on the spectral properties of discrete schrÖdinger operators

ON THE SPECTRAL PROPERTIES

OF DISCRETE SCHRODINGER OPERATORS:

THE MULTI-DIMENSIONAL CASE

ANNE BOUTET DE MONVEL and JAOUAD SAHBANI

Institut de Mathematiques de Jussieu, CNRS UMR 7586Physique mathematique et Geometrie

Universite Paris 7-Denis DiderotU.F.R. de Mathematiques, case 7012

Tour 45–55, 5-eme etage2, place Jussieu, 75251 Paris Cedex 05, France

E-mail : [email protected] : [email protected]

Received 23 February 1998

We use the method of the conjugate operator to prove the limiting absorption principleand the absence of the singular continuous spectrum for the discrete Schrodinger operator.We also obtain local decay estimates. Our results apply to a large class of perturbatingpotentials V tending arbitrarily slowly to zero at infinity.

1. Introduction

Our purpose in this work is to study the absolute continuity of the spectrum of

the discrete Schrodinger operators. We also investigate the propagation properties

of these operators. Our method works for a large class of arbitrarily slowly decaying

potentials V that will be explicitly described in our theorems.

Our study is based on the method of the conjugate operator. This theory shows,

in an abstract frame, that a Hamiltonian H has nice spectral and propagation

properties if it has a conjugate operator A, i.e. a self-adjoint operator A such that

the commutator [H, iA] is strictly positive in a convenient sense (see [1–3, 9] and

references therein). This theory was used efficiently for the spectral and scattering

theory of (pseudo) differential operators (see [1, 8] and references therein). It is

natural to apply it to the study of discrete operators.

The configuration space is the multidimensional lattice Zν for some integer

ν > 0. For a multi-index α = (α1, . . . , αν) ∈ Zν we set |α|2 = α21 + · · · + α2

ν .

Let us consider the Hilbert space

H = `2(Zν)

of square integrable sequences ψ = (ψ(α))α∈Zν . We are interested here in the dis-

crete Schrodinger operatorsH = H0+V acting inH, whereH0 is the finite difference

1061

Reviews in Mathematical Physics, Vol. 11, No. 9 (1999) 1061–1078c©World Scientific Publishing Company

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1062 A. BOUTET DE MONVEL and J. SAHBANI

operator defined by

(H0ψ)(α) = −∑β∈Zν|β−α|=1

ψ(β) , ∀ ψ ∈ H ,

and V is the multiplication operator by a real valued sequence (V (α))α∈Zν :

(V ψ)(α) = V (α)ψ(α) .

Using a Fourier transform, one sees that H0 is a bounded self-adjoint operator in

H, and that its spectrum is purely absolutely continuous:

σac(H0) = [−2ν, 2ν] ;(1.1)

σs(H0) = σsc(H0) ∪ σpp(H0) = ∅ .

Our purpose here is to study the essential stability of this spectral structure under

perturbation by a potential V that decays at infinity. By essential stability we

mean that the absolutely continuous spectrum of the perturbed operator H does

not change, but the singular spectrum that can occur is a countable set with only

±2ν as possible accumulation points, and in particularH has no singular continuous

spectrum.

Such questions have been studied by many authors and there is a large literature

about it. In particular, in the one-dimensional case (ν = 1) which we shall discuss

briefly in the rest of this introduction (see also [4]). It is not difficult to see that if

V is in `1(Zν), e.g. if for ε > 0,

|V (α)| ≤ C

|α|1+ε,

then the spectrum of the associated Schrodinger operator H is absolutely conti-

nuous. Indeed, V is trace class in this case.

On the other hand, if V decreases slower than 1/|α| then the absolute continuity

of the spectrum of H can be partially or completely destroyed. Indeed, in the one

dimensional case Simon has given in [10] an example of a potential V such that

|V (α)| ≤ C

|α| 12,

and that the associated Schrodinger operator has only point spectrum, so σac(H) =

∅. Nevertheless, the different components of the spectrum can coexist if

|V (α)| ≤ C

|α|δ , δ >1

2.

Indeed, again in the one-dimensional case, Kiselev [5] recently proved that the

absolutely continuous component of the spectrum fills the whole essential spectrum

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ON THE SPECTRAL PROPERTIES OF DISCRETE SCHRODINER OPERATORS 1063

[−2,+2], if δ > 34 . On the other hand, Naboko and Yakovlev [6] have constructed

a potential V such that

|V (α)| ≤ C(α)

α,

with C(α)→∞ for α→∞ (arbitrarily slowly)

for which the set of eigenvalues of the corresponding discrete Schrodinger operator

H is dense in [−2,+2].

Here we will describe a class of perturbations V decaying arbitrarily slowly to

zero at infinity which leave essentially stable the spectral structure (1.1). We give a

compromise between regularity and decay at infinity by allowing several components

in the perturbations, behaving each differently at infinity.

