Hilbert C*-modules and spectral analysis of many-body systems

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**Hilbert C*-modules and spectral analysis of many-body systems**

Mondher Damak, Vladimir Georgescu

To cite this version:

Mondher Damak, Vladimir Georgescu. Hilbert C*-modules and spectral analysis of many-body systems. 2008. �hal-00285199�

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Hilbert C

^∗

-modules and spectral analysis of many-body systems

Mondher DAMAK^∗ and Vladimir GEORGESCU^†

June 4, 2008

Abstract

We study the spectral properties of a class of many channel Hamiltonians which contains those of systems of particles interacting throughk-body and field type forces which do not preserve the number of particles. Our results concern the essential spectrum, the Mourre estimate, and the absence of singular continuous spectrum. The appropriate formalism involves gradedC^∗-algebras and HilbertC^∗-modules as basic tools.

1 Introduction and main results

In this section, after some general comments on the algebraic approach that we shall use, we describe our main results in a slightly simplified form. For notations and terminology, see Subsections 2.1, 3.1 and 5.1 1.1 An algebraic approach

By many-body systemwe mean a system of particles interacting between themselves through k-body forces with arbitraryk ≥ 1but also subject to interactions which allow the system to make transitions between states with different numbers of particles. The second type of interactions consists of creation- annihilation processes as in quantum field theory so we call them field type interactions.

We use the terminologyN-body systemin a rather loose sense. Strictly speaking this should be a system of N particles which may interact throughk-body forces with1 ≤ k ≤ N. However we also speak ofN-body system when we consider the following natural abstract version: the configuration space of the system is a locally compact abelian groupX, so the momentum space is the dual groupX^∗, and the

“elementary Hamiltonians” (cf. below) are of the formh(P) +P

Y vY(Q). Herehis a real function on X^∗, theY are closed subgroups ofX, andvY ∈ Co(X/Y). One can give a meaning to the numberN even in this abstract setting, but this is irrelevant here.

Similarly, we shall give a more general meaning to the notion of many-body system: these are systems obtained by coupling a certain number (possibly infinite) ofN-body systems. Our framework is abstract and allows one to treat quite general examples which, even if they do not have an immediate physical meaning, are interesting because they furnish Hamiltonians with a rich many channel structure. Note that here and below we do not use the word “channel” in the scattering theory sense, speaking about “phase structure” could be more appropriate.

The Hamiltonians we want to analyze are rather complicated objects and standard Hilbert space techniques seem to us inefficient in this situation. Instead, we shall adopt a strategy proposed in [GI1, GI2]

based on the observation that often the C^∗-algebra generated^† by the Hamiltonians we want to study (we call themadmissible) has a quite simple and remarkable structure which allows one to describe its quotient with respect to the ideal of compact operators in more or less explicit terms. And this suffices to get the qualitative spectral properties which are of interest to us. We shall refer to thisC^∗-algebra as the Hamiltonian algebra(orC^∗-algebra of Hamiltonians) of the system.

To clarify this we consider the case of N-body systems [DaG1]. Let X be a finite dimensional real vector space (the configuration space). LetT be a set of subspaces ofX. In the non-relativistic case an Euclidean structure is given onXand the simplest Hamiltonians are of the form

H = ∆ + X

Y∈T

vY(πY(x)) (1.1)

where∆is the Laplace operator,vY is a continuous function with compact support on the quotient space X/Y, andπY : X → X/Y is the canonical surjection (only a finite number ofvY is not zero). Such Hamiltonians should clearly be admissible. On the other hand, if a Hamiltonianh(P) +V is considered as admissible thenh(P+k) +V should be admissible too because the zero momentumk = 0should not play a special role. In other terms, translations in momentum space should leave invariant the set of admissible Hamiltonians. We shall now describe the smallestC^∗-algebraC_X(S)such that the operators

†A self-adjoint operatorHon a Hilbert spaceHis affiliated to aC^∗-algebraCof operators onHif(H+i)⁻¹∈C. IfEis a set of self-adjoint operators, the smallestC^∗-algebra such that allH∈Eare affiliated to it is theC^∗-algebra generated byE.

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(1.1) are affiliated to it and which is stable under translations in momentum space. LetS be the set of finite intersections of subspaces fromT and

CX(S) =Pc

Y∈SCo(X/Y)≡ norm closure of P

Y∈SCo(X/Y).

Note that one may think ofCX(S)as aC^∗-algebra of multiplication operators onL²(X). LetC^∗(X)be the groupC^∗-algebra ofX(see§3.1). Then Corollary A.4 gives:

C_X(S) =CX(S)·C^∗(X)≡closed linear subspace generated by theST withS∈ CX(S), T ∈C^∗(X).

It turns out that this algebra is canonically isomorphic with the crossed productCX(S)⋊X. This example illustrates our point: the Hamiltonian algebra of anN-body system is a remarkable mathematical object. Moreover,CX(S)contains the ideal of compact operators and its quotient with respect to it can be computed by using general techniques from the theory of crossed products [GI1]. On the other hand, CX(S)is equipped with anS-gradedC^∗-algebra structure [BG1, Ma1, Ma2] and this gives a method of computing the quotient which is more convenient in the framework of the present paper.

