A Mourre estimate for a Schrödinger operator

on a binary tree

C. Allard and R. Froese

Department of Mathematics University of British Columbia Vancouver, British Columbia, Canada

September 18, 2000

Résumé

Soit G un arbre binaire ayant pour noeuds l’ensemble V et soit H un op� erateur de Schr� odinger agissant sur ℓ^{2}(V). Une d� ecomposition de l’espace ℓ^{2}(V) en sous-espaces invariants est pr� esent� ee de sorte � a fournir une op� erateur conjugu� e A utilisable dans une estimation de Mourre. Nous montrons qu’une estimation de Mourre vaut pour des potentials q dont les diff� erences de premier ordre satisfaisont � a une condition de d� ecroissance.

Abstract

Let G be a binary tree with vertices V and let H be a Schr� odinger operator acting on ℓ^{2}(V). A decomposition of the space ℓ^{2}(V) into invariant subspaces is exhibited yielding a conjugate operator A for use in the Mourre estimate. We show that for potentials q satisfying a �rst order difference decay condition, a Mourre estimate for H holds.

Introduction

Let G = (V, E) be a graph with vertices V and edges E. The Laplace operator acts on functions de�ned on V. If ϕ : V → ℂ is such a function, then Δϕ is the function de�ned by

(Δϕ)(v) = ∑_{w:w�v} (ϕ(w) − ϕ(v)),

where w�v means that v and w are connected by an edge. We are interested in the spectral theory of −Δ and perturbations −Δ + q, acting in the Hilbert space ℓ^{2}(V) of square summable functions on V.

This is the space of functions ϕ satisfying

∑_{v∈V} |ϕ(v)|^{2} < ∞,

with inner product given by

⟨ϕ, ψ⟩ = ∑_{v∈V} ϕ̄(v)ψ(v).

Let L denote the off-diagonal part of Δ. Thus

(Lϕ)(v) = ∑_{w:w�v} ϕ(w).

If d(v) denotes the number of edges joined to the vertex v then

Δ = L − d,

where d is the operator of multiplication by d(v). The degree term d can be included in the potential as a perturbation, hence −Δ + q = −L + d + q can be considered as a perturbation of L. Both Δ and L are symmetric on ℓ^{2}(V). When d(v) is bounded, then both operators are also bounded operators, hence self-adjoint.

The goal of this paper is to prove a Mourre estimate and related bounds for the Schr� odinger operator −Δ + q when the underlying graph is a binary tree. Here q denotes multiplication by a potential function that tends to zero at in�nity. For a binary tree, d = 3 − d_{0}, where d_{0} is the potential with d_{0}(v) = 1 at the root of the tree and 0 otherwise. Hence −Δ + q = −L + 3 − d_{0} + q and the spectrum of −Δ + q is the same as that of L − q + d_{0} up to a shift and a re�ection about zero. In considering the Mourre estimate, the d_{0} term can be absorbed in q and the sign of the potential changed, since −q + d_{0} satis�es our decay assumptions whenever q does. Hence we aim at obtaining a Mourre estimate for L + q.

The operator L can be diagonalized explicitly. Its spectrum is absolutely continuous and equal to σ(L) = σ_{ac}(L) = [−2\sqrt{2}, 2\sqrt{2}]. This is also the essential spectrum of L. Since q is a compact operator, perturbation by q does not change the essential spectrum, and so σ_{ess}(L + q) = [−2\sqrt{2}, 2\sqrt{2}].

We will de�ne a self-adjoint conjugate operator A such that, under appropriate conditions on q (i) [L + q, iA] is bounded

(ii) [[L + q, iA], iA] is bounded

(iii) L + q and A satisfy a Mourre estimate at every point in (−2\sqrt{2}, 2\sqrt{2})

By de�nition, (iii) means that for every λ ∈ (−2\sqrt{2}, 2\sqrt{2}), there exists an interval I containing λ such that

E_{I}[L + q, iA]E_{I} ≥ αE_{I}^{2} + K

Here E_{I} = E_{I}(L+q) denotes the (possibly smoothed) spectral projection corresponding to the interval I, α is a positive number, and K is a compact operator. Precise statements can be found in Lemma 5, Lemma 6, Lemma 7 and Theorem 9 below.