Another goal of this work is to study the propagation properties of H. More

precisely, we establish estimates of local decay type, i.e. estimates of the form

‖e−iHtϕ‖X ≤ C‖ϕ‖Yfor some ϕ ∈ Hac in adequate Banach spaces X, Y . These estimates play a funda-

mental role in scattering theory.

The paper is organized as follows. Section 2 contains a detailed description of

our main results. In Sec. 3 we recall what we need of the method of the conjugate

operator. Section 4 contains the main step of our proofs, namely the construction

of the conjugate operator for the unperturbed hamiltonian H0. In Sec. 5 we prove

our results. In the appendix we give a criterion to verify the regularity requirements

of the abstract theory of the conjugate operator method.

2. Main Results

Let us consider N = (N1, . . . , Nν), where Nj is the diagonal operator in Hgiven by

(Njψ)(α) = αjψ(α) .

Nj is a self-adjoint operator with domain

D(Nj) =

{ψ ∈ H such that

∑α∈Zν

|αjψ(α)|2 <∞}.

For each real s we denote by Hs the Sobolev space associated to N and defined by

the norm ‖f‖s = ‖〈N〉sf‖, with 〈x〉 =√

1 + x2. By interpolation we obtain the

Besov space Hs,p:

Hs,p = (Hs1 ,Hs2)θ,p

for s1 < s2, 0 < θ < 1, s = θs1 + (1− θ)s2, 1 ≤ p ≤ ∞ .

We are specially interested in the space K := H 12 ,1

and its topological adjoint

K∗ = H− 12 ,∞. We also consider P = (P1, . . . , Pν), where Pj is the finite difference

operator given in H by

(Pjψ)(α) = ψ(α+ ej)− ψ(α) .

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We have denoted by ej the element of Zν whose components all vanish except the jth

component which is equal to 1. For a multi-index β ∈ Nν we set P β = P β1

1 . . . P βννand |β| = β1 + · · ·+ βν . Let us set C± = {z ∈ C| ± Im z > 0}.

Theorem 2.1. Assume that the potential V can be decomposed as

V = Vs + Vl + VM , where Vs, Vl and VM are real-valued sequences such that for

each j = 1, . . . , ν, and each β ∈ Zν , |β| = 1, 2 we have∫ ∞1

supr<|α|<2r

|Vs(α)|dr <∞, (2.1)

Vl(α)→ 0 as |α| → ∞ and

∫ ∞1

supr<|α|<2r

|(PjVl)(α)| dr <∞ , (2.2)

VM (α)→ 0 and |(P βVM )(α)| = O(|α|−|β|) as |α| → ∞ . (2.3)

Then:

(i) the set σp(H) of eigenvalues of H has no accumulation points except

(probably) ±2ν, and each eigenvalue is finitely degenerate;

(ii) the singular continuous spectrum of H is empty;

(iii) the holomorphic maps C± 3 z 7→ (H − z)−1 ∈ B(K,K∗) extend to a weak∗

continuous function on C± ∪ [(−2ν, 2ν) \ σp(H)].

Let us indicate what kind of behavior at infinity is allowed by our assumptions.

Example 2.1. For simplicity assume that V is radial, i.e. V (α) = V (|α|). The

assumption (2.1) is fulfilled if for some ε > 0 we have

|Vs(α)| ≤ C|α|−1 · (ln |α|)−1−ε .

Assumption (2.2) is satisfied for example if

Vl(α)→ 0 as |α| → ∞ and |(PjVl)(α)| ≤ C|α|−1 · (ln |α|)−1−ε .

Finally it is easy to see that

VM (α) = (ln ln |α|)−1

for |α| ≥ n0 satisfies the condition (2.3).

We see then our conditions allow potentials tending arbitrarily slowly to zero at

infinity.

Example 2.2. In order to compare with the results cited above we shall give

some examples in the one-dimensional case.

Let C be a bounded real-valued sequence such that |C(n) − c| = O(n−β), for

some β > 0 and some constant c. Let α be a positive number such that α+ β > 1,

and set

Vl(n) = n−α · C(n) , ∀ |n| > n0, for some n0 > 0 .

Clearly Vl satisfies assumption (2.2). Then Theorem 2.1 works for V = Vl (compare

with Simon’s result [10]).

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From this example one can deduce a class of real-valued sequences C tending to

infinity at infinity such that the potentials V of the form

V (n) = n−1 · C(n) ,

satisfy the condition (2.2) of Theorem 2.1 (compare with [6]).

The assertion (iii) of Theorem 2.1 is usually called the “limiting absorption

principle” and has important consequences in scattering theory. For example, it

allows to establish that for each ϕ ∈ C∞0 (µ(H)), for each ψ ∈ H, and each s > 12

we have ∫R‖(1 + |N |)−se−iHtϕ(H)ψ‖2dt ≤ C‖ψ‖2 .

