A LINEAR WEGNER ESTIMATE FOR ALLOY TYPE SCHRÖDINGER OPERATORS ON METRIC GRAPHS

MARIO HELM^{1} AND IVAN VESELIĆ^{1,2}

Abstract. We study spectra of alloy-type random Schrödinger operators on metric graphs. For finite edge subsets of general graphs we prove a Wegner estimate which is linear in the volume (i.e. the number of edges) and the length of the considered energy interval. The single site potential of the alloy-type model needs to have fixed sign, but the considered metric graph does not need to have a periodic structure. The second result we obtain is an exhaustion construction of the integrated density of states for ergodic random Schrödinger operators on metric graphs with a ℤ^{ν}-structure. For certain models the two above results together imply the Lipschitz continuity of the integrated density of states.

1. Introduction

In the present paper we study spectral properties of random Schrödinger operators on a metric graph. More precisely, we consider a countable metric graph and a random Hamiltonian with a so-called alloy type potential. Under suitable assumptions we are able to prove a Wegner estimate for the restriction of the random Hamiltonian to a finite part of the graph. Our Wegner estimate is optimal in the sense that it is linear in the energy and the volume.

In the case that the integrated density of states (IDS) of the random Schrödinger operator exists, the Wegner estimate implies that the IDS is Lipschitz continuous. For certain random operators on ℤ^{ν}-metric graphs we establish the existence of the IDS via an exhaustion procedure. More generally, the existence of an selfaveraging IDS is ensured by certain amenability assumptions on the metric graph and ergodicity assumptions on the random Hamiltonian, see Remark 4.

The literature on Wegner estimates for Schrödinger operators in the continuum, i.e. on ℝ^{ν}, is quite extensive. We refer to the textbook accounts [CFKS87, CL90, PF92, Sto01, Ves04, KM] and the references therein. For the application to spectral localization in random media, a quite weak form of Wegner estimate is sufficient. However, upper bounds which are linear both in the volume and the energy interval length are of independent interest, since they imply the regularity of the IDS. To obtain such a Wegner estimate, the proofs which are presented in the literature need some assumptions, beyond those necessary for a weak form of Wegner estimate. In particular, mostly the covering condition

∑_{k∈ℤ^{ν}} u(x − k) ≥ const. > 0 for all x ∈ ℝ^{ν}

was assumed. Here u is a compactly supported, non-negative single site potential. Papers which are devoted to the question how this assumption can be removed or at least weakened include [Klo95, Kir96, Ves96, CHN01, KV02, Gie, CHK03]. In the recent [CHK] a linear Wegner estimate is proposed without the abovementioned covering condition. However, the deterministic part of the Schrödinger operator needs to be translation invariant. The reason for this assumption is that it implies a uniform, translation invariant kind of unique continuation principle for the solutions of the Schrödinger equation, cf. Section 4 in [CHK03].

Key words and phrases. random Schrödinger operators, alloy type model, quantum graph, metric graph, integrated density of states, Wegner estimate.

In our setting, on metric graphs, the situation is somewhat simpler, since in certain aspects the Schrödinger operators behave as in the one dimensional case. In particular, on the (one dimensional) edges one has a stronger (and simpler) unique continuation principle at disposal. We use techniques developed for Wegner estimates for operators on L^{2}(ℝ) in [KV02]. (See also [Ves96, Gie, CHK03] for similar results for one dimensional random Schrödinger operators.)

This allows us to prove linear Wegner estimates for random operators on very general metric graphs. In particular, we do not need any periodicity condition on the graph. Our assumptions are general enough to include, for instance, operators on the metric graph associated to ℤ^{ν} or, more generally, the Cayley graph of a finitely generated discrete group, or a Penrose tiling graph.

Note that for quantum graphs there exists a certain dichotomy concerning the unique continuation principle. While along a single edge one has a quite strong version of this property, globally on the graph it does not hold. More precisely, a Laplacian (even without potential) on a metric graph with “free boundary conditions” at the vertices may exhibit compactly supported eigenfunctions. For ergodic Hamiltonians on discrete graphs it is well known (see for instance [DLM^{+}03, KLS03, Ves05, Ves06, LV]) that compactly supported eigenfunctions are related to discontinuities of the IDS. Thus, for a complete understanding of the continuity properties of the IDS a systematic analysis of compactly supported eigenfunctions is necessary. See also [GLV] for related results and discussion.

The structure of the paper is as follows: in the next section we formulate our results, Section 3 contains the bulk of the proof of the Wegner estimate, while arguments relating to boundary condition perturbations are deferred to Section 4, and those to the unique continuation principle to Section 5. The last section contains a proof of the construction of the IDS via an exhaustion procedure.

2. Model and Results

First we define the geometric structure of the metric graphs and the operators acting on the associated L^{2}-Hilbert space.

Definition 1. Let V and E be countable sets and G a map

G: E → V × V × [0, ∞), e ↦ (ι(e), τ(e), l_{e}).

We call the triple G = (V, E, G) a metric graph, elements of V = V (G) vertices, elements of E = E(G) edges, ι(e) the initial vertex of e, τ(e) the terminal vertex of e and l_{e} the length of e. Both ι(e) and τ(e) are called endvertices of e, or incident to e. The two endvertices of an edge are allowed to coincide. The number of edged incident to the vertex v is called the degree of v. We assume that the degree is finite for all vertices. The elements of V which have degree equal to one we call boundary vertices and denote their set by V^{∂}. The elements of V^{i} := V \ V^{∂} are called interior vertices.

We will consider the graph G = (V, E, G) as a topological space, more precisely as an one dimensional CW-complex, which we will again denote by G. The 0-skeleton of G is V and the collection of its one dimensional cells is given by E. Each onedimensional cell e ∈ E is attached either to one or to two zero dimensional cells v ∈ V, namely ι(e) and τ(e). This defines the topological structure of G.

We identify each edge e with the open interval (0, l_{e}), where the point 0 corresponds to the vertex ι(e) and l_{e} to τ(e). Assume that there exist constants 0 < l_{−}, l_{+} < ∞ such that for all e ∈ E

0 < l_{−} ≤ l_{e} ≤ l_{+} < ∞.