The estimates (i) (ii) and (iii) together with the abstract Mourre theory (see, for example [1]), have the following consequences.

(1) Eigenvalues of L + q not equal to ±2\sqrt{2} have �nite multiplicity and can only accumulate at

±2\sqrt{2}.

(2) The operator L + q has no singular continuous spectrum.

(3) Scattering for the pair L and L + q is asymptotically complete (see [2]).

Recently, it was noticed that the proof in [1] of the virial theorem�an essential step in the abstract Mourre theory�contains an error [3]. However, if H is bounded then e^{isA} trivially preserves the domain of H. This condition together with (i) ensure the validity of the virial theorem (see [3]).

Although we only treat the binary tree, the same method can be applied to related graphs, for example the Bethe Lattice, or trees with k-fold branching. Graph Laplacians and Schr� odinger operators on the Bethe Lattice are of interest in solid state physics, where they serve as a model for tightly bound electrons. Much effort has gone into studying operators with random potentials, and it is interesting to note that although the existence of dense point spectrum near band edges has been proven in many situations, the Bethe Lattice is the only model where it has been proved that for weak disorder, some absolutely continuous spectrum remains in the middle of the band [4]. From the purely mathematical point of view, the Bethe Lattice is the Cayley graph of a free group. It would be interesting to be able to say something about the continuous spectrum of the Laplace operator on the Cayley graph for a �nitely generated group that is not free, and to relate properties of the spectrum to properties of the group.

The subspace decomposition and conjugate operator used in this paper bear some similarity to the ones used in [5] in the case of exponentially large manifolds. However, the details are quite different. In particular, the calculation of the matrix elements of A and the method of estimating [q, iA] have no analogue. These results �rst appeared in [2]. Independent work on the commutator method for the graph ℤ^{n} has appeared in [6] and [7].

The operators Π and R

In this section we will let (V, E) be an arbitrary graph and introduce polar co-ordinates and some associated operators. Choose some 0 ∈ V to be the origin. De�ne |v| to be the distance in the graph from 0 to v. In other words, |v| is the length of the shortest path in the graph joining 0 to v. De�ne S_{r}, the sphere of radius r, to be the set of all vertices with |v| = r. Then V is a disjoint union

V = ⋃_{r=0}^{∞} S_{r}

and

ℓ^{2}(V) = ⨁_{r=0}^{∞} ℓ^{2}(S_{r})

In the case of a binary tree, see the �gure below.

Spheres in a binary tree

We will write v→w if v and w are connected by an edge and |w| = |v| + 1.

De�ne

(Πϕ)(v) = ∑_{w:w→v} ϕ(w)

The adjoint of Π can be computed by calculating

⟨ψ, Πϕ⟩ = ∑_{v} ∑_{w:w→v} ψ̄(v)ϕ(w)

This can be interpreted as a sum over all edges joining neighbouring spheres, where ψ̄ and ϕ are evaluated at the right and left endpoint of the edge respectively. We have chosen to label the edges by their right endpoint v, and the sum over w : w→v accounts for the possibility of several arrows having the same right endpoint. If we choose to label the edges by their left endpoints instead we �nd that the same sum can be written

⟨ψ, Πϕ⟩ = ∑_{v} ∑_{w:v→w} ψ̄(w)ϕ(v).

This shows that

(Π^{∗}ϕ)(v) = ∑_{w:v→w} ϕ(w)

Notice that Π^{∗}Π and ΠΠ^{∗} leave each ℓ^{2}(S_{r}) invariant. The action of Π^{∗}Π is given by

(Π^{∗}Πϕ)(v) = ∑_{w} ϕ(w)  (1)

where the sum is extended over w ∈ S_{|v|} that are joined to v by a path in the graph of length two, going from v to some element in S_{|v|+1} and then back to S_{|v|}. The formula for ΠΠ^{∗} is analogous, except that the path goes in the other direction to S_{|v|−1} and back.