In fact sharper propagation properties of the unitary group e−iHt generated by H

can be obtained, for example:

Theorem 2.2. Let s > 0 be a positive number and denote by [s] the largest

integer in s. Assume that V =∑

0≤k≤[s] Vk, where Vk is a real-valued sequence

tending to zero at infinity and such that

(P βVk)(α) = O(|α|−s+k−|β|) ∀ 0 ≤ |β| ≤ k . (2.4)

Let us set κ = 1−(2s−1)−1. Then for each σ ∈ [0, κ−1/2] and ϕ ∈ C∞0 ((−2ν, 2ν)\σp(H)) we have

‖〈N〉−σe−iHtϕ(H)〈N〉−σ‖ ≤ Const.〈t〉−κσ . (2.5)

3. The Conjugate Operator Method

As we have explained above the proofs of our main theorems are based on the

method of the conjugate operator described in [1–3] (see also [9]). In this section

we give a brief review on these considerations.

3.1. Holder Zygmund space

Let (E, ‖ · ‖) be a Banach space and f : R→ E a bounded continuous function.

Let ε > 0 and m ∈ N an integer. We define the modulus of continuity of order m

of f as

wm(f, ε) = supx∈R

∥∥∥∥∥∥m∑j=1

(−1)j(m

j

)f(x+ jε)

∥∥∥∥∥∥ . (3.1)

One says that f belongs to the Holder–Zygmund space Λα,p, α > 0, p ∈ [1,+∞) if

and only if there is an integer l > α such that the function

ε 7→ ε−αwl(ε) belongs to Lp((0, 1), ε−1dε) .

For p =∞ we set Λα = Λα,∞.

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3.2. Besov space associated to A

Let A be a self-adjoint operator in a Hilbert space H.

3.2.1. For each real s we denote by HAs the Sobolev space associated to A, defined

by the norm ‖f‖s = ‖〈A〉sf‖, with 〈x〉 =√

1 + x2. For all real numbers t ≤ s we

have a continuous dense embedding

HAs ⊂ HAt .

By interpolation we obtain the Besov space HAs,p associated to A, namely

HAs,p = (HAs1 ,HAs2

)θ,p

for s1 < s2, 0 < θ < 1, s = θs1 + (1− θ)s2, 1 ≤ p ≤ ∞ .

3.2.2. Let S be a bounded operator in H. For each integer k, we denote adkA(S)

the sesquilinear form on D(Ak) defined by induction as follows:

ad0A(S) = S ,

ad1A(S) = [S,A] = SA−AS and

adkA(S) = ad1A(adk−1

A (S)) =∑i,j≥0i+j=k

k!

i!j!(−1)iAiSAj .

We say that S is of class Ck(A) if the sesquilinear form adkA(S) has a continuous

extension to H, which we identify with the its associated bounded operator in H(from the Riesz Lemma) and we denote it by the same symbol. In this case one can

prove easily that the function

τ 7→ S(τ) := e−iτASe−iτA ∈ B(H)

is strongly of class Ck. Moreover

ikadkA(S) =dk

dτk

∣∣∣τ=0

S(τ) .

Using the continuity properties of the function S(τ) one can define another class of

regularity of operators. For s > 0 and 1 ≤ p ≤ ∞, we say that S is of class Cs,p(A)

if the function S(τ) is of class Λs,p. In the appendix we give an abstract tool which

will enable us to verify this regularity.

3.3. Mourre estimate

From now on let us assume that S is at least of class C1(A). In particular, the

commutator [S, iA] is a bounded operator in H. Then one may consider the real

open set µA(S) of points λ such that

E(J)[S, iA]E(J) ≥ aE(J) (3.2)

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for some number a > 0, and some neighborhood J of λ. Then we will say that A is

locally strictly conjugate to S on µA(S), and the estimate (3.2) is called the strict

Mourre estimate.

Similarly, we define the real open set µA(S) of points λ such that

E(J)[S, iA]E(J) ≥ aE(J) +K (3.3)

for some number a > 0, some compact operatorK inH and a suitable neighborhood

J of λ. Then we say that A is locally conjugate to S on µA(S), and the estimate

(3.3) is called the Mourre estimate.

Remark 3.1. In general one cannot found explicitly the set µA(S). For this

reason we have introduced the set µA(S), which one can describe rather explicitly

in many interesting situations. Another advantage of µA(S) is its stability under

weakly relatively compact perturbations. More precisely: If S, T are two bounded

and symmetric operators in H, which are of class C1u(A) (e.g. C1,1(A)) and such

that (S + i)−1 − (T + i)−1 is a compact operator in H, then

µA(S) = µA(T ) .

In the following proposition we describe the difference between the strict and

large Mourre estimates.

Proposition 3.1. Assume that S is of class C1(A). Then µA(S) and µA(S)

are open real sets, µA(S) ⊂ µA(S), and the set µA(S) \ µA(S) does not have accu-

mulation points in µA(S). Moreover, µA(S) \ µA(S) consists of eigenvalues of S of

finite multiplicities and the spectrum of H in µA(S) is purely continuous.

We do not know whether the C1(A) regularity property is sufficient for the

absence of singularly continuous spectrum of S in µA(S). But it is proved in Chap. 7

of [1] that the limiting absorption principle (L.A.P.) breaks down if S is not more

regular. The next theorem singles out a sufficient condition ensuring the L.A.P.

