Essential Spectrum of Multiparticle

Brown–Ravenhall Operators in External Field

Sergey Morozov

Received: February 4, 2008 Revised: March 7, 2008

Communicated by Heinz Siedentop

Abstract. The essential spectrum of multiparticle Brown– Ravenhall operators is characterized in terms of two–cluster decompositions for a wide class of external fields and interparticle interactions and for the systems with prescribed symmetries.

2000 Mathematics Subject Classification: 81V55, 81Q10

1 Introduction

It is well known that the eigenvalues of the one–particle Dirac operator are in much better accordance with the spectroscopic data then the eigenvalues of the Schrödinger operator. However, due to the presence of the negative continuum of positronic states the multiparticle Coulomb–Dirac operator has no eigenvalues and its essential spectrum is the whole real line. Coupling with the quantized electromagnetic field does not correct this situation. However, there are ways to construct a semibounded operator which will take the relativistic effects into account. Such models, although nonlocal, find their applications in numerical studies of heavy elements and cosmology, where the relativistic effects cannot be ignored. The most obvious choice of the kinetic energy (sometimes called Chandrasekhar or Herbst operator) given by \sqrt{p^{2}c^{2} + m^{2}c^{4}}, p and m being the momentum and mass of the particle, suffers from the lack of semiboundedness for nuclear charges exceeding 87, as shown in [9]. Most other operators considered in the literature are obtained by reducing the (multiparticle) Dirac operator onto some subspace on which it becomes semibounded. One of such models, extensively studied recently, is by Brown and Ravenhall [4], see also Bethe and Salpeter [3], Sucher [18, 19]. In this model every particle is confined the positive spectral subspace of the free Dirac operator. Since the multiplication by interaction potentials does not leave this subspace invariant, the potential energy terms are projected back by the corresponding projector. The mathematical study of the Brown–Ravenhall operator started from the one–particle case in the article of Evans, Perry, and Siedentop [7]. The authors have proved that the atomic Hamiltonian is semibounded from below for nuclear charges not exceeding 124. This makes the Brown–Ravenhall model applicable to all existing elements. It was also proved in [7] that the essential spectrum of the one–particle atomic Brown–Ravenhall operator is [mc^{2}, ∞) with m being the mass of the particle, and that the singular continuous spectrum is empty. Further studies of the Brown–Ravenhall operator include the improved lower bounds by Tix [21, 22] (see also Burenkov and Evans [5]) in the atomic case, the proof that the eigenvalues of Brown–Ravenhall operator are strictly bigger than those of the one–particle Dirac operator by Griesemer et al. [8], proofs of stability of one-electron molecule by Balinsky and Evans [2], the proof of stability of matter by Hoever and Siedentop [10], and the asymptotic result on the ground state energy for large atomic charges Z (with Z/c fixed) by Cassanas and Siedentop [6]. The essential spectrum of the multiparticle operator was characterized by Jakubaßa–Amundsen [12, 13], and in our joint work with S. Vugalter [16] in terms of two–cluster decompositions. This is usually referred to as HVZ theorem after the well known result for the multiparticle Schrödinger operator. In [11] an analogous result is proved in presence of the constant magnetic field. It is also proved in [16] that the neutral atoms or positively charged atomic ions have infinitely many bound states. In all these previous studies the nuclei were considered as fixed sources of the external field, the particles were assumed to be identical, and the interaction potentials were purely Coulombic. In this paper we generalize the HVZ theorem of [12, 13, 16] as follows: We allow any number of (massive) particles of the system to be identical. We allow quite general matrix interaction potentials. In particular, our result applies in the presence of the magnetic fields if the vector potential decays at infinity in some weak sense. Another problem we address is the reduction to any irreducible representations of the groups of rotation–reflection symmetry and permutations of identical particles. Note that such a reduction allows to analyze the eigenvalues of some irreducible representations even if they are embedded into the continuous spectrum of some other representations. For some particular models (including atoms and molecules in the Born–Oppenheimer approximation) the existence of such eigenvalues can be shown along the same lines as in [16]. From the technical point of view, the nonlocality of the model due to the presence of the spectral projections of the free Dirac operator is overcome with the same ideas as in [16]. One more complication should be stressed: for the Brown–Ravenhall operator the center of mass motion cannot be separated in the same way as it is usually done for Schrödinger operators, where the complete Hamiltonian without external field can be represented in suitable coordinates as

H = A ⊗ I + I ⊗ B,

where A describes the free motion of the center of mass and B is the internal Hamiltonian of the system (see [14]). Such a decomposition appears to be especially fruitful in the presence of rotation symmetries. Since it cannot be obtained for pseudorelativistic operators due to the form of kinetic energy, we have used a completely different approach based on the commutation of the Hamiltonian with the absolute value of the total momentum of the system. Note that the proof of the HVZ theorem for a system of particles described by the Chandrasekhar operator was till now not known for operators reduced to irreducible representations of the rotation–reflection symmetry group (see the article of Lewis, Siedentop and Vugalter [15] for the case without such reductions). Such a proof can now be obtained as a simplified modification of the proof given in this paper. In Section 2 we introduce the model and make the necessary assumptions. At the end of this section we formulate the main result in Theorem 6. The rest of the article contains the proof of this theorem.

2 Setup and Main Result

[A, B] = AB − BA is the commutator of two operators. ⟨·, ·⟩ and ∥ · ∥ stand for the inner product and the norm in L_{2}(ℝ^{3d}, ℂ^{4^{d}}), where d is the dimension of the underlying configuration space. Irrelevant constants are denoted by C. I_{Ω} is the indicator function of the set Ω. For a selfadjoint operator A we denote its spectrum and the corresponding sesquilinear form by σ(A) and ⟨A·, ·⟩ = ⟨·, A·⟩, respectively. We use the conventional units ℏ = c = 1. Sometimes we denote the unitary Fourier transform by ˆ ·. In the Hilbert space L_{2}(ℝ^{3}, ℂ^{4}) the Dirac operator describing a particle of mass m > 0 is given by

D_{m} = −iα · ∇ + βm,

where α := (α_{1}, α_{2}, α_{3}) and β are the 4 × 4 Dirac matrices [20]. The form domain of D_{m} is the Sobolev space H^{1/2}(ℝ^{3}, ℂ^{4}) and the spectrum is (−∞, −m] ∪ [m, +∞). Let Λ_{m} be the orthogonal projector onto the positive spectral subspace of D_{m}:

Λ_{m} := \frac{1}{2} + \frac{−iα · ∇ + βm}{2\sqrt{−Δ + m^{2}}}.  (2.1)

We consider a finite system of N particles with positive masses m_{n}, n = 1, …, N. To simplify the notation we write D_{n} and Λ_{n} for D_{m_{n}} and Λ_{m_{n}},

respectively. Let ℌ_{N} := ⊗^{N}_{n=1}Λ_{n}L_{2}(ℝ^{3}, ℂ^{4}) be the Hilbert space with the inner

product induced by those of ⊗^{N}_{n=1}L_{2}(ℝ^{3}, ℂ^{4}) ≅ L_{2}(ℝ^{3N}, ℂ^{4^{N}}). In this space the N–particle Brown–Ravenhall operator is formally given by

H_{N} = Λ^{N}(∑^{N} ; n=1(D_{n} + V_{n}) + ∑^{N} ; n<jU_{nj})Λ^{N},  (2.2)

with

Λ^{N} := ∏_{n=1}^{N} Λ_{n} = ⊗^{N}_{n=1}Λ_{n}.  (2.3)

Here and below the indices n and j indicate the particle, on whose coordinates the corresponding operator acts. In (2.2) V_{n} is the external field potential for the n^{th} particle, i.e., the operator of multiplication by a hermitian 4 × 4 matrix–function V_{n}(x_{n}), n = 1, …, N, and U_{nj} is the potential energy of the interaction between the n^{th} and j^{th} particles, given by the operator of multiplication by a hermitian 16 × 16 matrix–function U_{nj}(x_{n} − x_{j}), n < j = 1, …, N. More explicitly, if we let s_{j} ∈ {1, 2, 3, 4} be the spinor index of the j^{th} particle, then

\begin{matrix}(V_{n}ψ)(x_{1}, s_{1}; …; x_{n}, s_{n}; …; x_{N}, s_{N}) \\ := ∑ ; s̃_{n} V_{n}^{s_{n},s̃_{n}}(x_{n})ψ(x_{1}, s_{1}; …; x_{n}, s̃_{n}; …; x_{N}, s_{N}), \end{matrix}

and

(U_{nj}ψ)(x_{1}, s_{1}; …; x_{n}, s_{n}; …; x_{j}, s_{j}; …; x_{N}, s_{N})

:= ∑_{s̃_{n},s̃_{j}} U_{nj}^{s_{n}s_{j},s̃_{n}s̃_{j}}(x_{n} − x_{j})ψ(x_{1}, s_{1}; …; x_{n}, s̃_{n}; …; x_{j}, s̃_{j}; …; x_{N}, s_{N}).

Before we make other assumptions on the interaction potentials, let us consider possible decompositions of the system into two clusters. Let Z = (Z_{1}, Z_{2}) be a decomposition of the index set I := {1, …, N} into two disjoint subsets:

I = Z_{1} ∪ Z_{2}, Z_{1} ∩ Z_{2} = ∅.

Let

N_{j} := #Z_{j}, j = 1, 2  (2.4)

be the number of particles in each cluster. We will write n#j if n and j belong to different clusters. Let

H_{Z,1} := ∑_{n∈Z_{1}}(D_{n} + V_{n}) + ∑_{\begin{matrix}n,j∈Z_{1} \\ n<j \end{matrix}} U_{nj},  (2.5)

H_{Z,2} := ∑_{n∈Z_{2}} D_{n} + ∑_{\begin{matrix}n,j∈Z_{2} \\ n<j \end{matrix}} U_{nj}.  (2.6)

We omit H_{Z,j} if Z_{j} = ∅, j = 1, 2. Let us introduce the operators corresponding to noninteracting clusters, with the second cluster transferred far away from the sources of the external field:

H̃_{Z,j} := Λ_{Z,j}H_{Z,j}Λ_{Z,j}, in ℌ_{Z,j} := ⊗_{n∈Z_{j}}Λ_{n}L_{2}(ℝ^{3}, ℂ^{4}), j = 1, 2, (2.7)

where

Λ_{Z,j} := ∏_{n∈Z_{j}}Λ_{n}.

We make the following assumptions:

Assumption 1 There exists C > 0 such that for any Z and j = 1, 2

|⟨H_{Z,j}φ, ψ⟩| ⩽ C∥φ∥_{H^{1/2}}∥ψ∥_{H^{1/2}}, for any φ, ψ ∈ ⊗_{n∈Z_{j}}H^{1/2}(ℝ^{3}, ℂ^{4}). (2.8)

For Coulomb interaction potentials (2.8) follows from Kato’s inequality.

