ_{x} ⊗ g_{i}}, where x ∈ V_{1} and i = 1, .., |V_{2}|. Then, using (H1), we observe that

σ (m_{1}^{−1}(·) ⊗ Δ_{G_{2}}) = \overline{m_{1}^{−1}(V_{1}) × \{λ_{1}, …, λ_{|V_{2}|}\}} = m_{1}^{−1}(V_{1}) × \{λ_{1}, …, λ_{|V_{2}|}\}.

Besides, 0 ∈ σ (m_{1}^{−1}(·) ⊗ Δ_{G_{2}}) if and only if 0 is an eigenvalue of m_{1}^{−1}(·) ⊗ Δ_{G_{2}} of infinite multiplicity if and only if λ_{1} = 0 if and only if Hol_{θ_{2}} = 0, by Lemma 2.2. Moreover, recalling (H1), we see that all the eigenvalues of m_{1}^{−1}(·) ⊗ Δ_{G_{2}} which are not 0 are of finite multiplicity. Therefore, m_{1}^{−1}(·)⊗Δ_{G_{2}} has a compact resolvent if and only if Hol_{θ_{2}} ≠ 0. Combining the latter and (10), the min-max Principle yields the first point. We turn to the second point and assume that Hol_{θ_{2}} ≠ 0. The equality of the form-domains is given by (11). Taking in account (10), the min-max Principle ensures the asymptotic behavior of λ_{n} and the inequalities (14). ■

Remark 2.15 In the case when Hol_{θ_{2}} = 0, for instance when θ_{2} = 0, we see that the form-domain is m_{1}^{−1/2} ⊗ P_{ker(Δ_{G_{2},θ_{2}} )}^{⊥}. In particular, the form-domain is not that of deg_{G}(·). Indeed if the two form-domains are the same, the closed graph theorem yields the existence of c_{1} > 0 and c_{2} > 0 so that

c_{1} deg_{G}(·) − c_{2} ≤ m_{1}^{−1/2} ⊗ P_{ker(Δ_{G_{2},θ_{2}} )}^{⊥}, in the form sense on C_{c}(V). However, note that 0 ∈ σ_{ess} (m_{1}^{−1/2} ⊗ P_{ker(Δ_{G_{2},θ_{2}} )}^{⊥}), whereas deg(·) has a compact resolvent. This is a contradiction with the minmax Principle. We obtain:

D (Δ_{G,θ}^{1/2}) = D (deg^{1/2}(·)) ⇔ Hol_{θ_{2}} ≠ 0

⇔ Δ_{G,θ} has a compact resolvent.

In (13), we exhibit the behaviour of the eigenvalues in terms of an explicit and computable mean. We now aim at comparing the asymptotic with that of the degree, as in [Go, BGK]. The new phenomenon is that we are able to obtain a constant different from 1 in the asymptotic.

Corollary 2.16 Let G_{1} := (E_{1}, V_{1}, m_{1}), G_{2} := (E_{2}, V_{2}, m_{2}), and I ⊂ V_{2} nonempty such that G := G_{1}×_{I} G_{2} is a discrete cusp. Suppose that deg_{G_{2}} is constant on V_{2} and take θ_{2} such that Hol_{θ_{2}} ≠ 0. Then, for all a ∈ [1, +∞[, there exists G̃_{1} := (Ẽ_{1}, V_{1}, m̃_{1}) such that

1) G̃:= G̃_{1} ×_{I} G_{2} is a discrete cusp.

2) E_{1} and Ẽ_{1} have the same zero set.

3) degG̃_{1}(x) ≤ deg_{G_{1}}(x) for all x ∈ V_{1}.