Theorem 3.1. Assume S is of class C1,1(A). Then the boundary values of

the resolvent R(λ ± i0) = w∗- limµ→±0 R(λ ± iµ) exist in B(H 12 ,1,H− 1

2 ,∞) locally

uniformly for λ ∈ µA(S). In particular, the spectrum of H is purely absolutely

continuous on µA(S).

Theorem 3.1 remains valid if S is only locally of class C1+0(A), i.e. for each

ϕ ∈ C∞0 (R), the operator ϕ(H) is of class C1+0(A) (see [9]). This fact allows us to

study quite singular Hamiltonians without any gap in their spectrum (see [8]).

Another important consequence of the limiting absorption principle is the so-

called estimate of local decay type, namely that for each ϕ ∈ C∞0 (µA(S)) and ε > 0

we have ∫R‖〈A〉− 1

2−εe−iStϕ(S)ψ‖2dt ≤ C‖ψ‖2 ,

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which plays an important role in scattering theory. The point is that if S is more

regular, then we have more precise propagation properties.

Theorem 3.2. Let s > 0, and S be a self-adjoint operator of class Cs+ 12 (A). Let

us set κ = 1−(2s−1)−1. Then for each σ ∈ [0, κ−1/2] and for each ϕ ∈ C∞0 (µA(S))

there exists a constant C <∞ such that

‖〈A〉−σe−iStϕ(S)〈A〉−σ‖ ≤ C〈t〉−κσ , t ∈ R . (3.4)

We finish this section with some technical result which allows us to check the

Mourre estimate for some particular operators.

Proposition 3.2. Let S1 and S2 be two self-adjoint bounded operators in

Hilbert spaces H1, H2 respectively. Assume that there exist two self-adjoint oper-

ators A1, A2 in H1, H2 such that Si is of class C1(Ai), and that Ai is strictly

conjugate to Si on Ji. Then the operator S = S1⊗1 + 1⊗S2 is of class C1(A), with

A = A1 ⊗ 1 + 1⊗A2. Moreover, A is conjugate to S on J = {λ1 + λ2 | λi ∈ Ji}.

4. The Conjugate Operator

As is explained in the appendix, a key point of our proofs is the construction of

a suitable conjugate operator A for the free hamiltonian H0. This section is entirely

devoted to this fact.

4.1. The one-dimensional case

It is instructive to study the one-dimensional case (see also [4]). More precisely,

the Hilbert space is H = `2(Z), and H0 is given by

(H0ψ)(n) = −ψ(n+ 1)− ψ(n− 1) , ∀ ψ = (ψ(n))n∈Z ∈ H .

In this case N is given by

(Nψ)(n) = nψ(n)

and the finite difference operator P has only one component which we denote by

the same symbol P given in H by

(Pψ)(n) = ψ(n+ 1)− ψ(n) .

By straightforward computations one can see that

H0 = P ∗P − 2 .

Let us consider the self-adjoint operator A in H such that

iA = NP − P ∗N . (4.1)

Lemma 4.1. H0 is of class C∞(A) and µA(H0) = R \ {±2}.

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Proof. Let us calculate the first commutator

[H0, iA] = [P ∗P,NP − P ∗N ]

= [P ∗P,N ]P − P ∗[P ∗P,N ]

= [H0, N ]P − P ∗[H0, N ] . (4.2)

On the other hand, for any ψ ∈ H with compact support we have

〈ψ, [H0, N ]ψ〉 = 〈H0ψ,Nψ〉 − 〈Nψ,H0ψ〉

= −∑n∈Z

(ψ(n+ 1) + ψ(n− 1))nψ(n)

+∑n∈Z

nψ(n)(ψ(n+ 1) + ψ(n− 1))

=∑n∈Z

ψ(n)(ψ(n− 1)− ψ(n+ 1)) .

Then it is clear that this quadratic form can be extended to a continuous quadratic

form in H. Moreover, we can also obtain that

[H0, iA] = 4−H20 = (2−H0)(2 +H0) . (4.3)

So, H0 is of class C∞(A). On the other hand, since −2 ≤ H0 ≤ 2, A is strictly

conjugate to H0 on R \ {±2}. �

4.2. The multidimensional case

In this section we shall show how one passes to the multidimensional case in

Lemma 4.1. We start by recalling some obvious, but useful, commutation relations

between the operators Pi and Nj . We have

[Ni, Nj] = [Pi, Pj ] = 0

[Ni, Pj ] = 0 if i 6= j

[Pi, Ni] = τei , [Ni, P∗i ] = τ−ei

with (ταψ)(β) = ψ(α+ β).

By a straightforward calculation we obtain

P ∗P =ν∑i=1

P ∗i Pi = H0 + 2ν .

This operator is clearly a bounded self-adjoint operator in H, and it is purely

absolutely continuous with spectrum [0, 4ν].

Let us consider the self-adjoint operator A defined by

iA =ν∑i=1

NiPi − P ∗i Ni ≡ NP − P ∗N , (4.4)

with its natural domain D(A) = D(N).