Assumption 2 There exist C_{1} > 0 and C_{2} ∈ ℝ such that for any Z

\begin{matrix}⟨H̃_{Z,j}ψ, ψ⟩ ⩾ C_{1}⟨ ∑_{n∈Z_{j}} D_{n}ψ, ψ⟩ − C_{2}∥ψ∥^{2} & , \\ for any ψ ∈ ⊗_{n∈Z_{j}}Λ_{n}H^{1/2}(ℝ^{3}, ℂ^{4}) & , j = 1, 2. \end{matrix}  (2.9)

Remark 3 Note that for ψ ∈ ⊗_{n∈Z_{j}}Λ_{n}H^{1/2}(ℝ^{3}, ℂ^{4}) the metric

⟨ ∑_{n∈Z_{j}} D_{n}ψ, ψ⟩^{1/2} = ∥ ∑_{n∈Z_{j}} |D_{n}|^{1/2}ψ∥

is equivalent to the norm of ψ in ⊗_{n∈Z_{j}}H^{1/2}(ℝ^{3}, ℂ^{4}), since

Λ_{n}D_{n}Λ_{n} = Λ_{n}|D_{n}|Λ_{n} = Λ_{n}\sqrt{−Δ + m_{n}^{2}}Λ_{n}.  (2.10)

An equivalent formulation of Assumption 2 is that the operator H̃_{Z,j} is semibounded from below even if we multiply all the interaction potentials by 1 + ε with ε > 0 small enough. This is only slightly more restrictive than the semiboundedness of H̃_{Z,j}.

Assumption 4 For any R > 0 there exists a finite constant C_{R} ⩾ 0 such that

∑_{n=1}^{N} ( ∫_{|x|⩽R} |V_{n}(x)|^{2}dx)^{1/2} + ∑_{n<j}^{N} ( ∫_{|x|⩽R} |U_{nj}(x)|^{2}dx)^{1/2} ⩽ C_{R}. (2.11)

This means that the interaction potentials are locally square integrable.

Assumption 5 For any ε > 0 there exists R > 0 big enough such that for all n = 1, …, N

∥V_{n}I_{{|x_{n}|>R}}ψ∥ ⩽ ε∥|D_{n}|^{1/2}ψ∥, for all ψ ∈ H^{1/2}(ℝ^{3}, ℂ^{4}),  (2.12)

and for all n < j = 1, …, N

∥U_{nj}I_{{|x_{n}−x_{j}|>R}}φ∥ ⩽ ε min {∥|D_{n}|^{1/2}φ∥, ∥|D_{j}|^{1/2}φ∥},
for all φ ∈ H^{1/2}(ℝ^{6}, ℂ^{16}).

(2.13)

By Remark 3 this assumption is weaker then the decay of L_{∞} norms of the interaction potentials at infinity. It follows from (2.9) and Remark 3 that for any Z there exists a constant C > 0

such that for any ψ ∈ ⊗_{n∈Z_{j}}Λ_{n}H^{1/2}(ℝ^{3}, ℂ^{4})

∥ψ∥_{H^{1/2}}^{2} ⩽ C(⟨H̃_{Z,j}ψ, ψ⟩ + ∥ψ∥^{2}), j = 1, 2.  (2.14)

Hence by Assumptions 1 and 2, the quadratic forms of operators (2.7) (and, in particular, H_{N}) are semibounded from below and closed on ⊗_{n∈Z_{j}}Λ_{n}H^{1/2}(ℝ^{3}, ℂ^{4}). Thus these operators are well–defined in the form sense.

Some particles of the system (say, k^{th} and l^{th}) can be identical (in which case m_{k} = m_{l}, V_{k} = V_{l}, and U_{kj} = U_{lj} for all j). Then the operator H_{N} can be reduced to the subspace of functions which transform in a certain way under permutations of identical particles. The most physically motivated assumption is that any transposition of two identical particles should change the sign of the wave function ψ ∈ ℌ_{N} describing the system. This is the Pauli principle applied to the identical fermions (the model describes spin 1/2 particles, thus fermions). Let Π be the subgroup of the symmetric group S_{N} generated by transpositions of identical particles. We denote the number of elements of Π by h_{Π}. Let E be some irreducible representation of Π with dimension d_{E} and character ξ_{E}. For ψ ∈ ℌ_{N} let

P^{E}ψ := \frac{d_{E}}{h_{Π}} ∑_{π∈Π} \overline{ξ_{E}(π)}πψ,  (2.15)

where π is the operator of permutation:

(πψ)(x_{1}, s_{1}; …; x_{N}, s_{N}) = ψ(x_{π^{−1}(1)}, s_{π^{−1}(1)}; …; x_{π^{−1}(N)}, s_{π^{−1}(N)}).

Here s_{1}, …, s_{N} are the spinor coordinates of the particles. The operator P^{E} defined in (2.15) is the projector to the subspace of functions in ℌ_{N} which transform according to the representation E of Π. Since any π ∈ Π commutes with H_{N}, P^{E} reduces H_{N}. Let H_{N}^{E} be the corresponding reduced selfadjoint operator in

ℌ_{N}^{E} := P^{E}ℌ_{N}.

For a decomposition Z = (Z_{1}, Z_{2}) let Π_{j}^{Z} be the group generated by transpositions of identical particles inside Z_{j}, j = 1, 2. For any irreducible representation E_{j} of Π_{j}^{Z} with dimension d_{E_{j}} and character ξ_{E_{j}} the projection to the space of functions in ℌ_{Z,j} transforming according to E_{j} under the action of Π_{j}^{Z} is given by

P^{E_{j}}ψ := \frac{d_{E_{j}}}{h_{Π_{j}^{Z}}} ∑_{π∈Π_{j}^{Z}} \overline{ξ_{E_{j}}(π)}πψ, ψ ∈ ℌ_{Z,j},

where h_{Π_{j}^{Z}} is the cardinality of Π_{j}^{Z}. Projectors P^{E_{j}} reduce operators H̃_{Z,j}. We

introduce the reduced operators H̃_{Z,j}^{E_{j}} in

ℌ_{Z,j}^{E_{j}} := P^{E_{j}}ℌ_{Z,j}, j = 1, 2.

Given an irreducible representation E of Π and a decomposition Z = (Z_{1}, Z_{2}), we have

ℌ_{N}^{E} ⊂ ⊕_{(E_{1},E_{2})}(ℌ_{Z,1}^{E_{1}} ⊗ ℌ_{Z,2}^{E_{2}}),  (2.16)

where E_{1,2} are some irreducible representations of Π_{1,2}^{Z}. We write (E_{1}, E_{2}) ≺_{Z} E

if the corresponding term cannot be omitted on the r. h. s. of (2.16) without violation of the inclusion. Apart from permutations of identical particles the operator H_{N}^{E} can have some rotation–reflection symmetries. Let γ be an orthogonal transform in ℝ^{3}: the rotation around the axis directed along a unit vector n_{γ} through an angle φ_{γ}, possibly combined with the reflection x ↦ −x. The corresponding unitary operator O_{γ} acts on the functions ψ ∈ ℌ^{N} as (see [20], Chapter 2)

(O_{γ}ψ)(x_{1}, …, x_{N}) = ∏_{n=1}^{N} e^{−iφ_{γ}n_{γ}·S_{n}}ψ(γ^{−1}x_{1}, …, γ^{−1}x_{N}).

Here S_{n} = −\frac{i}{4}α_{n} ∧ α_{n} is the spin operator acting on the spinor coordinates of the n^{th} particle. The compact group of orthogonal transformations γ such that O_{γ} commutes with V_{n} and U_{nj} for all n, j = 1, …, N (and thus with H_{N}^{E}) we denote by Γ. Further, we decompose ℌ_{N}^{E} into the orthogonal sum

ℌ_{N}^{E} = ⊕_{α∈A}ℌ_{N}^{D_{α},E},  (2.17)

where ℌ_{N}^{D_{α},E} consists of functions which transform under O_{γ} according to some irreducible representation D_{α} of Γ, and A is the set indexing all such irreducible representations. The decomposition (2.17) reduces H_{N}^{E}. We denote the selfadjoint restrictions of H_{N}^{E} to ℌ_{N}^{D_{α},E} by H_{N}^{D_{α},E}. For any fixed irreducible representation D with dimension d_{D} and character ζ_{D} the orthogonal projector in ℌ_{N} onto the subspace of functions which transform according to D is

P^{D} := d_{D} ∫_{Γ} \overline{ζ_{D}(γ)}O_{γ}dμ(γ),

where μ is the invariant probability measure on Γ. For j = 1, 2 let D_{j} be some irreducible representations of Γ with dimensions d_{D_{j}} and characters ζ_{D_{j}}. The corresponding projectors in ℌ_{Z,j} are given by

P^{D_{j}} = d_{D_{j}} ∫_{Γ} \overline{ζ_{D_{j}}(γ)}O_{γ,j}dμ(γ),

where O_{γ,j} is the restriction of O_{γ} to ℌ_{Z,j}:

(O_{γ,j}ψ)(x_{n_{1}}, …, x_{n_{N_{j}}}) = ∏_{n∈Z_{j}} e^{−iφ_{γ}n_{γ}·S_{n}}ψ(γ^{−1}x_{n_{1}}, …, γ^{−1}x_{n_{N_{j}}}).

Given representations D_{j} and E_{j}, projector P^{D_{j}}P^{E_{j}} = P^{E_{j}}P^{D_{j}} reduces H̃_{Z,j}. We denote the reduced operators in

ℌ_{Z,j}^{D_{j},E_{j}} := P^{D_{j}}P^{E_{j}}ℌ_{Z,j}

by H̃_{Z,j}^{D_{j},E_{j}}. Let

κ_{j}(Z, D_{j}, E_{j}) := inf σ(H̃_{Z,j}^{D_{j},E_{j}}).  (2.18)

We write (D_{1}, E_{1}; D_{2}, E_{2}) ≺_{Z} (D, E) if the corresponding term cannot be omit-

ted on the r. h. s. of

ℌ_{N}^{D,E} ⊂ ⊕_{\begin{matrix}(D_{1},E_{1}) \\ (D_{2},E_{2}) \end{matrix}}(ℌ_{Z,1}^{D_{1},E_{1}} ⊗ ℌ_{Z,2}^{D_{2},E_{2}})

without violation of the inclusion. For Z_{2} ≠ ∅ let

\begin{matrix}κ(Z, D, E) \\ :=\begin{cases}inf\{κ_{1}(Z, D_{1}, E_{1})+κ_{2}(Z, D_{2}, E_{2}) : (D_{1}, E_{1}; D_{2}, E_{2}) ≺_{Z} (D, E)\}, & Z_{1} ≠ ∅, \\ κ_{2}(Z, D, E), & Z_{1} = ∅. \end{cases} \end{matrix} ; (2.19)

The main result of the article is

Theorem 6 Suppose Assumptions 1, 2, 4, and 5 hold true. For N ∈ ℕ let D be some irreducible representation of Γ, and E some irreducible representation of Π, such that P^{D}P^{E} ≠ 0. Then

σ_{ess}(H_{N}^{D,E}) = [κ(D, E), ∞),

where

κ(D, E) = min {κ(Z, D, E) : Z = (Z_{1}, Z_{2}), Z_{2} ≠ ∅}.  (2.20)

Remark 7 We only need Assumption 2 for the operators H̃_{Z,j}^{D_{j},E_{j}} which appear in (2.18), (2.19).