We will denote by R the operator of multiplication by |v|. When restricted to ℓ^{2}(S_{r}), the operator R is multiplication by r. An easy calculation shows that

[R, Π] = Π  (2)

We may write L in terms of Π and Π^{∗} by breaking the sum in the de�nition of L into three pieces. We obtain

L = Π + Π^{∗} + L_{S},

where the spherical Laplacian L_{S} is de�ned by

(L_{S}ϕ)(v) = ∑_{w: {w�v \atop |w|=|v|}} ϕ(w).

Diagonalization L and definition of A for a Binary Tree

In this section we will exhibit a diagonalization of the off-diagonal Laplacian L on a binary tree. Choose the origin to be the base of the tree and introduce polar co-ordinates. Since there are no

edges that connect vertices within each sphere, L_{S} = 0, and

L = Π + Π^{∗}.

We now construct invariant subspaces M_{n} for Π. Let Q_{0,0} = ℓ^{2}(S_{0}) and de�ne Q_{0,r} = Π^{r}Q_{0,0}. Let

M_{0} = ⨁_{r=0}^{∞} Q_{0,r}.

To de�ne Q_{n,r} and M_{n} for n > 0 we proceed recursively. Suppose that Q_{m,s} have been de�ned whenever m < n and s ≥ m. Let Q_{n,n} be the orthogonal complement in ℓ^{2}(S_{n}) of all previously de�ned subspaces,

Q_{n,n} = ℓ^{2}(S_{n}) ⊖ (Q_{0,n} ⊕ ⋯ ⊕ Q_{n−1,n}) .

For r = n + j, j ≥ 1, de�ne Q_{n,r} = Π^{j}Q_{n,n}, and

M_{n} = ⨁_{r=n}^{∞} Q_{n,r} = ⨁_{j=0}^{∞} Q_{n,n+j}.

Schematically, this gives

ℓ^{2}(S_{0})  ℓ^{2}(S_{1})  ℓ^{2}(S_{2})  ℓ^{2}(S_{3})
M_{0}  Q_{0,0}  Q_{0,1}  Q_{0,2}  Q_{0,3}  ⋯
M_{1}  Q_{1,1}  Q_{1,2}  Q_{1,3}  ⋯
M_{2}  Q_{2,2}  Q_{2,3}  ⋯
⋮  ⋮

Orthogonal subspace decomposition

Lemma 1 The Hilbert space ℓ^{2}(V) can be written as an orthogonal direct sum

ℓ^{2}(V) = ⨁_{n=0}^{∞} M_{n} = ⨁_{n=0}^{∞} ⨁_{r=n}^{∞} Q_{n,r}.

The subspaces M_{n} are invariant for L.

Proof: Since Π maps ℓ^{2}(S_{r}) into ℓ^{2}(S_{r+1}) if follows that each Q_{n,r} is contained in ℓ^{2}(S_{r}). Thus, if r ≠ s, then Q_{n,r} and Q_{m,s} are orthogonal, for all n and m.

For a binary tree, it follows from (1) that

Π^{∗}Π = 2I  (3)

This implies that if ϕ and ψ are orthogonal, then Πϕ and Πψ are orthogonal too. Since Q_{n,n} is orthogonal to Q_{m,n} for m < n by construction, it follows that Q_{n,r} and Q_{m,r}, for r ≥ n are orthogonal too.

By construction ⨁_{l=0}^{r} Q_{l,r} = ℓ^{2}(S_{r}), so it is clear that the subspaces add up to ℓ^{2}(V).