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Lemma 4.2. H0 is of class C∞(A) and µA(H0) = R \ {±2ν}.

Proof. For ν = 1, Lemma 4.2 is Lemma 4.1. In the two-dimensional case,

we have

H ≈ H1 ⊗H2, with H1 = H2 = `2(Z) ,

and

H0 = H0,1 ⊗ 1 + 2⊗H0,2 , (4.5)

A = A1 ⊗ 1 + 1⊗A2 , (4.6)

with for j = 1, 2

H0,j = P ∗j Pj − 2 , (4.7)

iAj = NjPj − P ∗j Nj , (4.8)

acting in the Hilbert spaceHj . On the other hand, H0,j is of class C∞(Aj) and Aj is

locally strictly conjugate to H0,j on R\{±2}. It follows then, from Proposition 3.1,

that H0 is of class C∞(A) and that A is locally strictly conjugate to H0 on R\{±4}.Now, we are done by an obvious induction. �

5. Proofs

Applying the theorems of Sec. 3, in order to prove our main results we must

check that H is sufficiently regular with respect to A and that the Mourre estimate

between H and A holds. More precisely, we shall prove the following proposition.

Proposition 5.1. Assume that the assumptions of Theorem 2.1 hold. Then

(i) the operator V (hence H) is of class C1,1(A);

(ii) A is locally conjugate to H on R \ {±2ν}, i.e.

µA(H) = µA(H0) = R \ {±2ν};

(iii) if V satisfies hypothesis (2.4) of Theorem 2.2 then V (hence H) is of class

Cs(A).

Proof of Theorems 2.1 and 2.2. Combining Theorem 3.1 and the two first

assertions of the preceding proposition one can easily conclude the proof of

Theorem 2.1.

Similarly, Theorem 2.2 follows from the second and third assertions of the

preceding proposition combined with Theorem 3.2. �

Proof of Proposition 5.1. (i) First we note that the second assertion of

Proposition 5.1 follows from the first. Indeed, V tends to zero at infinity. Then the

difference of the resolvents (H + i)−1 − (H0 + i)−1 is a compact operator in H. It

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follows from Remark 3.1 that if H is of class C1,1(A) (in fact we only need C1u(A));

thus

µA(H) = µA(H0) = R \ {±2ν} .

(ii) Now we prove the first assertion of Proposition 5.1 For this we shall treat each

component separately. More precisely, we shall use Theorem 6.1 of the Appendix

by taking for A the operator defined by (4.4), Λ = 〈N〉, G = G∗ = H, and for T one

of the component of V . This is possible because of the obvious lemma:

Lemma 5.1. For each positive number s > 0, 〈N〉−sAs is a bounded operator

in H.

(iii) Taking T = Vs in (ii), it is not difficult to see that assumption (2.2) is

equivalent to hypothesis (6.1) for s = p = 1. Then Vs is of class C1,1(A).

Now we shall deal with Vl. For this we have to calculate the first commutator

between Vl and A. We have

[Vl, iA] = [Vl, NP − P ∗N ]

=ν∑i=1

[Vl, NiPi − P ∗i Ni]

=ν∑i=1

Ni[Vl, Pi]− [Vl, P∗i ]Ni

=ν∑i=1

Ni[Vl, Pi] + (Ni[Vl, Pi])∗ .

On the other hand, it is easy to see that [Vl, Pi] is a bounded operator in H, for

each i = 1, . . . , ν, and that

([Vl, Pi]ψ)(α) = (Vl(α)− Vl(α+ ei))ψ(α + ei)

= (PiVl)(α)ψ(α + ei)

≡ Vl(α)ψ(α + ei)

or equivalently

Ti := [Vl, Pi] = Vl · τei .

Then we have

[Vl, iA] =ν∑i=1

NiTi + T ∗i Ni . (5.1)

It follows that the commutator [Vl, iA] is a bounded operator in H if NiTi also is a

bounded operator for each i = 1, . . . , ν. But

(NiTiψ)(α) = αi(Vl(α) − Vl(α+ ei))ψ(α + ei)

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defines a bounded operator in H if and only if (PiVl)(α) = O(|α|−1) at infinity. But

this property is contained in hypothesis (2.3), and so Vl is of class C1(A). Moreover,

hypothesis (2.3) implies condition (6.5) of Theorem 6.1 for k = s = p = 1, so Vl is

of class C1,1(A).

(iv) To establish the first assertion of Proposition 5.1, it remains to show that

VM is also regular. In fact, we shall prove that VM is of class C2(A) which is

more than we need. Now replacing Vl by VM in the preceding computations, one

concludes that [VM , iA] is a bounded operator in H, and so VM is of class C1(A).

We have to show that [[VM , iA], iA] is a bounded operator in H. For this it suffices

to compute the commutator [NiTi, iA], with (NiTiψ)(α) = αi(PiVM )(α)ψ(α + ei).