4) Δ_{G̃,θ} is with compact resolvent, and

lim_{λ→∞} \frac{N_{λ} (Δ_{G̃,θ})}{N_{λ} (degG̃(·))} = a.  (15)

Proof: We choose m̃_{1} and Ẽ_{1} later. We denote by {λ_{i}}_{i=1,...,|V_{2}|} the eigenvalues of Δ_{G_{2},θ_{2}}. Since Hol_{θ_{2}} ≠ 0, we have λ_{i} ≠ 0 for all i = 1, …, |V_{2}|. This yields:

N_{λ} (\frac{1}{m̃_{1}} (·) ⊗ Δ_{G_{2},θ_{2}}) = |\{(x, i), \frac{λ_{i}}{m̃_{1}(x)} ≤ λ\}| = ∑_{i=1}^{|V_{2}|} |(\frac{1}{m̃_{1}} )^{[−1]} ([0, \frac{λ}{λ_{i}} ])| ,

where [−1] denotes the reciprocal image. On the other hand,

N_{λ} (\frac{1}{m̃_{1}(·)} ⊗ deg_{G_{2}}) = |V_{2}| × |(\frac{1}{m̃_{1}} )^{[−1]} ([0, \frac{λ}{deg_{G_{2}}} ])| .

Moreover, from (9) we get

N_{λ−M}(m_{1}^{−1}(·) ⊗ deg_{G_{2}}) ≤ N_{λ}(degG̃(·)) ≤ N_{λ}(m_{1}^{−1}(·) ⊗ deg_{G_{2}}),  (16)

for all λ ≥ 0, where M is given by (8).

Step 1: We first aim at a = 1 in (15). Thanks to Lemma 2.18, we choose m̃_{1} and Ẽ_{1} such that the three first points are satisfied and

|\{x ∈ V_{1}, \frac{1}{m̃_{1}(x)} ≤ λ\}| ∼ ln(λ), as λ → ∞,

where ∼ stands for asymptotically equivalent. We obtain:

\frac{N_{λ} (\frac{1}{m̃_{1}(·)} ⊗ Δ_{G_{2},θ_{2}})}{N_{λ} (\frac{1}{m̃_{1}(·)} ⊗ deg_{G_{2}})} ∼ \frac{∑_{i=1}^{|V_{2}|}(ln(λ) − ln(λ_{i}))}{|V_{2}|(ln(λ) − ln(deg_{G_{2}}))} → 1, as λ → ∞. (17)

and for all c ∈ ℝ,

N_{λ−c} (\frac{1}{m̃_{1}(·)} ⊗ deg_{G_{2}}) ∼ |V_{2}| ln(λ − c) ∼ |V_{2}| ln(λ)
∼ N_{λ} (\frac{1}{m̃_{1}(·)} ⊗ deg_{G_{2}}) , as λ → ∞.  (18)

Combining the latter with (16), we infer that for all c ∈ ℝ

N_{λ−c} (\frac{1}{m̃_{1}(·)} ⊗ deg_{G_{2}}) ∼ N_{λ}(degG̃(·)), as λ → ∞.  (19)

Using now (17), this yields that for all c ∈ ℝ

N_{λ−c} (\frac{1}{m̃_{1}(·)} ⊗ Δ_{G_{2},θ_{2}}) ∼ N_{λ}(degG̃(·)), as λ → ∞.  (20)

Finally recalling (14), we infer that

N_{λ} (Δ_{G̃,θ}) ∼ N_{λ} (degG̃(·)) , as λ → ∞.

In other words, there are m̃_{1} and Ẽ_{1} such that the three first points are satisfied and such that (15) is satisfied with a = 1. Step 2: We turn to the case a > 1 in (15). Given α > 0,. Thanks to Lemma 2.18, we choose m̃_{1} and Ẽ_{1} such that the three first points are satisfied and

|\{x ∈ V_{1}, \frac{1}{m̃_{1}(x)} ≤ λ\}| ∼ λ^{α}, as λ → ∞,

We obtain:

\frac{N_{λ} (\frac{1}{m̃_{1}(·)} ⊗ Δ_{G_{2},θ_{2}})}{N_{λ} (\frac{1}{m̃_{1}(·)} ⊗ deg_{G_{2}})} ∼_{λ→∞} \frac{1}{|V_{2}|} ∑_{i=1}^{|V_{2}|} (\frac{deg_{G_{2}}}{λ_{i}} )^{α} =: F (α).