By construction, each M_{n} is invariant for Π. That they are invariant for Π^{∗} follows from (3) as follows. It suf�ces to show that each Q_{n,r} is mapped to M_{n} under Π^{∗}. Suppose that ϕ ∈ Q_{n,r} for r = n + j with j ≥ 1. Then ϕ = Π^{j}χ for χ ∈ Q_{n,n}. Hence Π^{∗}ϕ = 2Π^{j−1}χ ∈ Q_{n,r−1}. On the other hand, if ϕ ∈ Q_{n,n}, then Π^{∗}ϕ ∈ ℓ^{2}(S_{n−1}). Suppose that ψ ∈ Q_{l,n−1} for some l ≤ n−1. Since Πψ ∈ Q_{l,n} for some l ≤ n − 1 and ϕ ∈ Q_{n,n}, we have ⟨ψ, Π^{∗}ϕ⟩ = ⟨Πψ, ϕ⟩ = 0. Hence Π^{∗}ϕ is orthogonal to each Q_{l,n−1}, which implies that Π^{∗}ϕ = 0. Thus each M_{n} is invariant for Π and Π^{∗}, and hence for L.

Since each M_{n} is an invariant subspace for L we can decompose L = ⊕_{n=0}^{∞}L_{n}, where L_{n} is the restriction of L to M_{n}. We now diagonalize L_{n}.

We begin by writing a vector ϕ in M_{n} as

ϕ = ⊕_{j=0}^{∞}ϕ_{n+j}

where ϕ_{n+j} ∈ Q_{n,n+j}. We want to obtain an isomorphism between M_{n} and ℓ^{2}(ℤ^{+}, Q_{n,n}), the space of Q_{n,n} valued sequences.

We �rst note that (\frac{1}{\sqrt{2}}Π)^{j} (not Π^{j}) de�nes an isometry between Q_{n,n} and Q_{n,n+j} for all j and that any ϕ ∈ M_{n} can be written as

ϕ = ⊕_{j=0}^{∞} (\frac{1}{\sqrt{2}}Π)^{j} χ_{n+j}  (4)

for a sequence of vectors χ_{n}, χ_{n+1}, … ∈ Q_{n,n}.

Under this representation, M_{n} and ℓ^{2}(ℤ^{+}, Q_{n,n}) are isomorphic, since

⟨ϕ, ϕ⟩ = ∑_{j=0}^{∞}⟨ϕ_{n+j}, ϕ_{n+j}⟩ = ∑_{j=0}^{∞}⟨χ_{n+j}, χ_{n+j}⟩ = ⟨W ϕ, W ϕ⟩

by the above isometry, where W denotes the isomorphism.

In this representation, the operator \frac{1}{\sqrt{2}}Π acts as a shift to the right, while \frac{1}{\sqrt{2}}Π^{∗} is a shift to the left with kernel Q_{n,n}.

Let

U : M_{n} ≅ ℓ^{2}(ℤ^{+}, Q_{n,n}) → L_{odd}^{2}([−π, π], dθ)

denote the unitary map de�ned by

U((α_{n}, α_{n+1}, …)) = \frac{1}{\sqrt{π}} ∑_{j=0}^{∞} α_{n+j} sin((j + 1)θ)

Lemma 2

UL_{n}U^{∗} = 2\sqrt{2} cos(θ).

Proof: The proof is a straightforward computation.

This lemma shows that the spectrum of L_{n} is [−2\sqrt{2}, 2\sqrt{2}], and is absolutely continuous. Thus the spectrum of L is also [−2\sqrt{2}, 2\sqrt{2}], with in�nite multiplicity.