3 Commutator Estimates

3.1 One Particle Commutator Estimate

Lemma 8 Let χ ∈ C_{B}^{2}(ℝ^{3}) (i. e. a bounded twice–differentiable function with bounded derivatives). Then for m_{n} > 0 the commutator [χ, Λ_{n}] is a bounded operator from L_{2}(ℝ^{3}, ℂ^{4}) to H^{1}(ℝ^{3}, ℂ^{4}). There exists C(m) > 0 such that

∥[χ, Λ_{n}]∥_{L_{2}(ℝ^{3},ℂ^{4})→H^{1}(ℝ^{3},ℂ^{4})} ⩽ C(m_{n})(∥∇χ∥_{L_{∞}} + ∥∂^{2}χ∥_{L_{∞}}).  (3.1)

Here ∥∂^{2}χ∥_{L_{∞}} = max_{\begin{matrix}z∈ℝ^{3} \\ k,l∈\{1,2,3\} \end{matrix}}|∂_{kl}^{2}χ(z)|.

Proof. In the coordinate representation for f ∈ C_{0}^{1}(ℝ^{3}, ℂ^{4}) the operator Λ_{n} acts as

(Λ_{n}f)(x) = \frac{f(x)}{2} + \frac{im_{n}}{2π^{2}} lim_{ε→+0} ∫_{|y−x|⩾ε} \frac{α · (x − y)}{|x − y|^{3}}K_{1}(m_{n}|x − y|)f(y)dy

+ \frac{m_{n}^{2}}{4π^{2}} ∫_{ℝ^{3}} (β \frac{K_{1}(m_{n}|x − y|)}{|x − y|} + \frac{iα · (x − y)}{|x − y|^{2}}K_{0}(m_{n}|x − y|))f(y)dy,

where the limit on the r. h. s. is the limit in L_{2}(ℝ^{3}, ℂ^{4}) (see Appendix B of [16], where this formula is derived in the case m_{n} = 1). The rest of the proof is an obvious modification of the proof of Lemma 1 of [16], where the case m_{n} = 1 is considered. •

Remark 9 Since we only deal with a finite number of particles with positive masses, we will not trace the m-dependence of the constant in (3.1) any longer.

3.2 Multiparticle Commutator Estimate

Lemma 10 For any d, k ∈ ℕ there exists C > 0 such that for any χ ∈ C_{B}^{1}(ℝ^{d}) and u ∈ H^{1/2}(ℝ^{d}, ℂ^{k})

∥χu∥_{H^{1/2}(ℝ^{d},ℂ^{k})} ⩽ C(∥χ∥_{L_{∞}(ℝ^{d})} + ∥∇χ∥_{L_{∞}(ℝ^{d})})∥u∥_{H^{1/2}(ℝ^{d},ℂ^{k})}.  (3.2)

Proof of Lemma 10. We can choose the norm in H^{1/2}(ℝ^{d}, ℂ^{k}) as (see [1], Theorem 7.48)

∥u∥_{H^{1/2}(ℝ^{d},ℂ^{k})}^{2} := ∥u∥_{L_{2}(ℝ^{d},ℂ^{k})}^{2} + ∫∫ \frac{|u(x) − u(y)|^{2}}{|x − y|^{d+1}} dxdy.

Documenta Mathematica 13 (2008) 51–79

Then

∥χu∥_{H^{1/2}(ℝ^{d},ℂ^{k})}^{2} = ∥χu∥_{L_{2}(ℝ^{d},ℂ^{k})}^{2} + ∫∫ \frac{|χ(x)u(x) − χ(y)u(y)|^{2}}{|x − y|^{d+1}} dxdy

⩽ ∥χ∥_{L_{∞}}^{2}∥u∥_{L_{2}}^{2} + ∫∫ (\frac{|χ(x)|^{2}|u(x) − u(y)|^{2}}{|x − y|^{d+1}} + \frac{|χ(x) − χ(y)|^{2}|u(y)|^{2}}{|x − y|^{d+1}} )dxdy

⩽ ∥χ∥_{L_{∞}}^{2}∥u∥_{H^{1/2}}^{2} + sup_{y∈ℝ^{d}} ∫ \frac{|χ(x) − χ(y)|^{2}}{|x − y|^{d+1}} dx∥u∥_{L_{2}}^{2}.

(3.3)

The supremum on the r. h. s. of (3.3) can be estimated as

sup_{y∈ℝ^{d}} ∫ \frac{|χ(x) − χ(y)|^{2}}{|x − y|^{d+1}} dx ⩽ sup_{y∈ℝ^{d}} ∫_{|x−y|⩽1} \frac{|χ(x) − χ(y)|^{2}}{|x − y|^{d+1}} dx

+ sup_{y∈ℝ^{d}} ∫_{|x−y|>1} \frac{|χ(x) − χ(y)|^{2}}{|x − y|^{d+1}} dx ⩽ |𝕊^{d−1}|(∥∇χ∥_{L_{∞}}^{2} + 4∥χ∥_{L_{∞}}^{2}),

(3.4)

where |𝕊^{d−1}| is the area of (d − 1)–dimensional unit sphere. Substituting (3.4) into (3.3) we obtain (3.2). •

Lemma 11 For any χ ∈ C_{B}^{2}(ℝ^{3N}) the operator [χ, Λ^{N}] is bounded in H^{1/2}(ℝ^{3N}, ℂ^{4^{N}}), and for any ψ ∈ H^{1/2}(ℝ^{3N}, ℂ^{4^{N}}) we have

\begin{matrix}∥[χ, Λ^{N}]ψ∥_{H^{1/2}} ⩽ C(∥∇χ∥_{L_{∞}} + ∥∂^{2}χ∥_{L_{∞}})(∥χ∥_{L_{∞}} + ∥∇χ∥_{L_{∞}})∥ψ∥_{H^{1/2}} \\ (3.5) \end{matrix}

with C depending only on N and the masses of the particles.

Proof. Successively commuting χ with Λ_{n}, n = 1, …, N (see (2.3)) we obtain

[χ, Λ^{N}] = ∑_{n=1}^{N} ∏_{k=1}^{n−1} Λ_{k}[χ, Λ_{n}] ∏_{l=n+1}^{N} Λ_{l},  (3.6)

where the empty products should be replaced by identity operators. By (2.1) the operators Λ_{n} are bounded in H^{1/2} for any n = 1, …, N. This, together with (3.6) and Lemmata 8 and 10, implies (3.5). •

4 Lower Bound of the Essential Spectrum

In this section we prove that

inf σ_{ess}(H_{N}^{D,E}) ⩾ κ(D, E).  (4.1)

4.1 Partition of Unity

Lemma 12 There exists a set of nonnegative functions {χ_{Z}} indexed by possible 2–cluster decompositions Z = (Z_{1}, Z_{2}) satisfying

1. χ_{Z} ∈ C^{∞}(ℝ^{3N}) for all Z;
2. χ_{Z}(κX) = χ_{Z}(X) for all |X| = 1, κ > 1, Z_{2} ≠ ∅;
3. ∑_{Z} χ_{Z}^{2}(X) = 1, for all X ∈ ℝ^{3N};  (4.2)

4.

There exists C > 0 such that for any X ∈ supp χ_{Z}

min {|x_{j} − x_{n}| : x_{j} ∈ Z_{1}, x_{n} ∈ Z_{2}; |x_{n}| : x_{n} ∈ Z_{2}} ⩾ C|X|;

(4.3)

5.

χ_{Z}(γx_{1}, …, γx_{N}) = χ_{Z}(x_{1}, …, x_{N}) for any orthogonal transformation γ;

6. χ_{Z} is invariant under permutations of variables preserving Z_{1,2}.

Proof. The proof is essentially based on the modification of the argument given in [17], Lemma 2.4.

1. We first prove that for any X = (x_{1}, …, x_{N}) ∈ ℝ^{3N} with |X| = 1 there exists a 2–cluster decomposition Z = (Z_{1}, Z_{2}) such that

min {|x_{j} − x_{n}| : x_{j} ∈ Z_{1}, x_{n} ∈ Z_{2}; |x_{n}| : x_{n} ∈ Z_{2}} > N^{−3/2}.  (4.4)

Indeed, let k be such that |x_{k}| ⩾ |x_{j}| for all j = 1, …, N. Then, since |X| = 1,

|x_{k}| ⩾ N^{− \frac{1}{2}} .  (4.5)

Choose Cartesian coordinates in ℝ^{3} with the first axis passing through the origin and x_{k}, so that x_{k} = (|x_{k}|, 0, 0). Consider N regions

\begin{aligned}R_{1} & := \{x ∈ ℝ^{3} : x^{1} ⩽ |x_{k}|/N\}, \\ R_{l} & := \{x ∈ ℝ^{3} : x^{1} ∈ ((l − 1)|x_{k}|/N, l|x_{k}|/N]\}, l = 2, …, N. \end{aligned}

At least one of these regions does not contain x_{j} with j ≠ k. Let l_{0} be the maximal index of such regions. Let Z_{2} be the set of indices n such that x_{n} ∈ ∪_{l>l_{0}}R_{l}. Z_{2} is nonempty since x_{k} ∈ Z_{2}. Setting Z_{1} := I \ Z_{2} we observe that

min {|x_{j} − x_{n}| : x_{j} ∈ Z_{1}, x_{n} ∈ Z_{2}; |x_{n}| : x_{n} ∈ Z_{2}} > |x_{k}|/N,

which together with (4.5) implies (4.4).

2. Choose η ∈ C^{∞}(ℝ_{+}, [0, 1]) so that

η(t) ≡ \begin{cases}0, & t ∈ [0, 1], \\ 1, & t ∈ [2, ∞). \end{cases}

Let

ζ_{Z}(X) := \begin{cases}η(2|X|) ∏_{n∈Z_{2}} η(\frac{2|x_{n}|}{|X|N^{−3/2}} ) ∏_{j∈Z_{1}} η(\frac{2|x_{j} − x_{n}|}{|X|N^{−3/2}}), & Z_{2} ≠ ∅, \\ 1 − η(2|X|), & Z_{2} = ∅. \end{cases}

(4.6)

Functions (4.6) satisfy conditions 1, 2, 4 (with C = N^{−3/2}/2), 5, and 6 of Lemma 12. Moreover, by the first part of the proof

∑_{Z} ζ_{Z}(X) ⩾ 1, for all X ∈ ℝ^{3N}.

Hence all the conditions are satisfied by the functions

χ_{Z} := ζ_{Z}^{1/2}( ∑_{Z} ζ_{Z})^{−1/2}.

• Let

χ_{Z}^{R}(X) := χ_{Z}(X/R),  (4.7)

where the functions χ_{Z} are defined in Lemma 12. The derivatives of χ_{Z}^{R} decay as R tends to infinity:

∥∇χ_{Z}^{R}∥_{∞} ⩽ CR^{−1}, ∥∂^{2}χ_{Z}^{R}∥_{∞} ⩽ CR^{−2}.  (4.8)

To simplify the notation we omit the superscript R further on.