The identification of edges by intervals allows us to define in a natural way the length of a path between two points in the topological space G. Taking the infimum over the lengths of paths connecting two given points in G, one obtains a distance function d: G × G → [0, ∞). Since we assumed that each vertex of G has bounded degree, the map d is indeed a metric, cf. for instance Section 2.2 in [Sch06]. Thus we have turned G into a metric space (G, d).

For a finite subset Λ ⊂ E we define the subgraph G_{Λ} by deleting all edges e ∈ E \ Λ and the arising isolated vertices. We denote the set of vertices of G_{Λ} by V_{Λ}, the set of vertices v ∈ V_{Λ} of degree one (in G_{Λ}) by V_{Λ}^{∂}, and its complement V_{Λ} \ V_{Λ}^{∂} by V_{Λ}^{i}. Again, elements of V_{Λ}^{∂} are called boundary vertices of G_{Λ} and elements of V_{Λ}^{i} interior vertices of G_{Λ}. Similarly as above we may consider G_{Λ} as a sub-CW-complex of G with an induced topology and metric.

For any Λ ⊂ E the Hilbert spaces L_{2}(G_{Λ}) have a natural direct sum representation L_{2}(G_{Λ}) = ⊕_{e∈Λ}L_{2}(0, l_{e}). In particular for Λ = E we have L_{2}(G) = ⊕_{e∈E}L_{2}(0, l_{e}), and for Λ̃ ⊂ Λ ⊂ E we have L_{2}(GΛ̃) = ⊕_{e∈Λ̃}L_{2}(0, l_{e}) ⊂ L_{2}(G_{Λ}) = ⊕_{e∈Λ}L_{2}(0, l_{e}).

For a function ϕ: G → ℂ and an edge e ∈ E we denote by ϕ_{e} := ϕ|_{e} its restriction to e (which is identified with (0, l_{e})). We denote by C(G) the space of continuous, complex-valued functions on the metric space (G, d). Similarly, C(G_{Λ}) denotes the space of continuous, complex-valued functions on the metric sub-space (G_{Λ}, d). For each v ∈ V, any edge e incident to v, and function f ∈ W^{2,2}(e) ⊂ C^{1}(e) ≅ C^{1}(0, l_{e}) we define the derivatives

(1)  ∂_{e}f(v) := ∂_{e}f(0) := lim_{ϵ↘0} \frac{f(ϵ) − f(0)}{ϵ}  if v = ι(e)

and

(2)  ∂_{e}f(v) := ∂_{e}f(l_{e}) := lim_{ϵ↘0} \frac{f(l_{e}) − f(l_{e} − ϵ)}{ϵ}  if v = τ(e).

Note that, since f|_{e} ∈ W^{2,2}(e), by the Sobolev imbedding theorem the function f is not only continuously differentiable on the open segment (0, l_{e}), but also that its derivative has well defined limits at both boundaries 0 and l_{e}.

For any Λ ⊂ E it will be convenient to use the following Sobolev space

W^{2,2}(Λ) := ⊕_{e∈Λ}W^{2,2}(e) ⊂ C^{1}(Λ) := ⊕_{e∈Λ}C^{1}(e)

with the norm ∥ϕ∥_{W^{2,2}(Λ)}^{2} := ∑_{e∈Λ} ∥ϕ_{e}∥_{W^{2,2}(0,l_{e})}^{2}.

Note that this space is defined on the edge set only and does not see the graph structure of G. The operators which we consider will be defined on the space

D(Δ_{Λ}) := {f ∈ W^{2,2}(Λ) ∩ C(G_{Λ}) |

∀v ∈ V_{Λ}^{i} : ∑_{e∈E,ι(e)=v}∂_{e}f(v) = ∑_{e∈E,τ(e)=v}∂_{e}f(v), ∀v ∈ V_{Λ}^{∂} : f(v) = 0}. (3)

For each Λ ⊂ E we define a linear operator

−Δ_{Λ} : D(Δ_{Λ}) → L^{2}(Λ)

by the rule

(−Δ_{Λ}f)(x) := −\frac{∂^{2}f_{e}(x)}{∂x^{2}}

if x ∈ G is contained in the edge e. This way the function −Δ_{Λ}f is defined on the set E ⊂ G, whose complement V = G \ E in the metric space G has Hausdorff measure zero.

The operator −Δ_{Λ} is selfadjoint on the domain D(Δ_{Λ}), see for instance [Kuc04, KS99] or Section 3.3 in [Sch06]. At the boundary vertices it has clearly Dirichlet boundary conditions, while the type of boundary conditions it has at the interior vertices is called “free boundary conditions” by some authors and “Kirchhoff boundary conditions” by others.

Next we describe the potential energy part of the Hamiltonian. Since it is random, we need an appropriate probability space.

Let ω_{−} < ω_{+} be real numbers, Ω a probability space, and ℙ a probability measure on Ω. Denote by 𝔼 the mathematical expectation on Ω with respect to ℙ. Let W: Ω × G → [ω_{−}, ω_{+}] be a stochastic process which is jointly measurable in the variables ω ∈ Ω and x ∈ G. For a fixed ω ∈ Ω we denote by W (ω): L_{2}(G) → L_{2}(G) the multiplication operator W (ω)f(x) = (W (ω, x)f)(x), x ∈ G.

To derive our Theorems 2 and 3 below we will need more specific hypotheses on ℙ and W (ω). These are formulated in what follows. The first assumption describes random potentials of alloy type for which we are able to prove a Wegner estimate.

Assumption 1. Let c_{+} ≥ c_{−} > 0, s > 0 and c_{g} be real numbers. For each edge e ∈ E let μ_{e} be a probability measure which is absolutely continuous with respect to Lebesgue measure. More precisely, for e ∈ E let g_{e} ∈ L_{∞}[ω_{−}, ω_{+}], ∥g_{e}∥_{∞} ≤ c_{g}, and dμ_{e}(t) = g_{e}(t) dt on the interval [ω_{−}, ω_{+}]. Let the probability space Ω be given by the cartesian product ×_{e∈E}[ω_{−}, ω_{+}], and ℙ by the product ⊗_{e∈E}μ_{e}.