We have

[NiTi, iA] =ν∑j=1

[NiTi, NjPj − P ∗j Nj ]

=ν∑j=1

Ni[Ti, NjPj − P ∗j Nj ] + [Ni, NjPj − P ∗j Nj ]Ti .

But it is not difficult to see that

ν∑j=1

[Ni, NjPj − P ∗j Nj]Ti = −NiτeiTi − τ−eiNiTi

which, as we saw before, is a bounded operator in H. On the other hand, a simple

computation shows that

(ψ,Ni[Ti, NjPj − P ∗j Nj]ψ) =∑α

ψ(α) · αi[(αj + δij)(PiVM )(α)

− αj(PiVM )(α + ej)]ψ(α + ei + ej)

+∑α

ψ(α) · αi[(αj − 1)(PiVM )(α − ej)

− (αj + δij − 1)(PiVM )(α)]ψ(α + ei − ej)

where we have denoted by δij the Kronecker symbol. It follows that this expression

defines a bounded operator in H if

αi[(αj + δij)(PiVM )(α) − αj(PiVM )(α + ej)]

= αiαj(PjPiVM )(α) + δijαi(PiVM )(α)

is bounded. But this holds if

(PjVl)(α) = O(|α|−1) and (PjPiVl)(α) = O(|α|−2) as |α| → ∞ .

Consequently, VM is of class C2(A) if it satisfies assumption (2.3) of Theorem 2.1.

This finishes the proof of the first assertion of Proposition 5.1.

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(v) Similarly, one can prove that if V satisfies assumption (2.3) then V is of

class Cs(A). Indeed, by an induction argument one can show that if (P βW )(α) =

O(|α|−|β|) at infinity for each multi-index |β| ≤ k, then W is of class Ck(A). After

that, the second part of Theorem 6.1 allows us to finish the proof easily. �

6. Appendix

The efficiency of the method of the conjugate operator in applications closely

depends on our ability to verify the regularity hypothesis of the studied operator

with respect to its conjugate operator. The goal of this appendix is to develop

abstract tools which will enable us to check this. Since such result can be applied

to other situations (see for example [8]) we describe them in general form.

Let G,H be two Hilbert spaces such that G ⊂ H. Then via the Riesz identifica-

tion we have G ⊂ H ≈ H∗ ⊂ G∗. Let A be a self-adjoint operator in H such that

its associated group eiAt leaves invariant G and G∗. Hence it induces two strongly

continuous groups of bounded operators in G and G∗, which we still denote by the

same symbol eiAt. We then get an automorphism group on X := B(G,G∗) denoted

Wt and defined by

Wt[T ] =WtT = e−iAtTeiAt , ∀ T ∈ X .

In this context one can introduce new regularity classes of operators. In what follows

the numbers s, p, k are such that: s ≥ 0, 1 ≤ p ≤ ∞ and k is a non-negative integer.

Definition 6.1. (a) Let s > 0. An operator T ∈ X is of class Cs,p(A;G,G∗)(resp. Ck(A;G,G∗)) if the function t 7→ WtT ∈ X is of class Λs,p on R (resp. strongly

Ck).

(b) For s = 0 and p = 1, we say that T ∈ X is of class C0,1(A;G,G∗) ≡C+0(A;G,G∗) if the function t 7→ WtT ∈ X is Dini continuous.

It is not difficult to see that T is of class Cs,p(A;G,G∗) if and only if there exists

an integer l > s such that (with the usual convention if p =∞):[∫ 1

0

‖ε−s(Wε − 1)lT‖pXdε

ε

] 1p

<∞ . (6.1)

Similarly we define for operators the regularity classes Cs,p(A;G∗,G), Ck(A;G∗,G)

and C+0(A;G∗,G).

Theorem 6.1. Let Λ be a self-adjoint operator in H bounded from below by a

strictly positive constant such that

(i) eiΛτG ⊂ G and ‖eiΛτ‖B(G) ≤ C〈τ〉N with N <∞;

(ii) the operator AlΛ−l is continuous in G∗ for some integer l ≥ 1.

Let 0 ≤ σ < l. Then a bounded symmetric operator T ∈ X is of class

Cs,p(A;G,G∗) if there exists a function θ ∈ C∞0 (R) with θ(x) > 0 for 0 < a <

|x| < b <∞ such that (with the usual convention if p =∞):

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1074 A. BOUTET DE MONVEL and J. SAHBANI[∫ ∞1

‖rsθ(Λ/r)T‖pXdr

r

] 1p

<∞ . (6.2)

In particular, if p = 1 or ∞ and if the operator T is of class Ck(A;G,G∗) for an

integer 0 ≤ k ≤ s and[∫ ∞1

‖rs−kθ(Λ/r)Ak[T ]‖pXdr

r

] 1p

<∞ , (6.3)

then T is of class Cs,p(A;G,G∗).

Proof. (i) Let us denote W ′t, resp. W ′′t the operators defined on X by

W ′t[T ] = eiAtT, resp. W ′′t [T ] = TeiAt .