4.2 Cluster Decomposition and Lower Bound

We now estimate from below the quadratic form of H_{N}^{D,E} on a function ψ from

ℌ_{N}^{D,E} ∩ Λ^{N} ⊗^{N}_{n=1} H^{1/2}(ℝ^{3}, ℂ^{4}), which is the form domain of H_{N}^{D,E}.

\begin{aligned}⟨H_{N}^{D,E}ψ, ψ⟩ & = ⟨( ∑_{n=1}^{N}(D_{n} + V_{n}) + ∑_{n<j}^{N} U_{nj}) ∑_{Z} χ_{Z}^{2}ψ, ψ⟩ \\ & = ∑_{Z}⟨( ∑_{n=1}^{N}(D_{n} + V_{n}) + ∑_{n<j}^{N} U_{nj})χ_{Z}ψ, χ_{Z}ψ⟩. \end{aligned}

Here we have used (4.2) and the relation

∑_{Z}⟨f, ∑_{n=1}^{N} ∇_{n}(χ_{Z}^{2}g)⟩ = ∑_{Z}⟨χ_{Z}f, ∑_{n=1}^{N} ∇_{n}(χ_{Z}g)⟩ + ∑_{Z}⟨f, ∑_{n=1}^{N} ∇_{n}(\frac{χ_{Z}^{2}}{2} )g⟩ (4.9)

which holds for any f, g ∈ ⊗^{N}_{n=1}H^{1/2}(ℝ^{3}, ℂ^{4}). The last term on the r. h. s. of

(4.9) is equal to zero due to (4.2). Thus

\begin{aligned}⟨H_{N}^{D,E}ψ, ψ⟩ & = ∑_{Z=(Z_{1},Z_{2})} (⟨(H_{Z,1} + H_{Z,2})Λ^{N}χ_{Z}ψ, Λ^{N}χ_{Z}ψ⟩ \\ & + ⟨(H_{Z,1} + H_{Z,2})[χ_{Z}, Λ^{N}]ψ, Λ^{N}χ_{Z}ψ⟩ \\ & + ⟨(H_{Z,1} + H_{Z,2})χ_{Z}ψ, [χ_{Z}, Λ^{N}]ψ⟩ \\ & + ⟨ ∑_{n∈Z_{2}} V_{n}χ_{Z}^{2}ψ, ψ⟩ + ⟨∑_{\begin{matrix}n<j \\ n#j \end{matrix}} U_{nj}χ_{Z}^{2}ψ, ψ⟩). \end{aligned}  (4.10)

By (2.12), (2.13), (4.3), (4.7), and (2.14) the terms at the last line of (4.10) can be estimated as

⟨ ∑_{n∈Z_{2}} V_{n}χ_{Z}^{2}ψ, ψ⟩ + ⟨∑_{\begin{matrix}n<j \\ n#j \end{matrix}} U_{nj}χ_{Z}^{2}ψ, ψ⟩ ⩾ −ε_{1}(R)(⟨H_{N}^{D,E}ψ, ψ⟩ + ∥ψ∥^{2}) (4.11)

with ε_{1}(R) → 0 as R → ∞. The terms at the second and third lines of (4.10) can also be estimated as

\begin{matrix}⟨(H_{Z,1} + H_{Z,2})[χ_{Z}, Λ^{N}]ψ, Λ^{N}χ_{Z}ψ⟩ + ⟨(H_{Z,1} + H_{Z,2})χ_{Z}ψ, [χ_{Z}, Λ^{N}]ψ⟩ \\ ⩾ −ε_{2}(R)(⟨H_{N}^{D,E}ψ, ψ⟩ + ∥ψ∥^{2}), ε_{2}(R) ⟶_{R→∞} 0, \end{matrix}

according to (2.8), (3.2), (3.5), (4.8), and (2.14). For Z_{2} ≠ ∅ we estimate the terms at the first line of (4.10) in the following way (recall the definitions (2.18), (2.19) and (2.20)):

⟨(H_{Z,1} + H_{Z,2})Λ^{N}χ_{Z}ψ, Λ^{N}χ_{Z}ψ⟩

= ∑_{(D_{1},E_{1};D_{2},E_{2})≺_{Z}(D,E)}⟨(H_{Z,1}P^{D_{1}}P^{E_{1}} + H_{Z,2}P^{D_{2}}P^{E_{2}})Λ^{N}χ_{Z}ψ, Λ^{N}χ_{Z}ψ⟩

⩾ ∑_{(D_{1},E_{1};D_{2},E_{2})≺_{Z}(D,E)}⟨(κ_{1}(Z, D_{1}, E_{1})P^{D_{1}}P^{E_{1}}

+ κ_{2}(Z, D_{2}, E_{2})P^{D_{2}}P^{E_{2}})Λ^{N}χ_{Z}ψ, Λ^{N}χ_{Z}ψ⟩

⩾ κ(D, E)⟨Λ^{N}χ_{Z}ψ, Λ^{N}χ_{Z}ψ⟩

= κ(D, E)⟨χ_{Z}^{2}ψ, ψ⟩ + κ(D, E)⟨[Λ^{N}, χ_{Z}]ψ, χ_{Z}ψ⟩.

(4.12)

By (3.2), (3.5), (4.8), and (2.14) the last term on the r. h. s. of (4.12) can be estimated as

κ(D, E)⟨[Λ^{N}, χ_{Z}]ψ, χ_{Z}ψ⟩ ⩾ −ε_{3}(R)(⟨H_{N}^{D,E}ψ, ψ⟩ + ∥ψ∥^{2}), ε_{3}(R) ⟶_{R→∞} 0.

(4.13)

Substituting the estimates (4.11) — (4.13) into (4.10) we obtain

\begin{aligned}⟨H_{N}^{D,E}ψ, ψ⟩ & ⩾ κ(D, E)⟨ ∑_{\begin{matrix}Z=(Z_{1},Z_{2}) \\ Z_{2}≠∅ \end{matrix}} χ_{Z}^{2}ψ, ψ⟩ + ⟨H_{N}^{D,E}Λ^{N}χ_{(I,∅)}ψ, Λ^{N}χ_{(I,∅)}ψ⟩ \\ & − ε_{4}(R)(⟨H_{N}^{D,E}ψ, ψ⟩ + ∥ψ∥^{2}), ε_{4}(R) ⟶_{R→∞} 0. \end{aligned} ; (4.14)

4.3 Estimate Inside of the Compact Region

It remains to estimate from below the quadratic form of H_{N}^{D,E} on Λ^{N}χ_{(I,∅)}ψ. Note that according to Lemma 12 and (4.7) supp χ_{(I,∅)} ⊂ [−2R, 2R]^{3N}. To simplify the notation let

χ_{0} := χ_{(I,∅)}.

Lemma 13 For M > 0 let

W_{M} := {p ∈ ℝ^{3N} : |p_{i}| ⩽ M, i = 1, …, 3N}, W̃_{M} := ℝ^{3N} \ W_{M}.

There exists a finite set Q_{M} ⊂ L_{2}(ℝ^{3N}) such that for any f ∈ L_{2}(ℝ^{3N}) with suppf ⊂ [−2R, 2R]^{3N}, f⊥Q_{M} holds

∥f̂∥_{L_{2}(W̃_{M})} ⩾ \frac{1}{2}∥f̂∥_{L_{2}(ℝ^{3N})}.

The proof of Lemma 13 is analogous to the proof of Theorem 7 of [23] and is given in Appendix C of [16].

It follows from (2.9) that for any M > 0

\begin{matrix}⟨H_{N}^{D,E}Λ^{N}χ_{0}ψ, Λ^{N}χ_{0}ψ⟩ ⩾ C_{1}⟨∑^{N} ; n=1D_{n}IW̃_{M} Λ^{N}χ_{0}ψ, Λ^{N}χ_{0}ψ⟩ − C_{2}∥χ_{0}ψ∥^{2}. \\ (4.15) \end{matrix}

Here IW̃_{M} is the operator of multiplication by the characteristic function of

W̃_{M} in momentum space. We choose

M := 8(κ(D, E) + C_{2})C_{1}^{−1}  (4.16)

and assume henceforth that f := χ_{0}ψ is orthogonal to the set Q_{M} defined in Lemma 13. Since in momentum space the operator D_{n} acts on functions from Λ_{n}L_{2}(ℝ^{3}, ℂ^{4}) as multiplication by \sqrt{|p|^{2} + m_{n}^{2}}, by construction of W̃_{M} we have

⟨∑^{N} ; n=1D_{n}IW̃_{M} Λ^{N}χ_{0}ψ, Λ^{N}χ_{0}ψ⟩ ⩾ M∥IW̃_{M} Λ^{N}χ_{0}ψ∥^{2}.  (4.17)

Inequalities (4.15) and (4.17) imply

\begin{matrix}⟨H_{N}^{D,E}Λ^{N}χ_{0}ψ, Λ^{N}χ_{0}ψ⟩ ⩾ C_{1}M∥IW̃_{M} Λ^{N}χ_{0}ψ∥^{2} − C_{2}∥χ_{0}ψ∥^{2} \\ ⩾ C_{1}M(∥IW̃_{M} χ_{0}ψ∥ − ∥IW̃_{M} [Λ^{N}, χ_{0}]ψ∥)^{2} − C_{2}∥χ_{0}ψ∥^{2} \\ ⩾ C_{1}M(\frac{1}{2}∥IW̃_{M} χ_{0}ψ∥^{2} − ∥IW̃_{M} [Λ^{N}, χ_{0}]ψ∥^{2}) − C_{2}∥χ_{0}ψ∥^{2} \\ ⩾ 4(κ(D, E) + C_{2})∥IW̃_{M} χ_{0}ψ∥^{2} \\ − 8(κ(D, E) + C_{2})∥[Λ^{N}, χ_{0}]ψ∥^{2} − C_{2}∥χ_{0}ψ∥^{2}. \end{matrix}  (4.18)

At the last step we have used (4.16). The second term on the r. h. s. of (4.18) can be estimated analogously to (4.13) as

−8(κ(D, E)+C_{2})∥[Λ^{N}, χ_{0}]ψ∥^{2} ⩾ −ε_{5}(R)(⟨H_{N}^{D,E}ψ, ψ⟩+∥ψ∥^{2}), ε_{5}(R) ⟶_{R→∞} 0.

For the first term on the r. h. s. of (4.18) Lemma 13 implies

4∥IW̃_{M} χ_{0}ψ∥^{2} ⩾ ∥χ_{0}ψ∥^{2}.  (4.19)

As a consequence of (4.18) — (4.19), we have

\begin{matrix}⟨H_{N}^{D,E}Λ^{N}χ_{0}ψ, Λ^{N}χ_{0}ψ⟩ & ⩾ κ(D, E)∥χ_{0}ψ∥^{2} − ε_{5}(R)(⟨H_{N}^{D,E}ψ, ψ⟩ + ∥ψ∥^{2}), \\ ε_{5}(R) ⟶_{R→∞} 0. \end{matrix} ; (4.20)

4.4 Completion of the Proof

By (4.14), (4.20), and (4.2)

⟨H_{N}^{D,E}ψ, ψ⟩ ⩾ κ(D, E)∥ψ∥^{2} − ε_{6}(R)(⟨H_{N}^{D,E}ψ, ψ⟩ + ∥ψ∥^{2}), ε_{6}(R) ⟶_{R→∞} 0.

for any ψ in the form domain of H_{N}^{D,E} orthogonal to the finite set of functions (cardinality of this set depends on R). This implies the discreteness of the spectrum of H_{N}^{D,E} below κ(D, E) and thus (4.1).

5 Spectrum of the Free Cluster

In this section we characterize the spectrum of the cluster Z_{2} which does not interact with the external field.

