Asymptotics near +-m of the spectral shift function for Dirac operators with non-constant magnetic fields

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Asymptotics near +-m of the spectral shift function for Dirac operators with non-constant magnetic fields

Rafael Tiedra de Aldecoa

To cite this version:

Rafael Tiedra de Aldecoa. Asymptotics near +-m of the spectral shift function for Dirac operators with non-constant magnetic fields. 2009. �hal-00431895�

Asymptotics near±m of the spectral shift function for Dirac operators with non-constant magnetic fields

Rafael Tiedra de Aldecoa

Facultad de Matem´aticas, Pontificia Universidad Cat´olica de Chile, Av. Vicu˜na Mackenna 4860, Santiago, Chile

E-mail: rtiedra@mat.puc.cl

Abstract

We consider a3-dimensional Dirac operatorH0with non-constant magnetic field of constant direction, perturbed by a sign-definite matrix-valued potentialV decaying fast enough at infinity. Then we determine asymptotics, as the energy goes to+mand−m, of the spectral shift function for the pair(H0, H0+V). We obtain, as a by-product, a generalised version of Levinson’s Theorem relating the eigenvalues asymptotics of H0+V near+mand−mto the scattering phase shift for the pair(H0, H0+V).

Contents

1 Introduction 3

2 Unperturbed operator 4

3 Perturbed operator 5

4 Spectral shift function 7

5 Decomposition of the weighted resolvent 8

6 Proof of the main results 12

6.1 The case|λ|< m . . . 13 6.2 The case|λ|> m . . . 18

7 Appendix 22

1 Introduction

It is known [38] that the free Dirac HamiltonianHmacting in the Hilbert spaceH:=L²(R³;C⁴)is unitarily equivalent to the operatorh(P)⊕ −h(P), whereP :=−i∇andR³ ∋ξ 7→h(ξ) := (ξ²+m²)^1/2. For this reason, the set{±m} = h

(∇h)⁻¹({0})

of critical values ofhplays an important role in spectral analysis and scattering theory for Dirac operators. For instance, one cannot prove at±mthe usual limiting absorption principle for operatorsHm+V, even with V a regular perturbation of Hm, by using standard commutator methods. Both the statements and the proofs have to be modified (see e.g. [4, 19]).

In this paper, we provide a new account on the spectral analysis of Dirac operators at the critical values by discussing the behaviour at±mof the spectral shift function associated to sign-definite perturbations of Dirac operators with non-constant magnetic fields. Our work is closely related to [27] where G. D. Raikov treats a similar issue in the case of magnetic Pauli operators. It can also be considered as a complement of [33], where general properties of the spectrum of Dirac operators with variable magnetic fields of constant direction and matrix perturbations are determined. Other related results on the spectrum of3-dimensional magnetic Dirac operators can be found in [2, 3, 5, 7, 11, 13, 15, 16, 17, 20, 23, 34, 36, 37].

Let us describe the content of this paper. We consider a relativistic spin-¹₂particle evolving inR³in pres- ence of a variable magnetic field of constant direction. By virtue of the Maxwell equations, we may assume with no loss of generality that the magnetic field has the form

B(x~ 1, x2, x3) = 0,0, b(x1, x2) . The system is described inHby the Dirac operator

H0:=α1Π1+α2Π2+α3P3+βm,

whereβ ≡ α0, α1, α2, α3 are the usual Dirac-Pauli matrices,m > 0 is the mass of the particle andΠj :=

−i∂j−ajare the generators of the magnetic translations with a vector potential

~a(x1, x2, x3) = a1(x1, x2), a2(x1, x2),0

that satisfiesB=∂1a2−∂2a1. Sincea3= 0, we writeP3=−i∂3instead ofΠ3. We assume that the function b : R² → Ris continuous (see Section 2 for details), so thatH0, defined onC₀^∞(R³;C⁴), can be extended uniquely to a selfadjoint operator inHwith domainD(H0).