This representation motivates our choice of conjugate operator. For a general multiplication operator ω(θ), and natural conjugate operator is A_{ω} = \frac{i}{2}(ω′ \frac{d}{dθ} + \frac{d}{dθ} ω′), since

[ω, iA_{ω}] = ω^{′2}

which is positive away from the critical points of ω. In the present case the natural conjugate operator is therefore

UA_{n}U^{∗} = −i\sqrt{2} (sin(θ)\frac{d}{dθ} + \frac{d}{dθ} sin(θ))

and a calculation now shows that on M_{n}

iA_{n} = U^{∗}\sqrt{2} (sin(θ)\frac{d}{dθ} + \frac{d}{dθ} sin(θ)) U = (R − n + \frac{1}{2})Π − Π^{∗}(R − n + \frac{1}{2})

Therefore a natural conjugate operator for L is ⊕_{n=0}^{∞}A_{n}. If we let P_{n} denote the projection onto M_{n}, and de�ne

N = ∑_{n=1}^{∞} nP_{n}

the conjugate operator can be written as

iA = (R − N + \frac{1}{2})Π − Π^{∗}(R − N + \frac{1}{2}).

Matrix elements of A

We will need estimates on the matrix elements of A. Let δ_{w} denote the standard basis element in ℓ^{2}(V) de�ned by δ_{w}(v) = δ_{w,v}. We wish to estimate the matrix elements ⟨δ_{v}, iAδ_{w}⟩. Using the formula

Πδ_{w} = ∑_{w→z} δ_{z}

we �nd that

⟨δ_{v}, iAδ_{w}⟩

= ⟨δ_{v}, (R − N + \frac{1}{2})Π − Π^{∗}(R − N + \frac{1}{2})δ_{w}⟩

= \{ \begin{matrix}(|v| + \frac{1}{2}) δ(w → v) − ∑_{z:w→z}⟨δ_{v}, Nδ_{z}⟩ & if & |w| = |v| − 1 \\ − (|w| + \frac{1}{2}) δ(v → w) + ∑_{z:v→z}⟨δ_{z}, Nδ_{w}⟩ & if & |w| = |v| + 1 \\ 0 & otherwise \end{matrix}

(5)

Here δ(w → v) is equal to 1 if w → v and 0 otherwise.

To calculate the matrix elements of N appearing in this formula, we introduce, at this point, an explicit basis for each Q_{n,r}. When n = 0, we have that Q_{0,r} = Π^{r}ℓ^{2}(S_{0}) is one dimensional and consists of all vectors ϕ(v) in ℓ^{2}(S_{r}) such that ϕ(v) has the same value for all v. An orthonormal basis for Q_{0,r} is therefore the single vector

ρ_{0,r,0} = 2^{−r/2}[1, 1, …, 1]

The space Q_{1,1} is the orthogonal complement in ℓ^{2}(S_{1}) of Q_{0,1}. Thus Q_{1,1} is also one dimensional and has orthonormal basis

ρ_{1,1,0} = 2^{−1/2}[1, −1]

Pushing the vector forward along the tree using Π and normalizing gives

ρ_{1,r,0} = 2^{−r/2}[1, 1, …, 1, −1, −1, …, −1]

as a basis for Q_{1,r}, where half the entries are 1 and the other half −1.

The space Q_{2,2} is the orthogonal complement in ℓ^{2}(S_{2}) of Q_{0,1} ⊕ Q_{1,1}. Since dim(ℓ^{2}(S_{2})) = 4, the space Q_{2,2} is two dimensional. It has orthonormal basis

\begin{aligned}ρ_{2,2,0} & = 2^{−1/2}[1, −1, 0, 0] \\ ρ_{2,2,1} & = 2^{−1/2}[0, 0, 1, −1] \end{aligned}

Pushing these vectors forward along the tree using Π and normalizing yields ρ_{2,r,0} and ρ_{2,r,1}.

Continuing in this fashion we de�ne the orthonormal basis ρ_{n,r,k} with k = 0, …, 2^{max{n−1,0}} −1. When we �x the second index r, the vectors ρ_{n,r,k} are the Haar basis for ℓ^{2}(S_{r}). Upon de�ning a partial order on the Haar basis elements for ℓ^{2}(S_{r}) using inclusion of supports, the Haar basis functions, as illustrated below for r = 4, naturally form a binary tree with r levels, extended by an extra vertex at its base.