Proposition 14 For any irreducible representations D_{2}, E_{2} of rotation– reflection and permutation groups the spectrum of H̃_{Z,2}^{D_{2},E_{2}} is

σ(H̃_{Z,2}^{D_{2},E_{2}}) = σ_{ess}(H̃_{Z,2}^{D_{2},E_{2}}) = [κ_{2}(Z, D_{2}, E_{2}), ∞),

with some κ_{2}(Z, D_{2}, E_{2}) ∈ ℝ.

Proof. Let us introduce the new coordinates in the configuration space ℝ^{3N_{2}} of the cluster Z_{2} = {n_{1}, …, n_{N_{2}}}, in the same manner as it is done in [15]. Let M := ∑_{n∈Z_{2}} m_{n} be the total mass of the particles constituting the cluster. We introduce

\begin{aligned}y_{0} & := \frac{1}{M} ∑_{n∈Z_{2}} m_{n}x_{n}, \\ y_{k} & := x_{n_{k+1}} − x_{n_{1}}, k = 1, …, N_{2} − 1. \end{aligned}  (5.1)

The Jacobian of this variable change is one. Here y_{0} is the coordinate of the center of mass, whereby y_{k}, k = 1, …, N_{2} − 1 are the internal coordinates of the cluster. Accordingly,

\begin{aligned}x_{n_{1}} & = y_{0} − \frac{1}{M} ∑_{k=1}^{N_{2}−1} m_{n_{k+1}}y_{k}, \\ x_{n_{l+1}} & = y_{0} + y_{l} − \frac{1}{M} ∑_{k=1}^{N_{2}−1} m_{n_{k+1}}y_{k}, l = 1, …, N_{2} − 1. \end{aligned}  (5.2)

The momentum operators in the new coordinates are

p_{n_{1}} := −i∇_{x_{n_{1}}} = \frac{m_{n_{1}}}{M} P − ∑_{k=1}^{N_{2}−1}(−i∇_{y_{k}}),
p_{n_{k}} := −i∇_{x_{n_{k}}} = \frac{m_{n_{k}}}{M} P + (−i∇_{y_{k−1}}), k = 2, …, N_{2},

(5.3)

where P is the total momentum of the cluster:

P := ∑_{n∈Z_{2}} −i∇_{x_{n}} = −i∇_{y_{0}}.

Let F_{0} be the partial Forurier transform on ℌ_{Z,2}^{D_{2},E_{2}} defined by

(F_{0}f)(P, y_{1}, …, y_{N_{2}−1}) := \frac{1}{(2π)^{3/2}} ∫_{ℝ^{3}} f(y_{0}, y_{1}, …, y_{N_{2}−1})e^{−iPy_{0}}dy_{0}.

By (2.6) and (2.7) we have

H̃_{Z,2}^{D_{2},E_{2}} = F_{0}^{−1}Λ̂_{Z,2}Ĥ_{Z,2}^{D_{2},E_{2}} Λ̂_{Z,2}F_{0},

where in the new coordinates

Ĥ_{Z,2}^{D_{2},E_{2}} := ∑_{n∈Z_{2}}(α_{n} ·p_{n} +β_{n}m_{n})+ ∑_{k=2}^{N_{2}−1} U_{n_{1}n_{k}}(y_{k})+ ∑_{1<k<l⩽N_{2}−1} U_{n_{k}n_{l}}(y_{k} −y_{l}),

(5.4)

Λ̂_{Z,2} := ∏_{n∈Z_{2}} Λ̂_{n},  (5.5)

Λ̂_{n} := \frac{1}{2} + \frac{α_{n} · p_{n} + β_{n}m_{n}}{2\sqrt{p_{n}^{2} + m_{n}^{2}}} ,

operators p_{n} are given by (5.3), and P should now be interpreted as multiplication by the vector–function. The operators (5.4) and (5.5) obviously commute with 𝔓 := |P|. The operator F_{0}^{−1}𝔓F_{0} (unlike F_{0}^{−1}PF_{0}) is well–defined in ℌ_{Z,2}^{D_{2},E_{2}}, since it commutes with P^{D_{2}} an P^{E_{2}} in ℌ_{Z,2}. This implies that H̃_{Z,2}^{D_{2},E_{2}} commutes with F_{0}^{−1}𝔓F_{0}. Let ω := P/𝔓 ∈ S^{2}. We decompose the Hilbert space ℌ_{Z,2}^{D_{2},E_{2}} into the direct integral

ℌ_{Z,2}^{D_{2},E_{2}} = ∫_{0}^{∞} ⊕ℌ_{Z,2}^{D_{2},E_{2},𝔓}𝔓^{2}d𝔓.  (5.6)

The fibre space ℌ_{Z,2}^{D_{2},E_{2},𝔓} can be considered as a subspace of L_{2}(ℝ^{3N_{2}−3} × S^{2}, ℂ^{4^{N_{2}}} ) with the inner product

⟨f, g⟩_{∗} := ∫_{ℝ^{3(N_{2}−1)}×S^{2}}⟨f, g⟩_{ℂ^{4^{N_{2}}}} dy_{1} ⋯ dy_{N_{2}−1}dω.

For f ∈ ℌ_{Z,2}^{D_{2},E_{2}} the corresponding element of ℌ_{Z,2}^{D_{2},E_{2},𝔓} is given by

f_{𝔓} := F_{0}f|_{|P|=𝔓}.

We have

∥f∥^{2} = ∫_{0}^{∞} ∥f_{𝔓}∥_{∗}^{2}𝔓^{2}d𝔓  (5.7)

in compliance with (5.6). The form domain of H̃_{Z,2}^{D_{2},E_{2},𝔓} is

𝔇^{𝔓} := Λ_{Z,2}^{𝔓}P^{D_{2}}P^{E_{2}}H^{1/2}(ℝ^{3(N_{2}−1)} × S^{2}, ℂ^{4^{N_{2}}} ),

where Λ_{Z,2}^{𝔓} is given by (5.5) with the only difference that we should replace P by ω𝔓 in (5.3). The operators on fibres of the direct integral (5.6) are

H̃_{Z,2}^{D_{2},E_{2},𝔓} := Λ_{Z,2}^{𝔓}H_{Z,2}^{D_{2},E_{2},𝔓}Λ_{Z,2}^{𝔓},

where H_{Z,2}^{D_{2},E_{2},𝔓} is given by the r. h. s. of (5.4) with P replaced by ω𝔓 in (5.3). We thus have

H̃_{Z,2}^{D_{2},E_{2}} = ∫_{0}^{∞} ⊕H̃_{Z,2}^{D_{2},E_{2},𝔓}𝔓^{2}d𝔓.  (5.8)

The spectrum of H̃_{Z,2}^{D_{2},E_{2}} can be represented as

σ(H̃_{Z,2}^{D_{2},E_{2}}) = \overline{ess ⋃_{𝔓∈ℝ_{+}} σ(H̃_{Z,2}^{D_{2},E_{2},𝔓})},  (5.9)

Documenta Mathematica 13 (2008) 51–79

68 Sergey Morozov where the essential union is taken with respect to the Lebesgue measure in ℝ_{+}. The bottom of the spectrum of H̃_{Z,2}^{D_{2},E_{2},𝔓} is given by

μ(𝔓) := inf_{ψ∈𝔇^{𝔓}} \frac{⟨H̃_{Z,2}^{D_{2},E_{2},𝔓}ψ, ψ⟩_{∗}}{∥ψ∥_{∗}^{2}} .  (5.10)

Lemma 15 Function (5.10) is continuous on ℝ_{+}. Proof of Lemma 15. Let us fix 𝔓 ∈ ℝ_{+} and ε > 0. We will prove that |μ(𝔓 + 𝔭) − μ(𝔓)| < ε if |𝔭| is small enough. Choose ψ ∈ 𝔇^{𝔓} such that

|\frac{⟨H̃_{Z,2}^{D_{2},E_{2},𝔓}ψ, ψ⟩_{∗}}{∥ψ∥_{∗}^{2}} − μ(𝔓)| ⩽ \frac{ε}{2}.  (5.11)

Let

ϕ := Λ_{Z,2}^{𝔓+𝔭}ψ ∈ 𝔇^{𝔓+𝔭}.

We have

ϕ − ψ = (Λ_{Z,2}^{𝔓+𝔭} − Λ_{Z,2}^{𝔓})ψ = ∑_{k=1}^{N_{2}} ∏_{i<k} Λ_{n_{i}}^{𝔓+𝔭}(Λ_{n_{k}}^{𝔓+𝔭} − Λ_{n_{k}}^{𝔓}) ∏_{j>k} Λ_{n_{j}}^{𝔓}ψ. (5.12)

Let F be the unitary Fourier transform in L_{2}(ℝ^{3(N_{2}−1)} × S^{2}, ℂ^{4^{N_{2}}} ) defined by (Fξ)(ω, q_{1}, …, q_{N_{2}−1})

:= (2π)^{3(1−N_{2})/2} ∫_{ℝ^{3(N_{2}−1)}} ξ(ω, y_{1}, …, y_{N_{2}−1})e^{−i ∑_{k=1}^{N_{2}−1} q_{k}·y_{k}}dy_{1} ⋯ dy_{N_{2}−1}.

We can rewrite (5.12) as

ϕ − ψ = F^{−1} ∑_{k=1}^{N_{2}} ∏_{i<k} Λ̂_{n_{i}}^{𝔓+𝔭}(Λ̂_{n_{k}}^{𝔓+𝔭} − Λ̂_{n_{k}}^{𝔓}) ∏_{j>k} Λ̂_{n_{j}}^{𝔓}Fψ,  (5.13)

where Λ̂_{n}^{𝔓}, n ∈ Z_{2} are the operators of multiplication by the symbols

Λ̂_{n}^{𝔓} := \frac{1}{2} + \frac{α_{n} · p̂_{n} + β_{n}m_{n}}{2\sqrt{p̂_{n}^{2} + m_{n}^{2}}} ,  (5.14)
\begin{aligned}p̂_{n_{1}} & := \frac{m_{n_{1}}}{M} ω𝔓 − ∑_{k=1}^{N_{2}−1} q_{k}, \\ p̂_{n_{k}} & := \frac{m_{n_{k}}}{M} ω𝔓 + q_{k−1}, k = 2, …, N_{2}. \end{aligned}  (5.15)

Documenta Mathematica 13 (2008) 51–79

The matrix–functions (5.14) are uniformly continuous in 𝔓. Thus by (5.13)

∥ϕ−ψ∥_{H^{1/2}(ℝ^{3(N_{2}−1)}×S^{2},ℂ^{4^{N_{2}}} )} ⩽ C ∑_{k=1}^{N_{2}} ∥Λ̂_{n_{k}}^{𝔓+𝔭}−Λ̂_{n_{k}}^{𝔓}∥_{L_{∞}}∥ψ∥_{H^{1/2}} ⟶_{|𝔭|→0} 0. (5.16)

We write

\begin{matrix}⟨H̃_{Z,2}^{D_{2},E_{2},𝔓+𝔭}ϕ, ϕ⟩_{∗} = ⟨H̃_{Z,2}^{D_{2},E_{2},𝔓}ψ, ψ⟩_{∗} + ⟨H_{Z,2}^{D_{2},E_{2},𝔓}(ϕ − ψ), ψ⟩_{∗} \\ + ⟨H_{Z,2}^{D_{2},E_{2},𝔓}ϕ, (ϕ − ψ)⟩_{∗} + ⟨(H̃_{Z,2}^{D_{2},E_{2},𝔓+𝔭} − H_{Z,2}^{D_{2},E_{2},𝔓})ϕ, ϕ⟩_{∗}. \end{matrix}  (5.17)

The second and third terms on the r. h. s. of (5.17) tend to zero as |𝔭| → 0 according to (5.16) and (2.8). The last term also tends to zero for small |𝔭|, since the symbol of the difference is

F(H̃_{Z,2}^{D_{2},E_{2},𝔓+𝔭} − H_{Z,2}^{D_{2},E_{2},𝔓})F^{−1} = ∑_{n∈Z_{2}} \frac{m_{n}}{M} α_{n} · ω𝔭.