Then Wt =W ′′t W ′−t. Hence we get

Wε − 1 =W ′′εW ′−ε − 1 = (W ′′ε − 1)W ′−ε + (W ′−ε − 1) .

Now let us calculate the powers (Wε−1)l as follows. By applying Newton’s formula

we obtain

(Wε − 1)l =l∑

k=0

(l

k

)(W ′′ε − 1)kW ′k−ε(W ′−ε − 1)l−k .

More explicitly, using the definition of W ′ε and W ′′ε , we get

(Wε − 1)l[T ] =l∑

k=0

(l

k

)e−kiAε(e−iAε − 1)l−k · T · (eiAε − 1)k .

On the other hand,

(eiAε − 1)m = eiAmε2

[eiA

ε2 − e−iA ε

2

]m= (2i)meiAm

ε2 sinm

(Aε

2

).

Consequently, (6.1) follows from

[∫ 1

0

‖ε−s sinm(Aε) · T · sinn(Aε)‖pXdε

ε

] 1p

<∞ (6.4)

for any integers m,n such that m+ n = l. But if we set ϕ(x) = sinx+ i sinxx

, it is

easy to see that there exists a finite constant C which depends only on ϕ such that

‖(sinAε)T‖X ≤ ‖εA(εA+ i)−1T‖X‖ϕ(εA)‖B(G∗)

≤ C‖εA(εA+ i)−1T‖X .

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Then (6.4) is a consequence of[∫ ∞1

‖rs(A(A+ ir)−1)mT (A(A+ ir)−1)n‖pXdr

r

] 1p

<∞ , (6.5)

for any integers m,n such that m+ n = l.

(ii) Let us set Ar = A(A + ir)−1 and Λr = Λ(Λ + r)−1. It is clear that

I = Λr + r(Λ + r)−1 .

It follows that for each integer m we have

I = (Λr + r(Λ + r)−1)m =m∑i=0

m!

i!(m− i)!ri(Λ + r)−iΛm−ir .

Consequently we have in B(G∗) the identity

Amr =m∑i=0

m!

i!(m− i)!riAmr (Λ + r)−iΛm−ir

=m∑i=0

m!

i!(m− i)!Am−ir (r(A + ir)−1)iAiΛ−iΛmr

≡ BrΛmr .

Similarly Amr = Λmr Cr in B(G). Since eiAt defines a strongly continuous group of

bounded operators in G∗, ‖r(A+ ir)−1‖B(G∗) is bounded by a finite constant inde-

pendent of r, this is also the case for Ar = I−r(A+ir)−1 in B(G∗). Using condition

(ii) and an interpolation argument, we deduce that ‖Br‖B(G∗) is dominated by a

finite constant independent of r. Similarly ‖Cr‖B(G) ≤ C, independently of r. Then

(6.5) is a consequence of[∫ ∞1

‖rs(Λ(Λ + r)−1)mT (Λ(Λ + r)−1)n‖pXdr

r

] 1p

<∞ , (6.6)

for any integers m,n such that m+ n = l.

(iii) In (6.6) the terms given by m = 0 and n = l dominate all other terms:

Lemma 6.1. For a given T ∈ X , there exists a finite constant C independent

of r such that

‖ΛnrTΛmr ‖X ≤ C‖Λm+nr T‖X .

Proof of lemma. Lemma 6.1 is obtained by complex interpolation. For this

we have to define the powers Λzr for a complex number z and to estimate them

conveniently. From assumption (i) of Theorem 6.1, eiΛτ induces a continuous group

G with polynomially growth at infinity. Using Theorem 3.7.10 of [1], we see that

for each function ϕ ∈ BC∞(R) (i.e. ϕ is a bounded function of class C∞(R) with

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bounded derivatives) the bounded operator ϕ(Λ) in H leaves G invariant, so its

associated bounded operator in G, which we denote by the same symbol, satisfies

‖ϕ(Λ)‖B(G) ≤ C‖ϕ‖BCk

for some constant C and some integer k. We know by hypothesis that there exists

a number a > 0 such that Λ ≥ a, then σ(Λ) ⊂ [a,∞). Let 0 < a0 < a and let

η ∈ C∞(R) such that

η(x) =

{a0 if x <

a0

2x if x > a

then η(Λ) = Λ in H. Let us consider the function ϕ(x) = log(η(x)(η(x) + r)−1),

then ϕ(Λ) = log(Λr) in H. But ϕ belongs to BC∞ (and its norm is independent

of r) then ϕ(Λ) = log(Λr) in G also. It follows that Λr = exp(ϕ(Λ)) in G and

consequently Λzr = exp(zϕ(Λ)) for each z ∈ C. Clearly the function z 7→ Λzr ∈ B(G)

is holomorphic and

‖Λzr‖B(G) ≤ exp(cr|z|) for cr = ‖ϕ(Λ)‖B(G) .

Moreover when z = iy ∈ iR this estimate is uniform with respect to r. Indeed, for

−1 ≤ y ≤ 1 we have

Λiyr = (ϕ(Λ))iy = η(Λ)iy(η(Λ) + r)−iy .