Then we consider a bounded positive multiplication operatorV ∈C R³;B_h(C⁴)

, whereB_h(C⁴)is the set of4×4hermitian matrices, and define the perturbed HamiltonianH_± :=H0±V. SinceV is bounded and symmetric, the operatorH_±is selfadjoint inHand has domainD(H) =D(H0). We also assume that|V(x)| decays more rapidly than|x|⁻³as|x| → ∞and that

(H_±−z)⁻³−(H0−z)⁻³∈S1(H) for each z∈R\ {σ(H0)∪σ(H_±)}, (1.1) whereS1(H)denotes the set of trace class operators inH.

Under these assumptions, there exists a unique functionξ(·;H_±, H0)∈L¹ R; (1 +|λ|)⁻⁴dλ

such that the Lifshits-Krein trace formula

Tr

f(H_±)−f(H0)

= Z

R

dλ f^′(λ)ξ(λ;H_±, H0) (1.2) holds for eachf ∈C₀^∞(R)(see [39, Sec. 8.11]). The functionξ(·;H_±, H0)is called the spectral shift function for the pair(H_±, H0). It vanishes identically onR\ {σ(H0)∪σ(H_±)}, and can be related to the number of eigenvalues ofH_± in(−m, m)(see Remark 4.5). Morever, for almost everyλ ∈ σac(H0)the spectral shift function is related to the scattering matrixS(λ;H_±, H0)for the pair(H_±, H0)by the Birman-Krein formula

detS(λ;H_±, H0) = e^{−2πiξ(λ;H±,H0)}.

After identification ofξ(·;H_±, H0)with some representative of its equivalence class, our results are the following. In Proposition 4.4, we show that there exists a constantζ >0defined in terms ofb(cf. Proposition

2.1) such thatξ(·;H_±, H0)is bounded on each compact subset of(−p

m²+ζ,p

m²+ζ)\ {±m} and is continuous on(−p

m²+ζ,p

m²+ζ)\ {±m} ∪σp(H_±)

. In Theorem 6.5, we determine the asymptotic behaviour ofξ(λ;H_±, H0)asλ→ ±m,|λ|< m, and in Theorem 6.14, we determine the asymptotic behaviour ofξ(λ;H_±, H0)as λ → ±m,|λ| > m. In both cases, one hasξ(λ;H_±, H0) → ±∞ asλ → ∓m. The divergence of ξ(λ;H_±, H0) nearλ = ±mscales as the number of eigenvalues near 0 of certain Berezin- Toeplitz type operators. WhenV admits a power-like or exponential decay at infinity, or when it has a compact support, we give the first term of the asymptotic expansion ofξ(λ,;H_±, H0)nearλ =±m(see Proposition 6.10 and Corollary 6.17). In these cases, we show that the limits

εlimց0

ξ m+ε;H₋, H0

ξ m−ε;H₋, H0 and lim

εց0

ξ −m−ε;H+, H0

ξ −m+ε;H+, H0

exist and are equal to positive constants depending on the decay rate ofV at infinity (see Corollary 6.18 for a precise statement). This can be interpreted as a generalised version of Levinson’s Theorem for the pair(H_±, H0) (see [21, 22] for usual versions of Levinson’s Theorem for Dirac operators). The relation between the behaviour of the spectral shift function nearλ= +mand nearλ=−mis explained in Remark 6.15 by using the charge conjugation symmetry.

These results are similar to the results of [27] (where Pauli operators with non-constant magnetic fields are considered) and [12] (where Schr¨odinger operators with constant magnetic field are considered). Part of the interest of this work relies on the fact that we were able to exhibit a non-trivial class of matrix potentialsV satisfying (1.1) even thoughH0 is not a bounded perturbation of the free Dirac operator. We refer to Remark 3.3 and Section 7 for a discussion of this issue.