From (5.16) and (5.17) follows that

|\frac{⟨H̃_{Z,2}^{D_{2},E_{2},𝔓}ψ, ψ⟩_{∗}}{∥ψ∥_{∗}^{2}} − \frac{⟨H̃_{Z,2}^{D_{2},E_{2},𝔓+𝔭}ϕ, ϕ⟩_{∗}}{∥ϕ∥_{∗}^{2}} | ⩽ \frac{ε}{2},  (5.18)

if |𝔭| is small enough. Hence by (5.11) and (5.18) for any ε > 0

|μ(𝔓 + 𝔭) − μ(𝔓)| < ε

for |𝔭| small enough. • Now we prove that μ is semibounded from below and tends to infinity as |𝔓| → ∞. This, together with (5.9) and Lemma 15, implies that the spectrum of H̃_{Z,2}^{D_{2},E_{2}} is purely essential and is concentrated on a semi–axis. Proposition 14 will be thus proved. According to (2.9) for j = 2 and (2.10) we have

⟨H̃_{Z,2}^{D_{2},E_{2}}ψ, ψ⟩ ⩾ C_{1}⟨ ∑_{n∈Z_{2}} \sqrt{−Δ_{n} + m_{n}^{2}}ψ, ψ⟩ − C_{2}∥ψ∥^{2},

for any ψ ∈ P^{D}P^{E} ⊗_{n∈Z_{2}} Λ_{n}H^{1/2}(ℝ^{3}, ℂ^{4}).

(5.19)

Since all the operators corresponding to the quadratic forms involved in (5.19) commute with F_{0}^{−1}𝔓F_{0}, it follows from (5.8) that for almost all 𝔓 the inequality

⟨H̃_{Z,2}^{D_{2},E_{2},𝔓}ψ, ψ⟩_{∗} ⩾ C_{1}⟨ ∑_{n∈Z_{2}} \sqrt{p̂_{n}^{2} + m_{n}^{2}}Fψ, Fψ⟩_{∗} − C_{2}∥ψ∥_{∗}^{2}  (5.20)

holds for every ψ ∈ 𝔇^{𝔓}, where p̂_{n} are defined in (5.15). Thus μ is semibounded from below. Since by (5.15)

𝔓 = | ∑_{n∈Z_{2}} p̂_{n}|,

there exists n ∈ Z_{2} such that

|p̂_{n}| ⩾ \frac{𝔓}{N_{2}}

and hence

∑_{n∈Z_{2}} \sqrt{p̂_{n}^{2} + m_{n}^{2}} ⩾ \frac{𝔓}{N_{2}} .

Thus the r. h. s. of (5.20) tends to infinity as 𝔓 → ∞. •

6 Absence of Gaps

We are now ready to finish the proof of Theorem 6 by proving that

[κ(D, E), ∞) ⊆ σ(H_{N}^{D,E}).  (6.1)

Let us first fix a decomposition Z on which the minimum is attained in (2.20). Following the general strategy of [14], we will prove that for any irreducible representations (D_{1}, E_{1}; D_{2}, E_{2}) ≺_{Z} (D, E) any

λ ⩾ κ_{1}(Z, D_{1}, E_{1}) + κ_{2}(Z, D_{2}, E_{2})

belongs to σ(H_{N}^{D,E}). This will imply (6.1) according to the definition (2.19). Let

λ_{1} := λ − κ_{1}(Z, D_{1}, E_{1}) ⩾ κ_{2}(Z, D_{2}, E_{2}).  (6.2)

We will use the notation and results of Section 5. The following lemma is a slight modification of Theorem 8.11 of [14] and is proved along the same lines:

Lemma 16 Let A be a selfadjoint operator in a Hilbert space ℌ and U(γ) be a continuous representation of a compact group Γ by unitary operators in ℌ such that U(γ) Dom A ⊂ Dom A and U(γ)A = AU(γ) for any γ ∈ Γ. Then for any irreducible (matrix) representation D of Γ the corresponding subspace P^{D}ℌ reduces A. For every λ ∈ σ(A^{D}) where A^{D} is the reduced operator and every ε > 0 there exists a D–generating subspace G of Dom A such that

∥Au − λu∥ ⩽ ε∥u∥, for all u ∈ G.

Remark 17 Recall that a subspace G of ℌ is called D–generating if the operator U(γ)|G is unitary in G for all γ ∈ Γ and there exists an orthonormal base in G such that for every γ ∈ Γ the operator U(γ)|G is represented by the matrix D(γ).

Proof of Lemma 16. Let r be the dimension of the representation D : γ ↦ (D_{lk}(γ))_{l,k=1}^{r}. Let us introduce in ℌ the bounded operators P_{lk} by

P_{lk} := r ∫_{Γ} \overline{D_{lk}(γ)}U(γ)dμ(γ), l, k = 1, …, r, where μ is the invariant probability measure on Γ. It is shown in the proof of Theorem 8.11 of [14] that P_{ll} are orthogonal projections onto mutually orthogonal subspaces of ℌ and that

P^{D} = ∑_{l=1}^{r} P_{ll}.  (6.3)

In fact, P_{ll} is the projection on the subspace of function which belong to the l^{th} row of the representation D. Moreover, P_{lk} is a partial isometry between P_{kk}ℌ and P_{ll}ℌ. Since λ ∈ σ(A^{D}), there exists a vector u_{0} ∈ Dom A^{D} such that

∥A^{D}u_{0} − λu_{0}∥ ⩽ ε∥u_{0}∥.

It follows from (6.3) that there exists l ∈ {1, …, r} such that ∥P_{ll}u_{0}∥ ⩾ r^{−1}. We can thus define u_{l} := P_{ll}u_{0}/∥P_{ll}u_{0}∥ and then u_{k} := P_{kl}u_{l} for k = 1, …, r. The subspace G spanned by {u_{k}}_{k=1}^{r} satisfies the statement of the lemma. • Let

r_{j} := dim(D_{j} ⊗ E_{j}), j = 1, 2.  (6.4)

Since κ_{1}(Z, D_{1}, E_{1}) belongs to the spectrum of H̃_{Z,1}^{D_{1},E_{1}} (see definition (2.18)), by Lemma 16 we can choose a sequence of (D_{1} ⊗ E_{1})–generating subspaces {G_{q}}_{q=1}^{∞} of Dom(H̃_{Z,1}^{D_{1},E_{1}}) such that for all q ∈ ℕ

∥H̃_{Z,1}^{D_{1},E_{1}}ϕ_{q} − κ_{1}(Z, D_{1}, E_{1})ϕ_{q}∥_{ℌ_{Z,1}} ⩽ q^{−1}∥ϕ_{q}∥_{ℌ_{Z,1}}, for all ϕ_{q} ∈ G_{q}. (6.5)

Analogously, for any 𝔓 ⩾ 0 we can find a sequence {G_{q}^{𝔓}}_{q=1}^{∞} of (D_{2} ⊗ E_{2})– generating subspaces of Dom H̃_{Z,2}^{D_{2},E_{2},𝔓} such that

∥H̃_{Z,2}^{D_{2},E_{2},𝔓}ψ_{q}^{𝔓} − μ(𝔓)ψ_{q}^{𝔓}∥_{∗} ⩽ q^{−1}∥ψ_{q}^{𝔓}∥_{∗}, for all ψ_{q}^{𝔓} ∈ G_{q}^{𝔓}.  (6.6)

Moreover, we can choose an orthonormal basis {ψ_{q,l}^{𝔓}}_{l=1}^{r_{2}} in G_{q}^{𝔓} in such a way that for every q ∈ ℕ and l = 1, …, r_{2} ψ_{q,l} belongs to the l^{th} row of the representation (D_{2} ⊗ E_{2}) and satisfies (6.6). By Proposition 14, Lemma 15, and (6.2) we can choose 𝔓_{0} in such a way that

μ(𝔓_{0}) = λ_{1}.  (6.7)

We choose R_{q} > q so that (2.12) and (2.13) hold true for all n, j = 1, …, N, n < j with

ε := q^{−1}(N_{1} + 1)^{−1}N_{2}^{−1/2}C_{1}^{1/2}(C_{2} + |λ_{1}| + 2)^{−1/2},  (6.8)

where N_{1,2} are the numbers of elements in Z_{1,2}, and C_{1,2} are the constants in (2.9) for j = 2, and so that for some orthonormal base {ϕ_{q,k}}_{k=1}^{r_{1}} of G_{q}

∥(1 − ∏_{j∈Z_{1}} I_{\{|x_{j}|<R_{q}\}})ϕ_{q,k}∥_{L_{2}(ℝ^{3N_{1}} ,ℂ^{4^{N_{1}}} )} ⩽ \frac{ν_{0}}{4d_{E}^{2}r_{1}r_{2}} ,  (6.9)

where d_{E} is the dimension of E, r_{1,2} are defined in (6.4), and the constant ν_{0} > 0 depending only on E, E_{1}, E_{2} will be specified later in the proof of Lemma 21. By Assumption 4 and Lemma 15, we can choose a sequence of positive numbers {δ_{q}}_{q=1}^{∞} tending to zero in such a way that

|μ(𝔓) − λ_{1}| ⩽ q^{−1} for all 𝔓 ∈ [𝔓_{0}, 𝔓_{0} + δ_{q}],  (6.10)

\frac{1}{2π^{2}} (𝔓_{0} + δ_{q})^{2}δ_{q}C_{R_{q}} < q^{−2},  (6.11)

where C_{R_{q}} is the constant in (2.11), and

\frac{1}{2π^{2}} (𝔓_{0} + δ_{q})^{2}δ_{q} · \frac{4}{3}πR_{q}^{3} < \frac{ν_{0}^{2}}{16d_{E}^{4}r_{1}^{2}r_{2}^{2}} .  (6.12)

Let us choose a function f_{q} ∈ L_{2}(ℝ_{+}) with supp f_{q} ⊂ [𝔓_{0}, 𝔓_{0} + δ_{q}] so that

∫_{𝔓_{0}}^{𝔓_{0}+δ_{q}} |f_{q}(𝔓)|^{2}𝔓^{2}d𝔓 = 1.  (6.13)

Let

\begin{matrix}ψ_{q,l}(y_{0}, …, y_{N_{2}−1}) \\ := \frac{1}{(2π)^{\frac{3}{2}}} ∫_{𝔓_{0}}^{𝔓_{0}+δ_{q}}∫_{S^{2}} e^{i𝔓ωy_{0}}f_{q}(𝔓)ψ_{q,l}^{𝔓}(ω, y_{1}, …, y_{N_{2}−1})𝔓^{2}dωd𝔓, \end{matrix}  (6.14)

where {y_{0}, …, y_{N_{2}−1}} and {x_{n}}_{n∈Z_{2}} are related by (5.1) and (5.2). It follows from (6.13) and the choice of ψ_{q,l}^{𝔓} that

∥ψ_{q,l}∥_{ℌ_{Z,2}} = 1, l = 1, …, N_{2},  (6.15)

and that ψ_{q,l} belongs to the l^{th} row of (D_{2} ⊗ E_{2}). Clearly the linear subspace G̃_{q} spanned by {ψ_{q,l}}_{l=1}^{r_{2}} is a (D_{2} ⊗ E_{2})–generating subspace of Dom H̃_{Z,2}^{D_{2},E_{2}}.