Documenta Mathematica 22 (2017) 1709–1727

First note that

lim_{α→1^{+}} F (α) = 1.

Next, the sum of the eigenvalues (counted with multiplicity) of Δ_{G_{2},θ_{2}} is equal to |V_{2}| deg_{G_{2}}. Therefore, there exists at least one eigenvalue λ_{i}, with 1 ≤ i ≤ |V_{2}| so that deg_{G_{2}} > λ_{i}. In particular

lim_{α→+∞} F (α) = +∞.

Finally, by continuity of F, we obtain that for all a > 1 there is α > 1 such that F (α) = a. To conclude, repeating the end of step 1, we obtain that for all a > 1, there are m̃_{1} and Ẽ_{1} such that the three first points are satisfied and such that (15) is satisfied. ■

Remark 2.17 In [Go, BGK], the asymptotic in N_{λ} was not discussed since the estimates that they obtain seem too weak to conclude. Being able to compute N_{λ} in an explicit way, as in (15), is a new phenomenon.

We have used the following lemma:

Lemma 2.18 Let G_{1} := (E_{1}, V_{1}, m_{1}) be a graph satisfying (H1) and (H3) in Definition 2.5 and let f : [1, +∞) → [1, +∞) be a continuous and strictly increasing function that tends to +∞ at +∞. There exists G̃_{1} := (Ẽ_{1}, V_{1}, m̃_{1}) such that

1) E and Ẽ have the same zero set.

2) (H1) and (H3) are satisfied for G̃_{1}.

3) degG̃_{1}(x) ≤ deg_{G_{1}}(x) for all x ∈ V_{1}.

4) We have:

|\{x ∈ V_{1}, \frac{1}{m̃_{1}(x)} ≤ λ\}| ∼ f(λ), as λ → ∞,

where ∼ stands for asymptotically equivalent.

Proof: Without any loss of generality, one may suppose that f(1) = 1. Let ϕ : ℕ^{∗} → V_{1} be a bijection. Set:

m̃_{1}(ϕ(n)) := \frac{1}{f^{[−1]}(n)},

where [−1] denotes the reciprocal image. Note that (H1) is satisfied. Moreover,

|\{x ∈ V_{1}, \frac{1}{m̃_{1}(x)} ≤ λ\}| = |\{n ∈ ℕ^{∗}, n ≤ f(λ)\}| = ⌊f(λ)⌋ + 1 ∼ f(λ),

as λ → ∞. Finally, we set:

Ẽ_{1}(x, y) := E_{1}(x, y)\frac{min(m̃_{1}(x), m̃_{1}(y))}{max(m_{1}(x), m_{1}(y))}.

The first point is clear. For (H3), note that degG̃_{1}(x) ≤ deg_{G_{1}}(x) for all x ∈ V_{1}.

■

We end this section by proving the results stated in the introduction.

Proof of Theorem 1.1: Let us consider G_{1} := (E_{1}, V_{1}, m_{1}), where

V_{1} := ℕ, m_{1}(n) := exp(−n), and E_{1}(n, n + 1) := exp(−(2n + 1)/2),

for all n ∈ ℕ and G_{2} := (E_{2}, V_{2}, 1) a simple connected finite graph such that |V_{2}| = n. Set G := G_{1} ×_{V_{2}} G_{2}, θ_{1} := 0 and θ_{2} such that Hol_{θ_{2}} ≠ 0. In the spirit of [GM], we denote by P_{κ}^{le} the projection on ker(Δ_{G_{2},κθ_{2}}) and by P_{κ}^{he} is the projection on ker(Δ_{G_{2},κθ_{2}})^{⊥}. Here le stands for low energy and he for high energy. We have that Δ_{G,κθ} := Δ_{G,κθ}^{le} ⊕ Δ_{G,κθ}^{he}, where