For each e ∈ E let u_{e} ∈ L_{∞}(0, l_{e}) be a single site potential satisfying

c_{−} χ_{S_{e}} ≤ u_{e} ≤ c_{+} χ_{[0,l_{e}]},

where S_{e} ⊂ [0, l_{e}] is an interval of length |S_{e}| ≥ s. We imbed L_{∞}(0, l_{e}) ≅ L_{∞}(e) in L_{∞}(G) and thus consider u_{e} as an element of the latter space. Let the random potential W (ω) have the form

W (ω): L_{2}(G) → L_{2}(G), W (ω) = ∑_{e∈E} ω_{e}u_{e}.

For quantum graphs with a ℤ^{ν}-structure we will establish the existence of the integrated density of states in Theorem 3. The following hypothesis formulates precisely what type of ℤ^{ν}-structure we need for this result.

Assumption 2. The vertex set V consists of the points ℤ^{ν} ⊂ ℝ^{ν}. The set of edges E consists of segments e = [x, y] parallel to the coordinate axes in ℝ^{ν} where x, y ∈ V have Euclidean distance equal to one. The union V ∪ E ⊂ ℝ^{ν} inherits the metric structure of ℝ^{ν}. This structure coincides with the one defined earlier and we denote the corresponding metric space again by (G, d).

Assume that for each x ∈ ℤ^{ν} there is a measure preserving map τ_{x}: Ω → Ω such that {τ_{x}}_{x∈ℤ^{ν}} forms an additive group which acts ergodically on (Ω, F, ℙ).

Assume that the random potential transforms in the following way under translations: W (τ_{x}ω, y) = W (ω, y − x) for all x ∈ ℤ^{ν}, y ∈ G and ω ∈ Ω. Thus the Schrödinger operator is equivariant in the sense that

(4)  U_{x}H(ω)U_{x}^{∗} = H(τ_{x}ω) for all x ∈ ℤ^{ν} and ω ∈ Ω

where U_{x}f(y) = f(y − x), y ∈ G denotes the unitary translation operator by x ∈ ℤ^{ν}.

The restriction of the operator W (ω) to L_{2}(Λ) will be denoted by W_{Λ}(ω) or, if the set Λ is clear from the context, simply by W (ω) again. Note that the norm of the operator W_{Λ}(ω) is uniformly bounded in ω ∈ Ω and Λ ⊂ E. Thus the operator H_{Λ}(ω) := −Δ_{Λ} + W_{Λ}(ω) is selfadjoint on D(Δ_{Λ}) for any ω ∈ Ω and Λ ⊂ E. If the set Λ ⊂ E contains a single element e we write −Δ_{e}(ω) for −Δ_{Λ}(ω) and H_{e}(ω) for H_{Λ}(ω).

A metric graph G together with a Schrödinger H operator which is defined on it we call a quantum graph.

Our main result is the following:

Theorem 2. If Assumption 1 holds, then there exists for any λ ∈ ℝ a constant C such that for all ϵ ∈ [0, 1],

𝔼{tr χ_{[λ−ϵ,λ+ϵ]}(H_{Λ}(ω))} ≤ C · ϵ · ♯Λ.

If in the terminology of Assumption 1 we have S_{e} = [0, l_{e}] for all e ∈ E, then the constant appearing in Theorem 2 can be chosen uniformly in the energy parameter λ ∈ ℝ.

Now we turn to the situation where the quantum graph has a ℤ^{ν}-structure as formulated in Assumption 2, and describe how the integrated density of states can be defined by an exhaustion procedure.

For any l ∈ ℕ denote by Λ_{l} the set of edges which are contained in

{x ∈ ℝ^{ν} | x_{i} ∈ (0, l) for all i = 1, …, d},

and abbreviate V_{Λ_{l}} by V_{l}, and G_{Λ_{l}} by G_{l}. Denote by ∂G_{l} the boundary of G_{l} as a subset of the metric space G, i.e. the vertices in V_{l} which have degree one. Similarly abbreviate Δ_{l} := Δ_{Λ_{l}} and H_{l}(ω) := H_{Λ_{l}}(ω). These are restrictions of the operators Δ and H(ω) to a finite cube with sidelength l and with Dirichlet boundary conditions on the boundary of the cube. The finite cube Schrödinger operator H_{l}(ω) is again self-adjoint, lower bounded and has purely discrete spectrum. Let us enumerate the eigenvalues of H_{l}(ω) in ascending order λ_{1}(H_{l}(ω)) < λ_{2}(H_{l}(ω)) ≤ λ_{3}(H_{l}(ω)) ≤ … and counting multiplicities.

Thus for each λ ∈ ℝ and l ∈ ℕ, the counting function

F_{ω}^{l}(λ) := ♯{n ∈ ℕ | λ_{n}(H_{l}(ω)) ≤ λ}

is monotone increasing and right-continuous, i.e. a distribution function, which is associated to a pure point measure. Denote by N_{ω}^{l}(λ) := \frac{1}{l^{ν}} F_{ω}^{l}(λ) the volume-scaled version of F_{ω}^{l}(λ).

Theorem 3. Let Assumption 2 hold, then there exists a distribution function N: ℝ → ℝ and a subset Ω′ ⊂ Ω of measure one such that for all ω ∈ Ω and for all λ ∈ ℝ where N is continuous the convergence

(5)  lim_{l→∞} N_{ω}^{l}(λ) = N(λ)

holds.

Remark 4. Under certain additional assumptions it is possible to prove that the normalized finite volume eigenvalue counting functions converge uniformly in the energy parameter to the IDS, see [GLV]. For metric graphs whose isometry group does not exhibit a ℤ^{ν}-structure, but is merely amenable, it should be still possible to define the IDS of an ergodic Hamiltonian by an exhaustion, proceeding analogously as in the paper [LPV04].