Lemma 18 For any n ∈ Z_{2} and ψ ∈ G̃_{q} with ∥ψ∥ = 1 the one–particle density

ρ_{ψ,n}(x_{n}) := ∫_{ℝ^{3N_{2}−3}} |ψ(x_{n_{1}}, …, x_{n_{N_{2}}} )|^{2}(dx_{n_{1}} ⋯ dx_{n_{N_{2}}} )/dx_{n}

satisfies

∥ρ_{ψ,n}∥_{L_{∞}(ℝ^{3})} ⩽ \frac{1}{2π^{2}} (𝔓_{0} + δ_{q})^{2}δ_{q}.

Proof. By (6.14)

∥ρ_{ψ,n}∥_{L_{∞}(ℝ^{3})} ⩽ (2π)^{−3/2}∥ρ̂_{ψ,n}∥_{L_{1}(ℝ^{3})}

= \frac{1}{(2π)^{6}} ∫_{ℝ^{3}} | ∫_{ℝ^{3N_{2}}} ∫_{𝔓_{0}}^{𝔓_{0}+δ_{q}} ∫_{S^{2}} ∫_{𝔓_{0}}^{𝔓_{0}+δ_{q}} ∫_{S^{2}} e^{−ip(y_{0}+r_{n})}e^{−i𝔓ωy_{0}} \overline{f_{q}(𝔓)}

× ψ_{q}^{𝔓∗}(ω, y_{1}, …, y_{N_{2}−1})e^{i𝔓̃ω̃y_{0}}f_{q}(𝔓̃)ψ𝔓̃ ; q(ω̃, y_{1}, …, y_{N_{2}−1})𝔓^{2}𝔓̃^{2}

× dω̃ d𝔓̃ dω d𝔓 dy_{0} dy_{1} ⋯ dy_{N_{2}−1}|dp,

(6.16)

where r_{n} := x_{n} − y_{0}, see (5.2). Integrating the r. h. s. of (6.16) in y_{0} we obtain (2π)^{3}δ(p+𝔓ω−𝔓̃ω̃) from all the factors involving y_{0}. Estimating the absolute value of the integral by the integral of absolute value and taking into account that ∫ δ(p + …)dp = 1 we get

\begin{matrix}∥ρ_{ψ,n}∥_{L_{∞}(ℝ^{3})} ⩽ \frac{1}{(2π)^{3}} ∫_{ℝ^{3N_{2}−3}} ∫_{𝔓_{0}}^{𝔓_{0}+δ_{q}} ∫_{S^{2}} ∫_{𝔓_{0}}^{𝔓_{0}+δ_{q}} ∫_{S^{2}} |f_{q}(𝔓)||f_{q}(𝔓̃)| \\ × |ψ_{q}^{𝔓}(ω, y_{1}, …, y_{N_{2}−1})||ψ𝔓̃ ; q(ω̃, y_{1}, …, y_{N_{2}−1})|𝔓^{2}𝔓̃^{2} \\ × dω̃ d𝔓̃ dω d𝔓 dy_{1} ⋯ dy_{N_{2}−1} ⩽ \frac{1}{(2π)^{3}} 4π(𝔓_{0} + δ_{q})^{2}δ_{q}, \end{matrix} (6.17)

where at the last step we have used Schwarz inequality and ∥ψ∥ = 1. The formal calculation (6.16) — (6.17) is justified by the fact that the integral over ℝ^{3N_{2}} can be considered as a limit of integrals over expanding finite volumes, since ψ ∈ L_{2}(ℝ^{3N_{2}}). •

Corollary 19 For any W ∈ L_{2}(ℝ^{3}), n ∈ Z_{2}, and ψ ∈ G̃_{q} with ∥ψ∥ = 1 we have

∫_{ℝ^{3N_{2}}} |W (x_{n})ψ(x_{n_{1}}, …, x_{n_{N_{2}}} )|^{2}dx_{n_{1}} ⋯ dx_{n_{N_{2}}} ⩽ \frac{1}{2π^{2}} (𝔓_{0} + δ_{q})^{2}δ_{q}∥W ∥^{2}.

Let F_{q} be the subspace of ℌ_{N} spanned by the functions

\begin{matrix}φ_{q,k,l}(x_{1}, …, x_{N}) := ϕ_{q,k}(x_{j} : j ∈ Z_{1}) ⊗ ψ_{q,l}(x_{n} : n ∈ Z_{2}), \\ k = 1, …, r_{1}, l = 1, …, r_{2}, \end{matrix}  (6.18)

where {ϕ_{q,k}}_{k=1}^{r_{1}} and {ψ_{q,l}}_{l=1}^{r_{2}} are orthonormal bases of G_{q} and G̃_{q}, respectively. We obviously have ∥φ_{q,k,l}∥_{L_{2}(ℝ^{3N} ,ℂ^{4^{N}})} = 1.

Lemma 20 For any q ∈ ℕ F_{q} ⊂ Dom H_{N}. For any φ ∈ F_{q}

∥(H_{N} − λ)φ∥ ⩽ 5q^{−1}r_{1}^{1/2}r_{2}^{1/2}∥φ∥.

Proof. It is enough to show that the functions (6.18) belong to Dom H_{N} and satisfy

∥(H_{N} − λ)φ_{q,k,l}∥_{L_{2}(ℝ^{3N} ,ℂ^{4^{N}})} ⩽ 5q^{−1}.  (6.19)

Indeed, by triangle and Cauchy inequalities for

φ = ∑_{k=1}^{r_{1}} ∑_{l=1}^{r_{2}} c_{kl}φ_{q,k,l}  (6.20)

we have

\begin{aligned}∥(H_{N} − λ)φ∥ & ⩽ ∑_{k=1}^{r_{1}} ∑_{l=1}^{r_{2}} |c_{kl}|∥(H_{N} − λ)φ_{q,k,l}∥ \\ & ⩽ sup_{k,l}∥(H_{N} − λ)φ_{q,k,l}∥r_{1}^{1/2}r_{2}^{1/2}∥φ∥. \end{aligned}

The operator domain of H_{N} can be characterized as the set of functions

ξ from the form domain ⊗^{N}_{n=1}Λ_{n}H^{1/2}(ℝ^{3}, ℂ^{4}) on which the sesquilinear form

⟨H_{N}ξ, ·⟩ is a bounded linear functional in ℌ_{N}. Functions (6.18) belong to

⊗^{N}_{n=1}Λ_{n}H^{1/2}(ℝ^{3}, ℂ^{4}) by construction. By (2.2), (2.5), and (2.6) we have

H_{N} = H_{Z,1} + H_{Z,2} + Λ^{N}( ∑_{n∈Z_{2}} V_{n} + ∑_{\begin{matrix}n<j \\ n#j \end{matrix}} U_{nj})Λ^{N}.  (6.21)

The sesquilinear forms ⟨(H_{Z,1} + H_{Z,2})φ_{q,k,l}, ·⟩ are bounded linear functionals over L_{2}(ℝ^{3N}, ℂ^{4^{N}}), since ϕ_{q,k} ∈ Dom(H̃_{Z,1}^{D_{1},E_{1}}) and ψ_{q,l} ∈ Dom H̃_{Z,2}^{D_{2},E_{2}}. Moreover, by (6.5)

∥(H_{Z,1} − κ_{1}(Z, D_{1}, E_{1}))φ_{q,k,l}∥ = ∥(H̃_{Z,1}^{D_{1},E_{1}} − κ_{1}(Z, D_{1}, E_{1}))ϕ_{q,k}∥ ⩽ q^{−1},

and by (6.6), (6.7), (6.10), (6.14), and (6.15)

∥(H_{Z,2} − λ_{1})φ_{q,k,l}∥ = ∥(H̃_{Z,2}^{D_{2},E_{2}} − λ_{1})ψ_{q,l}∥ ⩽ 2q^{−1}.  (6.22)

In view of (6.21)—(6.22) and (6.2), to prove that φ_{q,k,l} ∈ Dom H_{N} and that (6.19) holds true it is enough to obtain that

∥( ∑_{n∈Z_{2}} V_{n} + ∑_{\begin{matrix}n<j \\ n#j \end{matrix}} U_{nj})φ_{q,k,l}∥ ⩽ 2q^{−1}.  (6.23)

To do this, we first note that by (2.12), (2.13), and Cauchy inequality

\begin{matrix}∥( ∑_{n∈Z_{2}} V_{n}I_{\{|x_{n}|>R_{q}\}} + ∑_{\begin{matrix}n<j \\ n#j \end{matrix}} U_{nj}I_{\{|x_{n}−x_{j}|>R_{q}\}})φ_{q,k,l}∥ \\ ⩽ ε(N_{1} + 1) ∑_{n∈Z_{2}} ∥|D_{n}|^{\frac{1}{2}} ψ_{q,l}∥ ⩽ ε(N_{1} + 1)N2^{\frac{1}{2}} ( ∑_{n∈Z_{2}} ∥|D_{n}|^{\frac{1}{2}} ψ_{q,l}∥^{2})^{\frac{1}{2}} . \end{matrix} ; (6.24)

By (2.9), (6.15), and (6.22),

\begin{aligned}∑_{n∈Z_{2}} ∥D_{n}^{1/2}ψ_{q,l}∥^{2} & ⩽ C_{1}^{−1}(∥(H̃_{Z,2}^{D_{2},E_{2}} − λ_{1})ψ_{q,l}∥ + C_{2} + |λ_{1}|) \\ & ⩽ C_{1}^{−1}(C_{2} + |λ_{1}| + 2q^{−1}). \end{aligned}  (6.25)

Thus by (6.24), (6.25) and (6.8) for q ⩾ 1 we obtain

∥( ∑_{n∈Z_{2}} V_{n}I_{\{|x_{n}|>R_{q}\}} + ∑_{\begin{matrix}n<j \\ n#j \end{matrix}} U_{nj}I_{\{|x_{n}−x_{j}|>R_{q}\}})φ_{q,k,l}∥ ⩽ q^{−1}.  (6.26)

Now the scalar functions

V_{n,q}(x) := |V_{n}(x)|I_{{|x|⩽R_{q}}}(x) and U_{nj,q}(x) := |U_{nj}(x)|I_{{|x|⩽R_{q}}}(x)

(6.27)

are square integrable by (2.11). By Corollary 19, for n ∈ Z_{2}

∥V_{n,q}φ_{q,k,l}∥^{2} = ∥V_{n,q}ψ_{q,l}∥^{2} ⩽ \frac{1}{2π^{2}} δ_{q}(𝔓_{0} + δ_{q})^{2}∥V_{n,q}∥_{L_{2}(ℝ^{3})}^{2}  (6.28)

and for n < j, n#j

∥U_{nj,q}φ_{q,k,l}∥^{2} ⩽ sup_{z∈ℝ^{3}}∥U_{nj,q}(· − z)ψ_{q,l}∥^{2} ⩽ \frac{1}{2π^{2}} δ_{q}(𝔓_{0} + δ_{q})^{2}∥U_{nj,q}∥_{L_{2}(ℝ^{3})}^{2}.