The Haar basis

Lemma 3 Let z, w ∈ S_{r}. If z ≠ w, let N(z, w) denote the largest value of n for which both z and w lie in the support of a single basis function. Then

⟨δ_{z}, Nδ_{w}⟩ = \{ \begin{matrix}r − 1 + 2^{−r} & if z = w \\ −2^{N(z,w)−r} + 2^{−r} & if z ≠ w \end{matrix}

Proof: Fix r and label the elements of the Haar basis by the vertices α in the associated (extended) binary tree. Then

Nδ_{w} = ∑_{α} n(α)ρ_{α}(w)ρ_{α}

where n(α) denotes the level of α in the tree and the sum is taken over the one α at each level for which

ρ_{α}(w) ≠ 0.

Suppose that z = w. Then, for the one α at level n(α) = n for which ρ_{α}(w) ≠ 0 we have

ρ_{α}(w)^{2} = 2^{−r+n−1} Thus

⟨δ_{w}, Nδ_{w}⟩ = ∑_{n=1}^{r} n2^{−r+n−1} = r − 1 + 2^{−r}

Now suppose that z ≠ w. Then for the one α at level n for which ρ_{α}(w) ≠ 0 we have

ρ_{α}(z) = \{ \begin{matrix}ρ_{α}(w) & if n < N(z, w) \\ −ρ_{α}(w) & if n = N(z, w) \\ 0 & if n > N(z, w) \end{matrix}

Thus

\begin{aligned}⟨δ_{z}, Nδ_{w}⟩ & = ∑_{n=1}^{N(z,w)−1} n2^{−r+n−1} − N(z, w)2^{−r+N(z,w)−1} \\ & = −2^{−r+N(z,w)} + 2^{−r} \end{aligned}

Lemma 4

∑_{w} |⟨δ_{v}, iAδ_{w}⟩| = O(|v|).

Proof: Using (5) we �nd

∑_{w} |⟨δ_{v}, iAδ_{w}⟩| ≤ ∑_{w:|w|=|v|−1} (|v| + \frac{1}{2}) δ(w → v) + ∑_{w:|w|=|v|−1} ∑_{z:w→z} |⟨δ_{v}, Nδ_{z}⟩|

+ ∑_{w:|w|=|v|+1} (|w| + \frac{1}{2}) δ(v → w) + ∑_{w:|w|=|v|+1} ∑_{z:v→z} |⟨δ_{z}, Nδ_{w}⟩|

(6)

Since there is only one w with w → v we have

∑_{|w|=|v|−1} (|v| + \frac{1}{2}) δ(w → v) = |v| + \frac{1}{2}.

Since there exactly two w with v → w,

∑_{|w|=|v|+1} (|w| + \frac{1}{2}) δ(v → w) = 2(|v| + 1) + 1.

To estimate the remaining two terms in (6), we begin with

∑_{z:|z|=|v|} |⟨δ_{v}, Nδ_{z}⟩| = |⟨δ_{v}, Nδ_{v}⟩| + ∑_{z: {|z|=|v| \atop z≠v}} |⟨δ_{v}, Nδ_{z}⟩|

= |v| − 1 + 2^{−|v|} + ∑_{z: {|z|=|v| \atop z≠v}} 2^{N(v,z)−|v|} − 2^{−|v|}

≤ |v| + ∑_{z: {|z|=|v| \atop z≠v}} 2^{N(v,z)−|v|}

Since there are 2^{|v|−N(v,z)} z’s associated with each value of N(v, z) = n,

∑_{z:|z|=|v|} |⟨δ_{v}, Nδ_{z}⟩| ≤ |v| + ∑_{n=1}^{|v|} 2^{n−|v|} ∑_{z:z≠v,|z|=|v|,N(v,z)=n} 1

= |v| + ∑_{n=1}^{|v|} 2^{n−|v|}2^{|v|−n}

= 2|v|.