3. Proof of the Wegner estimate

For the purposes of the proof it will be necessary to differentiate the spectral projection with respect to the energy parameter, which motivates the introduction of the following smooth ’switch function’.

Let ρ be a smooth, non-decreasing function such that on (−∞, −ϵ] it is identically equal to −1, on [ϵ, ∞) it is identically equal to zero and ∥ρ′∥_{∞} ≤ 1/ϵ. Then

χ_{(λ−ϵ,λ+ϵ)}(x) ≤ ρ(x − λ + 2ϵ) − ρ(x − λ − 2ϵ) = ∫_{−2ϵ}^{2ϵ} ρ′(x − λ + t) dt

Thus by the spectral theorem

χ_{(λ−ϵ,λ+ϵ)}(H_{Λ}(ω)) ≤ ∫_{−2ϵ}^{2ϵ} dt ρ′(H_{Λ}(ω) − λ + t)

in the sense of quadratic forms. Since B_{ϵ}(λ) = (λ−ϵ, λ+ϵ) is bounded and σ(H_{Λ}(ω)) discrete, the above operators are trace class and we may estimate:

tr [χ_{B_{ϵ}(λ)}(H_{Λ}(ω))] ≤ tr [ ∫_{−2ϵ}^{2ϵ} ρ′(H_{Λ}(ω) − λ + t) dt] = ∑_{n∈ℕ} ∫_{−2ϵ}^{2ϵ} ρ′(λ_{n}^{Λ}(ω) − λ + t) dt

where λ_{n}^{Λ}(ω) denotes the eigenvalues of H_{Λ}(ω) enumerated in non-decreasing order and counting multiplicities. Only a finite number of terms in the sum are non-zero.

Proof of Theorem 2. Let ρ be as above. Similarly as in [Kir96], p. 509, we estimate

𝔼{tr χ_{[λ−ϵ,λ+ϵ]}(H_{Λ}(ω))} ≤ ∫_{[ω_{−},ω_{+}]^{Λ}} ∑_{n∈ℕ} ∫_{−2ϵ}^{2ϵ} ρ′(λ_{n}^{Λ}(ω) − λ + t) dt ∏_{ẽ∈Λ} dμẽ.

To bound the right hand side we use Corollary 9 in Section 5 and the monotone convergence theorem to obtain the estimate

C_{1} ∫_{[ω_{−},ω_{+}]^{Λ}} ∑_{n∈ℕ} ∫_{−2ϵ}^{2ϵ} ∑_{e∈Λ} \frac{∂ρ(λ_{n}^{Λ}(ω) − λ + t)}{∂ω_{e}} dt ∏_{ẽ∈Λ} dμẽ(ωẽ)

≤ C_{1} ∑_{e∈Λ} ∫_{[ω_{−},ω_{+}]^{A}} ∏_{ẽ∈A}dμẽ(μẽ) ∑_{n∈ℕ} ∫_{−2ϵ}^{2ϵ} ∫_{ω_{−}}^{ω_{+}} \frac{∂ρ(λ_{n}^{Λ}(ω) −λ+ t)}{∂ω_{e}} dμ_{e}(ω_{e})dt. (6)

Here we used the abbreviation A := Λ \ {e}. Denote by H_{Λ}(e, ω_{+}) and H_{Λ}(e, ω_{−}) the operator H_{Λ}, where the random variable ω_{e} is set to its maximum respectively its minimum value. The n−th eigenvalues of this operators are abbreviated by λ_{n}^{Λ}(e, ω_{+}) and λ_{n}^{Λ}(e, ω_{+}). Using monotonicity, the sum over n in (6) can be estimated by

(7)  ∥g_{e}∥_{∞} ∑_{n∈ℕ} ∫_{−2ϵ}^{2ϵ}{ ρ(λ_{n}^{Λ}(e, ω_{+}) − λ + t) − ρ(λ_{n}^{Λ}(e, ω_{−}) − λ + t) } dt.

We introduce now the operators H_{Λ}^{∗}(e, ω), ∗ ∈ {D, N}, that coincide with H_{Λ}(ω) up to additional Dirichlet, respectively Neumann b.c. at the vertices ι(e) and τ(e). Their eigenvalues are denoted by λ_{n}^{Λ,∗}. By Lemma 7 on Dirichlet-Neumann bracketing in Section 4 we have

(8) ρ(λ_{n}^{Λ}(e, ω_{+}) − λ + t) − ρ(λ_{n}^{Λ}(e, ω_{−}) − λ + t)

≤ ρ(λ_{n}^{Λ,D}(e, ω_{+}) − λ + t) − ρ(λ_{n}^{Λ,N}(e, ω_{−}) − λ + t).

Because of the decoupling of the edge e, the latter operators can be written as the direct sum H_{Λ}^{∗}(e, ω) = H_{Λ}^{e,∗}(e, ω) ⊕ H_{Λ}^{c,∗}(e, ω) of operators acting on L_{2}(0, l_{e}) and L_{2}(G_{Λ}\e), respectively. For the eigenvalues of these operators we use the notation λ_{n}^{e,∗}(e, ω) and λ_{n}^{c,∗}(e, ω). Hence the sum over the terms in (8) can be separated in the corresponding parts:

∑_{n∈ℕ} ρ(λ_{n}^{Λ,D}(e, ω_{+}) − λ + t) − ρ(λ_{n}^{Λ,N}(e, ω_{−}) − λ + t) (9)

(10)  = ∑_{n∈ℕ} ρ(λ_{n}^{e,D}(e, ω_{+}) − λ + t) − ρ(λ_{n}^{e,N}(e, ω_{−}) − λ + t)
(11)  + ∑_{n∈ℕ} ρ(λ_{n}^{c,D}(e, ω_{+}) − λ + t) − ρ(λ_{n}^{c,N}(e, ω_{−}) − λ + t).