Let us fix the notations that are used in the paper. The norm and scalar product ofH ≡L²(R³;C⁴)are denoted byk · kandh ·,· i. The symbol⊗stands for the closed tensor product of Hilbert spaces andSp(H), p ∈ [1,∞], denotes the p-th Schatten-von Neumann class of operators inH(S_∞(H)is the set of compact operators inH). We denote byk · kpthe corresponding operator norm. The variablex∈R³is often written as x≡(x_⊥, x3), withx_⊥ ∈R²andx3 ∈R. The symbolQj,j = 1,2,3, denotes the multiplication operator by xjinH,Q:= (Q1, Q2, Q3), andQ_⊥:= (Q1, Q2). Sometimes, when the context is unambiguous, we consider the operatorsQjandPjas operators inL²(R)instead ofHwithout changing the notations. Given a selfadjoint operatorAin a Hilbert spaceG, the symbolE^A(·)stands for the spectral measure ofA.

2 Unperturbed operator

Throughout this paper we assume that the componentb:R²→Rof the magnetic fieldB~ ≡(0,0, b)belongs to the class of “admissible” magnetic fields defined in [27, Sec. 2.1]. Namely, we assume thatb=b0+eb, where b0>0is a constant while the functioneb:R²→Ris such that the Poisson equation

∆ϕe=eb

admits a solutionϕe : R² → R, continuous and bounded together with its derivatives of order up to two. We also defineϕ0(x_⊥) := ¹₄b0|x_⊥|²for eachx_⊥ ∈R² and setϕ:= ϕ0+ϕ. Then we obtain a vector potentiale

~a≡(a1, a2, a3)∈C¹(R²;R³)for the magnetic fieldB~ by putting

a1:=∂1ϕ, a2:=∂2ϕ and a3:= 0.

(changing, if necessary, the gauge, we shall always assume that the vector potential~ais of this form). We refer to [27] for further properties and examples of admissible magnetic fields.

Since the vector potential~abelongs toL^∞_loc(R²;R³), the magnetic Dirac operator H0=α1Π1+α2Π2+α3P3+βm

satisfies all the properties of [33, Sec. 2.1]. The operatorH0 is essentially selfadjoint onC0^∞(R³;C⁴), with domainD(H0)⊂ H^1/2loc(R³;C⁴), the spectrum ofH0satisfies

σ(H0) =σac(H0) = (−∞,−m]∪[m,∞), (2.1)

and we have the identity

H₀²=





H_⊥⁻⊗1+1⊗(P₃²+m²) 0 0 0

0 H⁺_⊥⊗1+1⊗(P₃²+m²) 0 0

0 0 H⁻_⊥⊗1+1⊗(P3²+m²) 0

0 0 0 H_⊥⁺⊗1+1⊗(P₃²+m²)



 (2.2)

with respect to the tensorial decompositionL²(R²)⊗L²(R)ofL²(R³). Here the operatorsH_⊥^±are the compo- nents of the Pauli operatorH_⊥:=H_⊥⁻⊕H_⊥⁺inL²(R²;C²)associated with the vector potential(a1, a2).

We recall from [27, Sec. 2.2] thatdim ker(H_⊥⁻) = ∞, that dim ker(H_⊥⁺) = 0 and that we have the following result.

Proposition 2.1. Letb be an admissible magnetic field with b0 > 0. Then0 = infσ(H_⊥)is an isolated eigenvalue of infinite multiplicity. More precisely, we have

dim ker(H_⊥) =∞ and (0, ζ)⊂R\σ(H_⊥), where

ζ:= 2b0e^−2osc(ϕ)e and osc(ϕ) := supe

x_⊥∈R²ϕ(xe _⊥)− inf

x_⊥∈R2ϕ(xe _⊥).

Finally, since(0, ζ)⊂R\σ(H_⊥), we know from [33, Thm. 1.2.(d)] that the limits

εlimց0hQ3i^−ν3/2(H0−λ∓iε)⁻¹hQ3i^−ν3/2, ν3>1, (2.3) exist for eachλ∈(−p

m²+ζ,p

m²+ζ)\ {±m}(note that we use the usual notationh·i:=p

1 +| · |²).

3 Perturbed operator

We consider now the perturbed operatorsH_± = H0 ±V, whereV ≡ {Vjk} is the multiplication operator associated to the following matrix-valued functionV.