Therefore

∑_{w:|w|=|v|−1} ∑_{z:w→z} |⟨δ_{v}, Nδ_{z}⟩| = ∑_{z:|z|=|v|} |⟨δ_{v}, Nδ_{z}⟩| ≤ 2|v|

and

∑_{w:|w|=|v|+1} ∑_{z:v→z} |⟨δ_{z}, Nδ_{w}⟩| = ∑_{z:v→z} ∑_{w:|w|=|z|} |⟨δ_{z}, Nδ_{w}⟩| ≤ ∑_{z:v→z} 2|v| = 4|v|.

Thus each term in (6) is O(|v|) and the proof is complete.

The Mourre estimate

We begin with the commutator formula for L. This is just a disguised form of the formula

[2\sqrt{2} cos(θ), \sqrt{2}(sin(θ)\frac{d}{dθ} + \frac{d}{dθ} sin(θ))] = 8 sin^{2}(θ).

Lemma 5

[L, iA] = 8 − L^{2}

Proof: Since Π and Π^{∗} commute with N, we have

[L, iA] = [Π + Π^{∗}, (R − N + \frac{1}{2})Π] + adjoint

= [Π, (R − N + \frac{1}{2})]Π + [Π^{∗}, (R − N + \frac{1}{2})]Π + (R − N + \frac{1}{2})[Π^{∗}, Π] + adjoint = [Π, R]Π + [Π^{∗}, R]Π + (R − N + \frac{1}{2})[Π^{∗}, Π] + adjoint = −Π^{2} + Π^{∗}Π + (R − N + \frac{1}{2})[Π^{∗}, Π] + adjoint, where adjoint refers to all previous terms. Here we used (2). Now notice that [Π^{∗}, Π] is a projection onto the sum of the initial subspaces Q_{n,n}, and that this is precisely the kernel of the operator R − N, since on this sum R = N. Thus (R − N)[Π^{∗}, Π] = 0, and

[L, iA] = −Π^{2} + Π^{∗}Π + \frac{1}{2}[Π^{∗}, Π] + adjoint
= −Π^{2} − (Π^{∗})^{2} + 3Π^{∗}Π − ΠΠ^{∗}
= 4Π^{∗}Π − (Π + Π^{∗})^{2}
= 8 − L^{2}

Lemma 6 Suppose that

sup_{w:|w|=|v|±1} |q(v) − q(w)| = o(|v|^{−1}),

as |v| → ∞. Then [q, iA] is compact. Proof: Let Λ_{n} denote the projection onto ⊕_{r=0}^{n}ℓ^{2}(S_{r}). We will show that ∥[q, iA] − [q, iA]Λ_{n}∥ = ∥[q, iA](1 − Λ_{n})∥ → 0 as n → ∞. This shows that [q, iA] is approximated in norm by the �nite rank operators [q, iA]Λ_{n}, and hence compact.

The matrix elements of [q, iA](1 − Λ_{n}) are given by

⟨δ_{v}, [q, iA](1 − Λ_{n})δ_{w}⟩ = (q(v) − q(w))⟨δ_{v}, iAδ_{w}⟩

provided |w| > n, and 0 if |w| ≤ n. Using Schur’s lemma (the ℓ^{1}�ℓ^{∞} bound), the fact that the matrix elements of ⟨δ_{v}, iAδ_{w}⟩ are non-zero only for |w| = |v| ± 1, the decay hypothesis on q, and Lemma 4 we �nd that

∥[q, iA](1 − Λ_{n})∥ ≤ sup_{v} ∑_{w:|w|>n} |q(v) − q(w)||⟨δ_{v}, iAδ_{w}⟩|

≤ sup_{v:|v|>n−1} o(|v|^{−1}) ∑_{w} |⟨δ_{v}, iAδ_{w}⟩|

≤ sup_{v:|v|>n−1} o(|v|^{−1})O(|v|).

This tends to zero for large n.