Δ_{G,κθ}^{le} := Δ_{G_{1},0} ⊗ P_{κ}^{le},

on (1 ⊗ P_{κ}^{le})ℓ^{2}(V, m), and

Δ_{G,κθ}^{he} := Δ_{G_{1},0} ⊗ P_{κ}^{he} + \frac{1}{m_{1}(·)} ⊗ P_{κ}^{he}Δ_{G_{2},κθ_{2}},

on (1 ⊗ P_{κ}^{he})ℓ^{2}(V, m). By Lemma 2.2, Corollary 2.4, and Remark 2.15, there exists ν > 0 such that

P_{κ}^{le} = 0 ⇔ Hol_{κθ_{2}} ≠ 0
⇔ κ ≠ 0 in ℝ/νℤ ⇔ D (Δ_{G,κθ}^{1/2}) = D (deg_{G}^{1/2}(·)) .

The proof of Theorem 2.14 gives the first point. Assume that κ ∈ ℝ/νℤ. Let U : ℓ^{2}(ℕ, m_{1}) → ℓ^{2}(ℕ, 1) be the unitary map given by Uf(n) := \sqrt{m_{1}(n)}f(n). We see that:

UΔ_{G,κθ}^{le}U^{−1} = Δ_{ℕ,0} + (e^{−1/2} − 1)δ_{0} + e^{1/2} + e^{−1/2} − 2 in ℓ^{2}(ℕ),

where Δ_{ℕ,0} is related to the simple graph of ℕ. By using for instance some Jacobi matrices techniques, it is well-known that the essential spectrum of Δ_{G,κθ}^{le} is purely absolutely continuous and equal to

σ_{ac}(Δ_{G,κθ}^{le}) = [e^{1/2} + e^{−1/2} − 2, e^{1/2} + e^{−1/2} + 2],

with multiplicity one, e.g., [We]. It has a unique eigenvalue and it is negative.

We turn to the high energy part. Denote by {λ_{i}}_{i=1,...,n}, with λ_{i} ≤ λ_{i+1}, the eigenvalues of Δ_{G_{2},κθ_{2}}. Recall that λ_{1} = 0 due to the fact that Hol_{κθ_{2}} = 0. By (10),

\frac{1}{m_{1}(·)} ⊗ Δ_{G_{2},κθ_{2}}P_{κ}^{he} ≤ Δ_{G,κθ}(1 ⊗ P_{κ}^{he}) ≤ 2M + \frac{1}{m_{1}(·)} ⊗ Δ_{G_{2},κθ_{2}}P_{κ}^{he}.

Hence, Δ_{G,κθ}(1 ⊗ P_{κ}^{he}) has a compact resolvent and

N_{λ−2M} (m_{1}^{−1}(·) ⊗ Δ_{G_{2},κθ_{2}}P_{κ}^{he}) ≤ N_{λ} (Δ_{G,κθ}(1 ⊗ P_{κ}^{he}))
≤ N_{λ} (m_{1}^{−1}(·) ⊗ Δ_{G_{2},κθ_{2}}P_{κ}^{he}) ,

for all λ ≥ 0. Finally:

\frac{N_{λ}(\frac{1}{m_{1}(·)} ⊗ Δ_{G_{2},κθ_{2}}P_{κ}^{he})}{N_{λ} (\frac{1}{m_{1}(·)} ⊗ deg_{G_{2}})} ∼ \frac{∑_{i=2}^{n} ln(λ) − ln(λ_{i})}{n(ln(λ) − ln(deg_{G_{2}}))} → \frac{n − 1}{n}, as λ → ∞.

We conclude with (18) for a = 1. ■

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Sylvain Golénia Univ. Bordeaux Bordeaux INP, CNRS, IMB, UMR 5251 F-33400 Talence, France Françoise Truc Grenoble University Institut Fourier Unité mixte de recherche CNRS-UJF 5582 BP 74, 38402-Saint Martin d’Hères Cedex France 1728

Documenta Mathematica 22 (2017)