We estimate first the sum in (10). The difference H_{Λ}^{e,D} − H_{Λ}^{e,N} is a perturbation of rank 2 in resolvent sense, see for instance [Sim95]. By Lemma 6 in Section 4 the first term in (10) obeys the bound

ρ(λ_{n}^{e,D}(e, ω_{+}) − λ + t) ≤ ρ(λ_{n+2}^{e,N}(e, ω_{+}) − λ + t).

By a telescoping argument we obtain the estimate

∑_{n∈ℕ} ρ(λ_{n+2}^{e,N}(e, ω_{+}) − λ + t) − ρ(λ_{n}^{e,N}(e, ω_{+}) − λ + t) ≤ 2

where we used that the total variation of ρ equals one. Thus we are left with estimating

(12) ∑_{n∈ℕ} ρ(λ_{n}^{e,N}(e, ω_{+}) − λ + t) − ρ(λ_{n}^{e,N}(e, ω_{−}) − λ + t)

= tr [ρ(H_{Λ}^{e,N}(e, ω_{+}) − λ + t) − ρ(H_{Λ}^{e,N}(e, ω_{−}) − λ + t)].

Note that by definition of ρ we have ρ(x−λ+t) ≤ χ_{[λ−3ϵ,∞)}(x) and −ρ(x−λ+t) ≤ χ_{(λ+3ϵ,∞)}(x) for all t ∈ [−2ϵ, 2ϵ]. Thus the trace in (12) is bounded by

tr [χ_{[λ−3ϵ,∞)}(H_{Λ}^{e,N}(e, ω_{+})) − χ_{(λ+3ϵ,∞)}(H_{Λ}^{e,N}(e, ω_{−}))]

≤ tr [χ_{[λ−3ϵ,∞)}(H_{Λ}^{e,N}(e, ω_{−}) + (ω_{+} − ω_{−})∥u_{e}∥_{∞}) − χ_{(λ+3ϵ,∞)}(H_{Λ}^{e,N}(e, ω_{−}))]
= 2 + tr (χ_{[λ−3ϵ−(ω_{+}−ω_{−})∥u_{e}∥_{∞},λ+3ϵ)}(H_{Λ}^{e,N}(e, ω_{−})))
≤ C_{2}.

Here C_{2} is a constant which is independent of Λ and depends only on the considered energy interval. In fact, a careful look reveals that C_{2} depends only on the length (6ϵ + (ω_{+} − ω_{−})∥u_{e}∥_{∞}) of the energy interval, but not on λ ∈ ℝ. See for instance Section 3 in [GLV].

Next we want to estimate the sum in (11). Let d̃:= deg(ι(e)) + deg(τ(e)) − 2. Similarly as above, one sees that the difference H_{Λ}^{c,D}(e, ω_{+}) − H_{Λ}^{c,N}(e, ω_{+}) is a perturbation of rank d̃in resolvent sense. Consequently, the first term in (11) obeys the bound

ρ(λ_{n}^{c,D}(e, ω_{+}) − λ + t) ≤ ρ(λ_{n+d̃}^{c,N}(e, ω_{+}) − λ + t).

A telescoping argument bounds the whole sum in (11) by d̃times the total variation of ρ, i.e. by d̃.

Hence the following upper bound on (6) completes the proof:

C_{1}∑_{e∈Λ}∥g_{e}∥_{∞} ∫_{−2ϵ}^{2ϵ}(d̃+ C_{2}) dt ≤ 4C_{1}(C_{2} + d̃) c_{g} · ϵ · ♯Λ.

□

4. Changing boundary conditions

In this section we show how to control the shifting of eigenvalues when one modifies boundary conditions. This is used in the proof of Theorem 2 above.

Lemma 6 shows that changing the boundary conditions from the Dirichlet operator H_{Λ}^{e,D}(e, ω_{−}) to the Neumann one H_{Λ}^{e,N}(e, ω_{−}) shifts the eigenvalue index at most by the rank of the perturbation, i.e. by 2. The proof of this fact is based on the min-max formula for eigenvalues, see e.g. [RS78], Theorem XIII.1.

Lemma 7 is a monotonicity statement about boundary conditions known as Dirichlet-Neumann-bracketing. By quadratic form considerations one sees that the passage from Neumann to Dirichlet boundary condition shifts the eigenvalues up.

First we proof an auxiliary lemma for bounded operators.

Lemma 5. Let S, T = S+V be bounded, selfadjoint operators on a separable Hilbert space H with dim (ker V)^{⊥} = d < ∞. Assume that the spectra of S and T below inf σ_{ess}(S), respectively inf σ_{ess}(T), consist of an infinite, discrete set of eigenvalues.

Then for the m-th eigenvalues of S and T, counted in ascending order including multiplicities, we have λ_{m}(T) ≤ λ_{m+d}(S) (∀m ∈ ℕ).

Proof. Let L̃ := (ker V)^{⊥}. By the min-max principle one has

λ_{m}(T) = max_{dim L=m−1} min_{∥φ∥=1, φ⊥L} (φ | T φ)
≤ max_{dim L=m−1} min_{∥φ∥=1, φ⊥L+L̃} [(φ | Sφ) + \underbrace{(φ | V φ)}_{=0}]
≤ max_{dim L̂=m+d−1} min_{∥φ∥=1, φ⊥L̂} (φ | Sφ)
= λ_{m+d} (S).
□

The lemma above can of course not be applied to the Hamiltonians under consideration because they are unbounded operators. But there are some related operators for which the statement is true - one can simply compare suitable resolvents and rearrange the eigenvalues by the spectral theorem.

Lemma 6. For the operators H_{Λ}^{e,D}(e, ω_{−}) and H_{Λ}^{e,N}(e, ω_{−}) we have

(13)  λ_{m}(H_{Λ}^{e,D}(e, ω_{−})) ≤ λ_{m+2}(H_{Λ}^{e,N}(e, ω_{−})).

Proof. We shift the spectrum of both operators by addition of a suitable constant and work in the following with the two arising strictly positive operators H_{1} (Dirichlet case) and H_{2} (Neumann case).