Assumption 3.1. The functionV ∈ C R³;B_h(C⁴)

satisfies for eachx ≡ (x_⊥, x3) ∈ R³and eachj, k ∈ {1, . . . ,4}

V(x)≥0 and |Vjk(x)| ≤Const.hx_⊥i^−ν⊥hx3i^−ν3 for someν_⊥>2andν3>1. (3.1) The potentialV in Assumption 3.1 is short-range alongx3. So we know from [33, Thm. 1.2] that

(i) σess(H_±) =σess(H0) = (−∞,−m]∪[m,∞).

(ii) The point spectrum ofH_± in −p

m²+ζ,p

m²+ζ

\ {±m}is composed of eigenvalues of finite multiplicity and with no accumulation point.

(iii) H_±has no singular continuous spectrum in −p

m²+ζ,p

m²+ζ

. In particular,H0andH_±have a common spectral gap in(−m, m).

Using the formula

(A+λ)^−γ = Γ(γ)⁻¹ Z _∞

0

dt t^γ−1e^−t(A+λ), A:D(A)→ H, A≥0, λ, γ >0,

the diamagnetic inequality [1, Thm. 2.3], and the compactness criterion [9, Thm. 5.7.1], we find that

|Vjk|^1/2 P

ℓ≤3Π^∗_ℓΠℓ+m²−1/4

∈S_∞[L²(R³)].

Sincebis bounded this implies that

|H0|^−1/2V|H0|^−1/2≤ |H0|^−1/2 P

j,k≤4|Vjk|

|H0|^−1/2∈S_∞(H).

So|H0|^−1/2V|H0|^−1/2 also belongs toS_∞(H), sinceS_∞(H)is an hereditaryC^∗-subalgebra ofB(H)[24, Cor. 3.2.3]. One has in particular

V^1/2(|H0|+ 1)^−1/2∈S_∞(H). (3.2)

The standard criterion [31, Thm. XI.20] shows that

|Vjk|^1/2 −∆ +m²−γ

∈Sq[L²(R³)] ifq∈[2,∞)andγq >3/2.

This together with arguments as above implies that

V^1/2|H0|^−γ ∈Sq(H) ifq≥2is even andγq >3. (3.3) So we have in particular that

V^1/2E^H0(B)∈S2(H) for any bounded borel setB ⊂R. (3.4) In the sequel we shall need a more restrictive assumption onV. For this, we recall that there exists numbers z∈R\ {σ(H0)∪σ(H_±)}sinceH0andH_±have a common spectral gap in(−m, m). We also setR0(z) :=

(H0−z)⁻¹andR_±(z) := (H_±−z)⁻¹forz∈C\σ(H0)andz∈C\σ(H_±), respectively.

Assumption 3.2. The functionV ∈C R³;Bh(C⁴)

satisfies for eachx∈R³and eachj, k∈ {1, . . . ,4} V(x)≥0 and |Vjk(x)| ≤Const.hxi^−ν for some constantν >3. (3.5) Furthermore,V is chosen such that

R³_±(z)−R³₀(z)∈S1(H) for eachz∈R\ {σ(H0)∪σ(H_±)}. (3.6) Note that (3.5) implies (3.1) if one takesν3∈ (1, ν−2)andν_⊥ :=ν−ν3. Note also that the choice of functionλ7→(λ−z)⁻³in the trace class condition (3.6) has been made for convenience. Many other choices would also guarantee the existence of the spectral shift function for the pair(H_±, H0)(see e.g. [39, Sec. 8.11]).