Lemma 7 Suppose that

sup_{z,w:|w|=|v|±1,|z|=|w|±1} |q(v) + q(z) − 2q(w)| = O(|v|^{−2}),

as |v| → ∞. Then [[q, iA], iA] is bounded. Proof: The matrix elements of [[q, iA], iA] are given by

⟨δ_{v}, [[q, iA], iA]δ_{z}⟩ = ⟨δ_{v}, [q, iA]iA − iA[q, iA]δ_{z}⟩

= ∑_{w}⟨δ_{v}, [q, iA]δ_{w}⟩⟨δ_{w}, iAδ_{z}⟩ − ⟨δ_{v}, iAδ_{w}⟩⟨δ_{w}, [q, iA]δ_{z}⟩

= (q(v) + q(z) − 2q(w))⟨δ_{v}, iAδ_{w}⟩⟨δ_{w}, iAδ_{z}⟩ Thus, as in Lemma 6,

∥[[q, iA], iA]∥ ≤ sup_{v} ∑_{z} ∑_{w} |q(v) + q(z) − 2q(w)||⟨δ_{v}, iAδ_{w}⟩||⟨δ_{w}, iAδ_{z}⟩|

≤ sup_{v} O(|v|^{−2}) ∑_{w} |⟨δ_{v}, iAδ_{w}⟩| ∑_{z} |⟨δ_{w}, iAδ_{z}⟩|

= sup_{v} O(|v|^{−2})O(|v|)O(|v|) ≤ C

Lemma 8 Suppose that q(v) → 0 as |v| → ∞. Let E denote a smoothed out spectral projection. Then E(L) − E(L + q) is compact.

Proof: This follows from the compactness of (L − z)^{−1} − (L + q − z)^{−1} = (L − z)^{−1}q(L + q − z)^{−1} and a Stone�Weierstra� approximation argument (see [1]).

Now we can prove the Mourre estimate for L + q and A.

Theorem 9 Suppose that q(v) → 0 as |v| → ∞. Assume that

sup_{w:|w|=|v|±1} |q(v) − q(w)| = o(|v|^{−1}),

as |v| → ∞. Let E denote a smoothed out spectral projection whose support is properly contained in the interval (−2\sqrt{2}, 2\sqrt{2}). Then there exists a compact operator K and a positive number α such that

E(L + q)[L + q, iA]E(L + q) ≥ αE^{2}(L + q) + K.

Proof: By the compactness of [q, iA] and E(L + q) − E(L) we have

\begin{aligned}E(L + q)[L + q, iA]E(L + q) & = E(L)[L, iA]E(L) + K \\ & = E(L)(8 − L^{2})E(L) + K \end{aligned}

On the support of E(L), 8 − L^{2} ≥ α for some positive α, which gives

E(L + q)[L + q, iA]E(L + q) ≥ αE^{2}(L + q) + K,

where the compact term α(E^{2}(L) − E^{2}(L + q)) has been added into K. This completes the proof.

References

[1] Cycon, H.L., Froese, R.G., Kirsch, W. and Simon, B., Schrödinger Operators, Texts and Monographs in Physics, Springer, 1987.

[2] Allard, C., Asymptotic Completeness via Mourre Theory for a Schrödinger Operator on a Binary Tree Graph, Master’s thesis, University of British Columbia, April 1997.

[3] Georgescu, V. and G� erard, G., On the Virial Theorem in Quantum Mechanics, Preprint.

[4] Klein, A., Extended states in the Anderson model on the Bethe lattice, Adv. Math. 133 (1998), no. 1, 163�184.

[5] Froese, R.G. and Hislop, P., Spectral analysis of second order elliptic operators on noncompact manifolds, Duke Math J., Vol. 58, No. 1 (1989), 103�129

[6] Boutet de Monvel, A. and Sahbani, J., On the spectral properties of discrete Schrödinger operators, to appear in Rev. Math. Phys.

[6] Boutet de Monvel, A. and Sahbani, J., On the spectral properties of discrete Schrödinger operators, C. R. Acad. Sci. Paris S� er. I Math. 326, no. 9 (1998), 1145�1150.