Let D_{0} := D(H_{1}) ∩ D(H_{2}). Then one has

(14)  H_{1}^{−1} − H_{2}^{−1} |_{H_{2}D_{0}} = H_{1}^{−1}(H_{2} − H_{1})H_{2}^{−1} |_{H_{2}D_{0}}.

By definition, H_{1}|_{D_{0}} − H_{2}|_{D_{0}} = 0, such that by (14) and continuity we get

H_{1}^{−1} − H_{2}^{−1} |\overline{H_{2}D_{0}} = 0,

i.e. ker (H_{1}^{−1} − H_{2}^{−1})^{⊥} ⊂ H_{2}D_{0}^{⊥}.

We want to apply Lemma 5 to the bounded operators −H_{1}^{−1} and −H_{2}^{−1}. So we have to show that dim ker(H_{1}^{−1} − H_{2}^{−1})^{⊥} ≤ 2, for what in turn dim H_{2}D_{0}^{⊥} ≤ 2 is sufficient. For this purpose we show next that D_{0} and D(H_{2}) differ by a 2dimensional subspace L.

Let ϕ_{1}, ϕ_{2} ∈ C^{∞}(0, l_{e}) be such that ϕ_{1} ≡ 1 in a neighborhood of 0 and ϕ_{1} ≡ 0 in a neighborhood of l_{e}, and similarly let ϕ_{2} ≡ 0 in a neighborhood of 0 and ϕ_{2} ≡ 1 in a neighborhood of l_{e}. Thus ϕ_{1}, ϕ_{2} are linearly independent vectors and moreover elements of D(H_{2}) \ D_{0}.

Let ψ an arbitrary element of D(H_{2}), c_{1} := ψ(0) and c_{2} := ψ(l_{e}). Then

(ψ − c_{1}ϕ_{1} − c_{2}ϕ_{2})′(0) and (ψ − c_{1}ϕ_{1} − c_{2}ϕ_{2})′(l_{e}) = 0

as well as

(ψ − c_{1}ϕ_{1} − c_{2}ϕ_{2})(0) = 0 and (ψ − c_{1}ϕ_{1} − c_{2}ϕ_{2})(l_{e}) = 0.

Hence ψ − c_{1}ϕ_{1} − c_{2}ϕ_{2} ∈ D_{0} what implies that dim D(H_{2}) \ D_{0} = 2.

Now, H_{2} : D(H_{2}) → H is one to one, and we get

H = H_{2}D(H_{2})
= H_{2}(D_{0} + L)
= H_{2}D_{0} + H_{2}L.

Hence dim H_{2}D_{0}^{⊥} ≤ dim H_{2}L ≤ 2.

So Lemma 5 is applicable to −H_{1}^{−1} and −H_{2}^{−1}, and inequality (13) follows by the spectral theorem. □

It can be seen easily that the operators H_{Λ}^{c,D}(e, ω_{−}) and H_{Λ}^{c,N}(e, ω_{−}) in the proof of Theorem 2 can be treated in the same way.

Lemma 7. Let ω ∈ Ω, Λ ⊂ E finite, and e ∈ Λ be arbitrary. Consider the operators H_{Λ}(e, ω), H_{Λ}^{D}(e, ω) and H_{Λ}^{N}(e, ω) and their eigenvalues λ_{n}^{Λ}(e, ω), λ_{n}^{Λ,D}(e, ω) and λ_{n}^{Λ,N}(e, ω) (defined in Section 6). Then the following inequalities hold for all n ∈ ℕ

λ_{n}^{Λ,N}(e, ω) ≤ λ_{n}^{Λ}(e, ω) ≤ λ_{n}^{Λ,D}(e, ω). (15)

Proof. Denote by h_{Λ}(e, ω), h_{Λ}^{D}(e, ω) and h_{Λ}^{N}(e, ω) the quadratic forms associated to the operators H_{Λ}(e, ω), H_{Λ}^{D}(e, ω) and H_{Λ}^{N}(e, ω). Then by Section 4.2 in [Sch06] the quadratic form domains obey

h_{Λ}^{N}(e, ω) ⊃ h_{Λ}(e, ω) ⊃ h_{Λ}^{D}(e, ω).

Thus the statement of the lemma follows immediately if one applies the quadratic form version of the min-max formula for eigenvalues, see e.g. [RS78], Theorem XIII.2. □

5. Single site potentials of small support

In this section we prove a uniform lower bound on the sum of derivatives of eigenvalues which is formulated in

Lemma 8.

(16)  ∑_{e∈Λ} \frac{∂λ_{n}^{Λ}(ω)}{∂ω_{e}} ≥ C_{1}(I) > 0

for all eigenvalues λ_{n}^{Λ} of H_{Λ}(ω) inside a bounded energy interval I. The bound C(I) does not depend on the set of edges Λ ⊂ E and on the eigenvalue index n ∈ ℕ.

We infer immediately:

Corollary 9. Let ρ : ℝ → [0, 1] be a smooth, monotone function with ρ = −1 on (−∞, −ϵ] and ρ = 0 on [ϵ, ∞). Then, for the n−th eigenvalue λ_{n}^{Λ} of H_{Λ}(ω) we have

ρ′(λ_{n}^{Λ}(ω) − λ + t) ≤ C_{1}(I) ∑_{e∈Λ} \frac{∂ρ(λ_{n}^{Λ}(ω) − λ + t)}{∂ω_{e}}.

Proof. By the chain rule

∑_{e∈Λ} \frac{∂ρ(λ_{n}^{Λ}(ω) − λ + θ)}{∂ω_{e}} = ρ′(λ_{n}^{Λ}(ω) − λ + θ) ∑_{e∈Λ} \frac{∂λ_{n}^{Λ}(ω)}{∂ω_{e}} ,

(16) implies the estimate:

ρ′(λ_{n}^{Λ}(ω) − λ + θ) ≤ C_{1}(I)^{−1} ∑_{e∈Λ} \frac{∂ρ(λ_{n}^{Λ}(ω) − λ + θ)}{∂ω_{e}}.