Remark 3.3. Since the operatorH0is not a bounded perturbation of the free Dirac operator, we cannot apply the results of [40, Sec. 4] to prove the inclusion (3.6) under the condition (3.5). In general, one has to impose additional assumptions onV to get the result. For instance, ifV verifies (3.5), and

(i) [V, α1] = [V, α2] = 0,

(ii) for eachx∈R³and eachj, k, ℓ∈ {1, . . . ,4}, one has|(∂ℓVjk)(x)| ≤Const.hxi^−ςfor someς >3, (iii) for eachj, k, ℓ∈ {1, . . . ,4}, one has(∂ℓ∂3Vjk)∈L^∞(R³),

then (3.6) is satisfied. Furthermore, if V is scalar, then the same is true without assuming (iii) (and (i) is trivially satisfied). The proof of these statements can be found in the appendix. Here, we only note that a matrix V∈B_h(C⁴)satisfying (i) is necessarily of the form

V=

v₁ 0 v₃ 0 0 v₂ 0 v₃ v₃ 0 v₂ 0

0 v₃ 0 v₁

! , withv₁,v₂∈Randv₃∈C.

4 Spectral shift function

In this section we recall some results due to A. Pushnitski on the representation of the spectral shift function for a pair of not semibounded selfadjoint operators.

Given a a Lebesgue measurable setB⊂R, we setµ(B) := _π¹R

B dt

1+t², and note thatµ(R) = 1. Further- more, ifT =T^∗is a compact operator in a separable Hilbert spaceG, we set

n_±(s;T) := rankE^±T (s,∞)

fors >0.

Then we have the following estimates.

Lemma 4.1 (Lemma 2.1 of [26]). LetT1 = T₁^∗ ∈ S_∞(H)andT2 = T₂^∗ ∈ S1(H). Then one as for each s1, s2>0 Z

R

dµ(t)n_±(s1+s2;T1+tT2)≤n_±(s1;T1) + 1

πs2kT2k1. Forz∈C\σ(H0), we define the usual weighted resolvent

T(z) :=V^1/2(H0−z)⁻¹V^1/2 and the corresponding real and imaginary parts

A(z) :=ReT(z) and B(z) :=ImT(z).

Then the next lemma is direct consequence of the inclusions (3.2)-(3.4) and [25, Prop. 4.4.(i)].

Lemma 4.2. LetV satisfy Assumption 3.1. Then, for almost everyλ∈R, the limitsA(λ+i0) := limεց0A(λ+

iε)andB(λ+i0) := limεց0B(λ+iε)≥0exist inS4(H).

Next theorem follows from the inclusions (3.2), (3.4), (3.6), from the equations (1.9), (8.1), (8.2) of [25], and from Theorem 8.1 of [25].

Theorem 4.3. LetV satisfy Assumption 3.2. Then, for almost everyλ∈R,ξ(λ;H_±, H0)exists and is given by ξ(λ;H_±, H0) =±

Z

R

dµ(t)n_∓ 1;A(λ+i0) +tB(λ+i0)

. (4.1)

We know from (2.3) thatA(λ+i0)andB(λ+i0)exist inB(H)for eachλ∈(−p

m²+ζ,p

m²+ζ)\ {±m}. In Propositions 5.2-5.3 and Corollary 5.5 below we show that in factA(λ+i0)∈S4(H)andB(λ+i0)∈ S1(H)for eachλ∈(−p

m²+ζ,p

m²+ζ)\ {±m}. Hence, by Lemma 4.1, the r.h.s. of (4.1) will turn out to be well-defined for everyλ ∈ (−p

m²+ζ,p

m²+ζ)\ {±m}. In the next proposition we state some regularity properties of the function

(−p

m²+ζ,p

m²+ζ)\ {±m} ∋λ7→ξ(λ;e H_±, H0) :=± Z

R

dµ(t)n_∓ 1;A(λ+i0) +tB(λ+i0) . The proof (which relies on Propositions 5.2-5.3, Lemma 5.4, Corollary 5.5 and the stability result [14, Thm. 3.12]) is similar to the one of [6, Sec. 4.2.1].

Proposition 4.4. LetV satisfy Assumption 3.1. Thenξ(e ·;H_±, H0) is bounded on each compact subset of (−p

m²+ζ,p

m²+ζ)\ {±m}and is continuous on(−p

m²+ζ,p

m²+ζ)\ {±m} ∪σp(H_±) . In the sequel, we identify the functionsξ(e ·;H_±, H0)andξ(·;H_±, H0)since they are equal for almost everyλ∈Rdue to Theorem 4.3 (see [35] for a study where the r.h.s. of (4.1) is directly treated as a definition ofξ(λ;H_±, H0)).