Since ψ(x) = (η(x) + r)iy is of class BC∞ and all its derivatives have a supremum

independent of r, the norm of ψ(Λ) in B(G) is bounded by a constant independent of

r. Then (for more details see [1, p. 329]) there exists a constant c <∞ independent

of r such that

‖Λiyr ‖B(G) ≤ c ∀ y ∈ [−1, 1] .

It follows that there exists a similar constant c such that

‖Λiyr ‖B(G) ≤ cec|y| ∀ y ∈ R .

Let us set M = Λr. For g ∈ G let us consider

z 7→ F (z) = 〈Mz∗g, TM l−zg〉ez2

,

which is holomorphic in the strip {x + iy | y ∈ R, x ∈ (0, l)}, and is continuous

on the closure of this strip; |F (z)| ≤ Ce−y2/2 with a constant C independent of r.

Then

〈Mng, TM l−ng〉en2

= |F (n)| ≤ max

{supy∈R|F (iy)|, sup

y∈R|F (l + iy)|

}

≤ supy∈R

e−y2‖M iy‖2B(G)‖M lT‖X‖g‖2G

≤ C‖M lT‖X‖g‖2G .

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This implies that there exists a constant C independent of r such that for n =

1, . . . , l − 1, we have

‖MnTM l−n‖X ≤ C‖M lT‖X .

This ends the proof of the lemma. �

(iv) Consequently (6.6) follows from[∫ ∞1

‖rs(Λ(Λ + r)−1)lT‖pXdr

r

] 1p

<∞ . (6.7)

But Theorem 3.5.11(b) of [1], p. 144 shows that (6.7) follows from (6.2):[∫ ∞1

‖rsθ(Λ/r)T‖pXdr

r

] 1p

<∞ .

This finishes the proof of the first part of our theorem.

(v) Now let us consider the case p = 1. Let us recall that for 0 < k < s,

1 ≤ p ≤∞, we have

T ∈ Cs,p(A;G,G∗)⇔{T ∈ Ck(A;G,G∗) and

T ∈ Cs−k,p(A;G,G∗)(6.8)

If condition (6.3) of the theorem is satisfied for an integer 0 < k < s then the first

part implies that Ak[T ] (G → G∗) is of class Cs−k,1(A;G,G∗) and (6.8) finishes the

proof in this case.

It remains to prove our assertion in the case where s is an integer and k = s.

In this case, T is of class Ck(A;G,G∗), so Ak[T ] ∈ X . Moreover, condition (6.3)

coincides with (6.2) for s = 0 and with Ak[T ] instead of T . We deduce from the first

part that AkT is of class C+0(A;G,G∗), i.e. T is of class Ck+0(A;G,G∗). It suffices

to note that Ck+0(A;G,G∗) ⊂ Ck,1(A;G,G∗). The case p =∞ is similar. Note that

in this case, the result is trivial for s = k because Ck(A;G,G∗) ⊂ Ck(A;G,G∗). �

This theorem has been proved in [7] (see also [1] where this theorem is proved

in the case σ = 1).

References

[1] W. Amrein, A. Boutet de Monvel and V. Georgescu, C0-Groups, Commutator Methodsand Spectral Theory of N-Body Hamiltonians, Birkhauser, Progress in Math. Ser. 135,Basel 1996.

[2] A. Boutet de Monvel, V. Georgescu and J. Sahbani, “Boundary values of resolventfamilies and propagation properties”, C. R. Acad. Sci. Paris Ser. I Math. 322 (1996)289–294.

[3] A. Boutet de Monvel, V. Georgescu and J. Sahbani, “Higher order estimates in theconjugate operator theory”, Helv. Phys. Acta 71 (1998) 518–553 & preprint Institutde Mathematiques de Jussieu, no. 59, 1996.

[4] A. Boutet de Monvel and J. Sahbani, “On the spectral properties of discreteSchrodinger operators”, C. R. Acad. Sci. Paris Ser. I Math. 326 (1998) 1145–1150.

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[5] A. Kiselev, “Absolutely continuous spectrum of one-dimensional Schrodinger opera-tors and Jacobi matrices with slowly decreasing potentials”, Commun. Math. Phys.179 (1996) 377–400.

[6] S. N. Naboko and S. I. Yakovlev, “On the point spectrum of discrete Schrodingeroperator”, Func. Analys. Appl. 26 (1992) 145–147.

[7] J. Sahbani, “Theoremes de propagation, Hamiltoniens localement reguliers et appli-cations”, PhD thesis, Univ. Paris 7, July 1996.

[8] J. Sahbani, “Propagation theorems for some classes of pseudo-differential operators”,J. Math. Anal. Appl. 211 (1997) 481–497.

[9] J. Sahbani, “The conjugate operator method for locally regular Hamiltonians”,J. Operator Theory 38 (1997) 297–322.

[10] B. Simon, “Some Jacobi matrices with decaying potential and dense point spectrum”,Commun. Math. Phys. 87 (1982) 253–258.

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on the spectral properties of discrete schrÖdinger operators

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