□

To infer the lower bound (16), set S = ⋃_{e∈Λ} S_{e} and apply the Hellman-Feynman theorem, i.e. first order perturbation theory. For a normalized eigenfunction ψ_{n} corresponding to λ_{n}^{Λ}(ω) we have:

∑_{e∈Λ} \frac{∂λ_{n}^{Λ}(ω)}{∂ω_{e}} = ∑_{e∈Λ}(ψ_{n} | u_{e}ψ_{n}) ≥ ∫_{S} |ψ_{n}|^{2}.

If the integral on the rightern side would extend over the whole of Λ, it would be equal to 1 due to the normalization of ψ_{n}. A priori the integral over S could be much smaller, but the following Lemma shows that we can control the ratio of the two integrals.

Lemma 10. Let I be a bounded interval and S_{e} ⊂ [0, l_{e}] a non-degenerate interval. There exists a constant C_{1}(I) > 0 such that

(17)  ∫_{S_{e}} |ψ|^{2} ≥ C_{1}(I) ∫_{0}^{l_{e}} |ψ|^{2}

for all Λ ⊂ E finite, all e ∈ Λ and for any eigenfunction ψ corresponding to an eigenvalue λ ∈ I of H_{Λ}(ω).

Thus ∫_{S} |ψ|^{2} ≥ C_{1}(I) ∫_{G_{Λ}} |ψ|^{2} with the same constant as in (17). Hence Lemma 10 implies directly Lemma 8.

Proof. For y ∈ ℝ denote by S_{e} + y := {x ∈ (0, l_{e}) | x − y ∈ S_{e}} the translations of the set S_{e} along the edge e. The derivative of the function

(18)  ϕ(y) := ∫_{S_{e}+y} |ψ(x)|^{2} dx = ∫_{S_{e}} |ψ(x − y)|^{2} dx

satisfies

|\frac{∂}{∂y} ϕ(y)| = |∫_{S_{e}} [\frac{∂}{∂y} ψ(x − y)] \overline{ψ(x − y)} dx + ∫_{S_{e}} ψ(x − y) \frac{∂}{∂y} \overline{ψ(x − y)} dx|
≤ 2 ∥ψ∥_{L^{2}(S_{e}+y)} ∥ψ′∥_{L^{2}(S_{e}+y)} .

Sobolev norm estimates (e.g. Theorems 7.25 and 7.27 in [GT83]) imply

∥ψ′∥_{L^{2}(S_{e}+y)} ≤ C_{3} ∥ψ∥_{L^{2}(S_{e}+y)} + ∥ψ′′∥_{L^{2}(S_{e}+y)}.

By the eigenvalue equation we have

(19)  |\frac{∂}{∂y} ϕ(y)| ≤ C_{4} ∥ψ∥_{L^{2}(S_{e}+y)}^{2} = C_{4} ϕ(y),  C_{4} = C_{4}(∥W − λ∥_{∞}).

Gronwall’s Lemma implies ϕ(y) ≤ exp(C_{4}|y|) ϕ(0) and thus

∫_{0}^{l_{e}} |ψ|^{2} ≤ e^{C_{4}l_{e}} \frac{l_{e}}{|S_{e}|} ∫_{S_{e}} |ψ|^{2}.

□

6. Proof of Theorem 3

The proof of Theorem 3 on the existence of the integrated density of states follows the arguments of [KM82]. There the convergence (5) of the IDS was established for all rational energies λ ∈ ℚ, whereas we establish the same fact for all λ which are continuity points of the IDS. The main step consists in proving that a superadditive ergodic theorem of the paper [AK81] is applicable.

In the whole of this section we assume that the quantum graph has a ℤ^{ν}-structure as described in Assumption 2.

Let us first describe the type of superadditive processes considered in [AK81]. Denote by T the semigroup of measurable transformations given by τ_{x}, x ∈ ℕ_{0}^{ν}, where ℕ_{0} := ℕ ∪ {0}, and by I the class of sets of the form

{x ∈ ℝ^{ν} | a_{i} ≤ x_{i} < b_{i}, for all i = 1, …, d}

where a, b ∈ ℕ_{0}^{ν}. For x ∈ ℕ_{0}^{ν} and Q ∈ I denote by Q + x := {y ∈ ℝ^{ν} | y − x ∈ Q} the translation of the set Q by x.

Definition 11. A function F: I → L^{1}(Ω, ℙ) which satisfies

(i) F_{Q} ∘ τ_{x} = F_{Q+x} for all Q ∈ I and x ∈ ℕ_{0}^{ν}, (ii) if Q_{1}, …, Q_{n} ∈ I are disjoint sets and if their union Q = ∪_{i=1}^{n}Q_{i} is again an

element of I, then

F_{Q} ≥ ∑_{i=1}^{n} F_{Q_{i}},

(iii) γ(F) := sup_{Q∈I,|Q|>0} \frac{1}{|Q|} 𝔼\{F_{Q}\} < ∞

is called a (discrete) superadditive process.

We state Theorem 2.4 from [AK81]:

Theorem 12. If F is a discrete superadditive process and Q_{l} := {x | 0 ≤ x_{i} < l for all i = 1, …, d}, then

lim_{l→∞} \frac{F_{Q_{l}}(ω)}{l^{ν}}

exists for almost all ω ∈ Ω.

In fact, in the case that the semigroup T acts ergodically on the probability space (Ω, ℙ) one can identify the limit, see the remark on page 59 in [AK81]:

lim_{l→∞} l^{−d} F_{Q_{l}}(ω) = γ(F) almost surely.

In Section 6.2 of [Kre85] the above statements are extended to the case that it is not the semigroup T which acts ergodically on Ω, but rather the full group τ_{x}, x ∈ ℤ^{ν}.

To apply the superadditive ergodic theorem we consider for arbitrary, fixed λ ∈ ℝ the process given by the eigenvalue counting functions of the Schrödinger operator with Dirichlet boundary conditions. For Q ∈ I denote by Λ the set of edges e ∈ E ⊂ ℝ^{ν} which are contained in the interior of Q and set

F_{Q} := F_{Q}(λ, ω) := ♯{n| λ_{n}(H_{Λ}(ω)) < λ}, Q ∈ I.