(6.29)

Hence by (6.27), (6.28), (6.29), (2.11), and (6.11)

∥( ∑_{n∈Z_{2}} V_{n}I_{\{|x_{n}|⩽R_{q}\}} + ∑_{\begin{matrix}n<j \\ n#j \end{matrix}} U_{nj}I_{\{|x_{n}−x_{j}|⩽R_{q}\}})φ_{q,k,l}∥ ⩽ q^{−1}.  (6.30)

It remains to add (6.26) and (6.30) to obtain (6.23), finishing the proof of the lemma. • The subspace F_{q} spanned by the functions (6.18) is D_{1} ⊗ E_{1} ⊗ D_{2} ⊗ E_{2}– generating. Since (D_{1}, E_{1}; D_{2}, E_{2}) ≺_{Z} (D, E), F_{q} contains some nontrivial D–

generating subspace. Hence the subspace K_{q} := P^{D}F_{q} is not equal to {0} and is contained in F_{q}.

Lemma 21 There exists a constant C_{E} > 0 such that for every q ∈ ℕ

∥P^{E}φ∥ ⩾ C_{E}∥φ∥, for all φ ∈ F_{q}.  (6.31)

Proof. Projector (2.15) can be written as

P^{E} = \frac{d_{E}}{h_{Π}} ∑_{π∈Π_{1}^{Z}×Π_{2}^{Z}} \overline{ξ_{E}(π)}π + \frac{d_{E}}{h_{Π}} ∑_{π∈Π\(Π_{1}^{Z}×Π_{2}^{Z})} \overline{ξ_{E}(π)}π.  (6.32)

We will denote the first term in (6.32) by Q^{E}, and the second by R^{E}. Then

∥P^{E}φ∥^{2} = ⟨φ, P^{E}φ⟩ = ⟨φ, Q^{E}φ⟩ + ⟨φ, R^{E}φ⟩.  (6.33)

Relation (D_{1}, E_{1}; D_{2}, E_{2}) ≺_{Z} (D, E) implies that the representation E|Π_{1}^{Z} × Π_{2}^{Z}

is unitarily equivalent to a sum ⊕^{k}_{i=0}n_{i}E^{(i)}, where n_{i} > 0 are multiplicities of the

irreducible representations E^{(i)} of the group Π_{1}^{Z} × Π_{2}^{Z} with E^{(0)} = E_{1} ⊗ E_{2}. For the corresponding characters this gives

ξ_{E}(π) = ∑_{i=0}^{k} n_{i}ξ^{(i)}(π), for all π ∈ Π_{1}^{Z} × Π_{2}^{Z}.

Hence

Q^{E} = ∑_{i=0}^{k} ν_{i}P_{i},

where ν_{i} > 0 and P_{i} is the projector corresponding to the representation E^{(i)}. By construction, P_{0}φ = φ for any φ ∈ F_{q}, hence P_{i}φ = 0 for i = 1, …, k. Thus for any φ ∈ F_{q}

⟨φ, Q^{E}φ⟩ = ν_{0}∥φ∥^{2}, ν_{0} > 0.  (6.34)

We will now estimate the second term on the r. h. s. of (6.33). For any n ∈ Z_{2} and any ψ ∈ G̃_{q} with ∥ψ∥ = 1 by Corollary 19 and (6.12) we have

∥I_{\{|x_{j}|<R_{q}\}}ψ∥^{2} ⩽ \frac{ν_{0}^{2}}{16d_{E}^{4}r_{1}^{2}r_{2}^{2}} .  (6.35)

For any functions (6.18) and any π ∈ Π inequality (6.9) implies that

|⟨φ_{q,k,l}, πφ_{q,k̃,l̃}⟩| ⩽ ⟨ ∏_{j∈Z_{1}} I_{\{|x_{j}|<R_{q}\}}|φ_{q,k,l}|, π|φ_{q,k̃,l̃}|⟩_{L_{2}(ℝ^{3N})} + \frac{ν_{0}}{4d_{E}^{2}r_{1}r_{2}} .

Now if π ∈ Π \ (Π_{1}^{Z} × Π_{2}^{Z}), then there exists j_{0} ∈ Z_{1} such that πj_{0} ∈ Z_{2}. Hence by (6.35)

⟨ ∏_{j∈Z_{1}} I_{\{|x_{j}|<R_{q}\}}|φ_{q,k,l}|, π|φ_{q,k̃,l̃}|⟩ ⩽ ⟨|φ_{q,k,l}|, I_{\{|x_{j_{0}} |<R_{q}\}}π|φ_{q,k̃,l̃}|⟩ ⩽ \frac{ν_{0}}{4d_{E}^{2}r_{1}r_{2}} .

Thus

|⟨φ_{q,k,l}, πφ_{q,k̃,l̃}⟩| ⩽ \frac{ν_{0}}{2d_{E}^{2}r_{1}r_{2}} , π ∈ Π \ (Π_{1}^{Z} × Π_{2}^{Z}).  (6.36)

Any φ ∈ F_{q} can be written as (6.20). By (6.36) and Cauchy inequality for any π ∈ Π \ (Π_{1}^{Z} × Π_{2}^{Z})

|⟨φ, πφ⟩| ⩽ ∑_{k,l,k̃,l̃}|c_{kl}||c_{k̃l̃}||⟨φ_{q,k,l}, πφ_{q,k̃,l̃}⟩| ⩽ \frac{ν_{0}}{2d_{E}^{2}} ∥φ∥^{2}.  (6.37)

Since the number of elements of Π \ (Π_{1}^{Z} × Π_{2}^{Z}) does not exceed d_{Π} and for any π |ξ_{E}(π)| ⩽ d_{E} as a trace of unitary matrix of dimension d_{E}, (6.37) implies that

|⟨φ, R^{E}φ⟩| ⩽ ν_{0}∥φ∥^{2}/2.

By (6.33) and (6.34) we conclude that (6.31) holds with C_{E} = \sqrt{ν_{0}/2}. • Lemmata 20 and 21 imply that L_{q} := P^{E}K_{q} is a nontrivial subspace of Dom H_{N}^{D,E} and for every f = P^{E}φ ∈ L_{q}

∥(H_{N}^{D,E} − λ)f∥ ⩽ ∥(H_{N} − λ)φ∥ ⩽ 5q^{−1}r1^{\frac{1}{2}} r2^{\frac{1}{2}} ∥φ∥ ⩽ 5q^{−1}r1^{\frac{1}{2}} r2^{\frac{1}{2}} C_{E}^{−1}∥f∥, q ∈ ℕ.

This implies that λ ∈ σ(H_{N}^{D,E}), and thus finishes the proof of Theorem 6.

Acknowledgement. The author was supported by the DFG grant SI 348/12–2. Part of this work was done during the stay at the Erwin Schrödinger Institute, Vienna.

References

[1] Robert A. Adams. Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65.

[2] A. A. Balinsky and W. D. Evans. Stability of one-electron molecules in the Brown-Ravenhall model. Comm. Math. Phys., 202(2):481–500, 1999.

[3] Hans A. Bethe and Edwin E. Salpeter. Quantum mechanics of one- and two-electron atoms. Springer-Verlag, Berlin, 1957.

[4] G. E. Brown and D. G. Ravenhall. On the interaction of two electrons. Proc. Roy. Soc. London Ser. A., 208:552–559, 1951.

[5] V. I. Burenkov and W. D. Evans. On the evaluation of the norm of an integral operator associated with the stability of one-electron atoms. Proc. Roy. Soc. Edinburgh Sect. A, 128(5):993–1005, 1998.

[6] Roch Cassanas and Heinz Siedentop. The ground-state energy of heavy atoms according to Brown and Ravenhall: absence of relativistic effects in leading order. J. Phys. A, 39(33):10405–10414, 2006.

[7] William Desmond Evans, Peter Perry, and Heinz Siedentop. The spectrum of relativistic one-electron atoms according to Bethe and Salpeter. Comm. Math. Phys., 178(3):733–746, 1996.

[8] Marcel Griesemer, Roger T. Lewis, and Heinz Siedentop. A minimax principle for eigenvalues in spectral gaps: Dirac operators with Coulomb potentials. Doc. Math., 4:275–283 (electronic), 1999.

[9] Ira W. Herbst. Spectral theory of the operator (p^{2} + m^{2})^{1/2} − Ze^{2}/r. Comm. Math. Phys., 53(3):285–294, 1977.

[10] Georg Hoever and Heinz Siedentop. Stability of the Brown-Ravenhall operator. Math. Phys. Electron. J., 5:Paper 6, 11 pp. (electronic), 1999.

[11] D. H. Jakubaßa–Amundsen. Preprint: The HVZ theorem for the brown–ravenhall operator with constant magentic field. 2007. http://arxiv.org/abs/0709.1034, to appear in Journal of Mathematical Physics.

[12] D. H. Jakubassa-Amundsen. Localization of the essential spectrum for relativistic N-electron ions and atoms. Doc. Math., 10:417–445 (electronic), 2005.

[13] Doris H. Jakubaßa-Amundsen. The HVZ theorem for a pseudo-relativistic operator. Ann. Henri Poincaré, 8(2):337–360, 2007.

[14] Konrad Jörgens and Joachim Weidmann. Spectral properties of Hamiltonian operators. Springer-Verlag, Berlin, 1973. Lecture Notes in Mathematics, Vol. 313.

[15] Roger T. Lewis, Heinz Siedentop, and Simeon Vugalter. The essential spectrum of relativistic multi-particle operators. Ann. Inst. H. Poincaré Phys. Théor., 67(1):1–28, 1997.

[16] Sergey Morozov and Semjon Vugalter. Stability of atoms in the Brown-Ravenhall model. Ann. Henri Poincaré, 7(4):661–687, 2006.

[17] Barry Simon. Geometric methods in multiparticle quantum systems. Comm. Math. Phys., 55(3):259–274, 1977.

[18] J. Sucher. Foundations of the relativistic theory of many-electron atoms. Phys. Rev. A (3), 22(2):348–362, 1980.

[19] J. Sucher. Relativistic many-electron hamiltonians. Phys. Scripta, 36:271– 281, 1987.

[20] Bernd Thaller. The Dirac equation. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1992.

[21] C. Tix. Lower bound for the ground state energy of the no-pair Hamiltonian. Phys. Lett. B, 405(3-4):293–296, 1997.

[22] C. Tix. Strict positivity of a relativistic Hamiltonian due to Brown and Ravenhall. Bull. London Math. Soc., 30(3):283–290, 1998.

[23] Semjon Vugalter and Timo Weidl. On the discrete spectrum of a pseudorelativistic two-body pair operator. Ann. Henri Poincaré, 4(2):301–341, 2003.

Sergey Morozov Mathematisches Institut LMU München Theresienstr. 39 80333 Munich, Germany morozov@mathematik.uni-muenchen.de 80

Documenta Mathematica 13 (2008)