Remark 4.5. In the interval(−m, m),H0has no spectrum and the spectrum ofH_± is purely discrete. Thus the spectral shift functionξ(·;H_±, H0)can be related to the number of eigenvalues of H_± as follows: for λ1, λ2∈(−m, m)\σ(H_±)withλ1< λ2, we have (see [25, Thm. 9.1])

ξ(λ1;H_±, H0)−ξ(λ2;H_±, H0) = rankE^H± [λ1, λ2) .

5 Decomposition of the weighted resolvent

In this section we decompose the weighted resolvent

T(z) =V^1/2(H0−z)V^1/2, z∈C\σ(H0),

into a sumT(z) = T^div(z) +T^bound(z), whereT^div(z)(respectivelyT^bound(z)) corresponds to the diverging (respectively non-diverging) part ofT(z)as z → ±m. Then we estimate the behaviour, in suitable Schatten norms, of each term asz→ ±m. We refer to [12, Sec. 4] and [27, Sec. 4.2] for similar approaches in the case of the Schr¨odinger and Pauli operators.

Letaanda^∗be the closures inL²(R²)of the operators given by

aϕ:= (Π1−iΠ2)ϕ and a^∗ϕ:= (Π1+iΠ2)ϕ,

forϕ∈C₀^∞(R²). Then one has (see [38, Sec. 5.5.2] and [28, Sec. 5]) H0=

m 0 1⊗P3 a⊗1 0 m a∗⊗1−1⊗P3

1⊗P3 a⊗1 −m 0 a∗⊗1−1⊗P3 0 −m

!

, (5.1)

with

ker(a^∗) = ker(aa^∗) = ker(H_⊥⁻)⊂L²(R²). (5.2) Now, let

P:=

_{P0 0 0}

0 0 0 0 0 0P 0 0 0 0 0

be the orthogonal projection onto the union of the eigenspaces ofH0 corresponding to the valuesλ = ±m.

SinceP ≡p⊗1is the orthogonal projection ontoker(H_⊥⁻)⊗L²(R), the equations (5.1) and (5.2) imply that H0andPcommute:

H₀⁻¹P=PH₀⁻¹. (5.3)

In fact, by using (2.2) and (5.1), one gets for eachz∈C\σ(H0)the equalities (H0−z)⁻¹P

= (H0+z) H₀²−z²₋1

P

=

p⊗R(z²−m²) ^{(z+m) 0}_{0 0} ⁰₀ ⁰₀

0 0 (z−m) 0

0 0 0 0

! +

p⊗P3R(z²−m²)_{0 0 1 0}

0 0 0 0 1 0 0 0 0 0 0 0

,

whereR(z) := P₃²−z₋1

,z∈C\[0,∞), is the resolvent ofP₃²inL²(R). This allows us to decomposeT(z) asT(z) =Tdiv(z) +Tbound(z), with

T^div(z) :=V^1/2

p⊗R(z²−m²) ^{(z+m) 0}_{0 0} ⁰₀ ⁰₀

0 0 (z−m) 0

0 0 0 0

! V^1/2, T^bound(z) :=V^1/2

p⊗P3R(z²−m²)_{0 0 1 0}

0 0 0 0 1 0 0 0 0 0 0 0

V^1/2+V^1/2(H0−z)⁻¹P^⊥V^1/2 (P^⊥ := 1−P).

One may note that this decomposition ofT(z)differs slightly from the simpler decomposition T(z) =V^1/2(H0−z)PV^1/2+V^1/2(H0−z)P^⊥V^1/2,

since the first term in T^bound(z)is associated to the projectionPand not the projection P^⊥. This choice is motivated by the will of distinguishing clearly the contributionT^div(z), that diverge as z → ±m, from the contributionT^bound(z), that stays bounded asz→ ±m.