The Dirichlet Schrödinger operator H_{Λ}(ω) has been defined in Section 2. We will show that F_{Q}, Q ∈ I, is a superadditive process.

Lemma 13. For any fixed energy value λ, the process F_{Q} is superadditive.

Proof. We have to check that the properties (i) – (iii) in Definition 11 hold.

Since Λ = int Q ∩ E we have also Λ + x = (int Q + x) ∩ E for all x ∈ ℤ^{ν}. The equivariance property (4) of the random operators carries over to the spectral projections and thus to the eigenvalue counting functions. Hence property (i) holds.

Property (ii) can bee seen using Lemma 7 on Dirichlet-Neumann bracketing in Section 4. For Q and Q_{1}, …, Q_{n} ∈ I as in (ii), the set Λ contains ⋃_{i=1}^{n} Λ_{i} and finitely many edges e_{1}, …, e_{N} which lie in int Q \ ⋃_{i=1}^{n} int Q_{i}. By introducing Dirichlet boundary conditions at finitely many vertices one obtains from the operator H_{Λ}(ω) the direct sum

(20)  ⨁_{i=1}^{n} H_{Λ_{i}}(ω) ⊕ ⨁_{j=1}^{N} H_{e_{j}}(ω).

Hence the eigenvalue counting function of H_{Λ}(ω) is an upper bound of the one of the direct sum operator (20). Obviously the counting function of ⨁_{i=1}^{n} H_{Λ_{i}}(ω) can be estimated from above by the one of the operator (20). Now property (ii) follows.

Denote the eigenvalue counting function of the negative Dirichlet Laplacian on e = (0, 1) by

n_{0}(λ) := ♯{n ∈ ℕ | λ_{n}(−Δ_{e}) ≤ λ},

Obviously the counting function of

(21)  − ⨁_{e∈Λ} Δ_{e}

equals |Λ| · n_{0}(λ). Since the operators −Δ_{Λ} and (21) differ by Dirichlet boundary conditions at 2 |Λ|, or even less, vertices, the counting function of −Δ_{Λ} is bounded by

|Λ| · n_{0}(λ) + 4d · |Λ|.

For this conclusion we use an argument analogous to Lemma 6 in Section 4.

Since the random potentials we are considering are uniformly bounded by a constant, say K, we have

F_{Q}(λ, ω) ≤ |Λ| · n_{0}(λ + K) + 4d · |Λ| for all λ ∈ ℝ, Q ∈ I.

The number of edges in the set Λ which is associated to a box Q ∈ I is linearly bounded by the volume of Q. Hence (iii) is proven.

□

Now we can complete the

Proof of Theorem 3. For a fixed λ ∈ ℝ one applies the ergodic theorem of [AK81] to the superadditive process F_{Q}(λ, ω), Q ∈ I. Let us denote the corresponding γ(F) by γ(λ). By definition F_{Q}(λ, ω) ≤ F_{Q}(λ̃, ω) for all λ ≤ λ̃ ∈ ℝ and all ω ∈ Ω, Q ∈ I, thus λ ↦ γ(λ) is a non-decreasing function. In particular, it has at most a countable set of discontinuity points. Denote the complement of this set by C and choose a dense countable set S ⊂ C. Hence γ is continuous at each λ ∈ S.

Since in our case the transformation group is ergodic, for each λ there is a set Ω_{λ} of measure one on which the convergence lim_{l→∞} l^{−d} F_{Q_{l}}(ω) = γ(λ) holds.

Since S is countable, Ω′ = ∩_{λ∈S}Ω_{λ} still has full measure and the convergence statement of the superadditive theorem holds for all λ ∈ S and ω ∈ Ω′. Now define the distribution function N(λ) := lim_{S∋λ̃↘λ} γ(λ̃). Thus on the set C the functions γ and N coincide.

The monotonicity of λ ↦ F_{Q_{l}}(λ, ω) and the continuity of N on the set C imply the convergence (5). To see this, choose a sequence λ_{n} ∈ S, λ_{n} ≥ λ, lim_{n→∞} λ_{n} = λ. Then we have

l^{−d}F_{Q_{l}}(λ, ω) − N(λ) ≤ l^{−d}F_{Q_{l}}(λ_{n}, ω) − N(λ_{n}) + N(λ_{n}) − N(λ).

For arbitrary ω ∈ Ω′ and ϵ > 0 we choose first n sufficiently large such that N(λ_{n})− N(λ) ≤ ϵ/2 and then l sufficiently large such that l^{−d}F_{Q_{l}}(λ_{n}, ω) − N(λ_{n}) ≤ ϵ/2. Thus one sees that

lim sup_{l→∞} l^{−d}F_{Q_{l}}(λ, ω) ≤ N(λ).

Similarly one can choose a sequence λ_{n} ∈ S, λ_{n} ≤ λ, lim_{n→∞} λ_{n} = λ and then show that lim inf_{l→∞} l^{−d}F_{Q_{l}}(λ, ω) ≥ N(λ). Thus the theorem is proven. □

Acknowledgements: Illuminating discussions with D. Lenz which led to an improvement of the manuscript are gratefully acknowledged. We thank C. Schubert for making a draft of the thesis [Sch06] available to us. The second named author was financially supported by the DFG under grant Ve 253/2-2 within the Emmy-Noether-Programme.

Note added: A few days before we finished this manuscript, the paper Anderson Localization for radial tree-like random quantum graphs by P. Hislop and O. Post, containing results not unrelated to ours, was made available at www.arxiv.org.

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(^{1}) Fakultät für Mathematik, 09107 TU-Chemnitz, Germany

E-mail address: Mario.Helm@mathematik.tu-chemnitz.de

(^{2}) Emmy-Noether-Programme of the Deutsche Forschungsgemeinschaft

URL: www.tu-chemnitz.de/mathematik/schroedinger/members.php