The main difficulty in this algebraic approach is to isolate the correctC^∗-algebra. Of course, we could accept an a priori givenC as C^∗-algebra of energy observables but we stress that a correct choice is of fundamental importance: if the algebraC we start with is too large, then its quotient with respect to the compacts will probably be too complicated to be useful. On the other hand, if it is too small then physically relevant Hamiltonians will not be affiliated to it. We refer to [GI1, GI2, GI4, Geo] for examples of Hamiltonian algebras of physical interest.

The basic object of this paper is the C^∗-algebra C defined in Theorem 1.1. This is the Hamiltonian algebra of interest here, in fact for us a many-body Hamiltonian is just a self-adjoint operator affiliated toC. We shall see that this is a very large class. On the other hand, it turns out thatC is generated by a rather small class of “elementary” Hamiltonians involving only quantum field like interactions, analogs in our context of the Pauli-Fierz Hamiltonians.

As in theN-body case [ABG] the natural framework for the study of many-body Hamiltonians is that ofC^∗-algebras graded by semilattices. In fact, we are able to make a systematic spectral analysis of the self-adjoint operators affiliated toC becauseCis graded with respect to a certain semilatticeS. We shall see that the channel structure and the formulas for the essential spectrum and the threshold set which appears in the Mourre estimate are completely determined byS, cf. Remark 1.19.

HilbertC^∗-modules play an important technical role in the construction ofC, for example the component C_XY ofC is a HilbertC_Y-module whereC_Y is anN-body type algebra (i.e. a crossed product as above).

But they also play a more fundamental role in a kind of second quantization formalism, see§1.7.

We mention that the algebraC is not adapted to symmetry considerations, in particular in applications to physical systems consisting of particles one has to assume them distinguishable. The Hamiltonian algebra for systems of identical particles interacting through field type forces (both bosonic and fermionic case) is constructed in [Geo].

1.2 The HamiltonianC^∗-algebraC

LetSbe a set of locally compact abelian (lca) groups such that forX, Y ∈ S:

(i) ifX ⊃Y then the topology and the group structure ofY coincide with those induced byX, (ii) X∩Y ∈ S,

(iii) there isZ∈ Ssuch thatX∪Y ⊂ZandX+Y is closed inZ, (iv) X )Y ⇒X/Y is not compact.

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If the first three conditions are satisfied we say thatS is aninductive semilattice of compatible groups.

Condition (iii) is not completely stated, a compatibility assumption should be added (see Definition 6.1).

However, this supplementary assumption is automatically satisfied if all the groups areσ-compact (count- able union of compact sets).

The groupsX ∈ S should be thought as configuration spaces of physical systems and the purpose of our formalism is to provide a mathematical framework for the description of the coupled system. If the systems are of the standardN-body type one may think that theX are finite dimensional real vector spaces. This, however, will not bring any significative simplification of the proofs.

The following are the main examples one should have in mind.

1. LetX be aσ-compact lca group and letSbe a set of closed subgroups ofX withX ∈ Sand such that ifX, Y ∈ SthenX∩Y ∈ S,X+Y is closed, andX/Y is not compact ifX)Y.

2. One may takeSequal to the set of all finite dimensional vector subspaces of a vector spaceover an infinite locally compact field: this is the main example in the context of the many-body problem.

3. The natural framework for thenonrelativistic many-body problemis: X is a real prehilbert space andSa set of finite dimensional subspaces ofXsuch that ifX, Y ∈ SthenX∩Y ∈ SandX+Y is included in a subspace ofS(there is a canonical choice, namely the set ofallfinite dimensional subspaces ofX). Then eachX∈ Sis an Euclidean space hence much more structure is available.

4. One may consider an extension of the usualN-body problem by taking asX in example 1 above a finite dimensional real vector space. In the standard framework [DeG1] the semilatticeS consists of linear subspaces ofX or here we allow them to be closed additive subgroups. We mention that the closed additive subgroups of X are of the formX = E+LwhereE is a vector subspace of X andLis a lattice in a vector subspaceF of X such thatE ∩F = {0}. More precisely, L=P

kZfkwhere{fk}is a basis inF. ThusF/Lis a torus and ifGis a third vector subspace such thatX =E⊕F ⊕Gthen the spaceX/X≃(F/L)⊕Gis a cylinder withF/Las basis.

We assume that eachX ∈ S is equipped with a Haar measure, so the Hilbert spaceH(X)≡L²(X)is well defined: this is the state space of the system withX as configuration space. We define the Hilbert space of the total system as the Hilbertian direct sum

H ≡ HS =⊕XH(X). (1.2)

IfO={0}is the zero group we takeH(O) =C. There is no particle number observable like in the Fock space formalism but there is a remarkableS-valued observable [ABG,§8.1.2] defined by associating to X ∈ Sthe orthogonal projectionΠXofHonto the subspaceH(X).

We shall identifyΠ^∗_Xwith the canonical embedding ofH(X)intoH. We abbreviate^†

L_XY =L(H(Y),H(X)), K_XY =K(H(Y),H(X)), and L_X =L_XX, K_X=K_XX. One may think of an operatorT on Has a matrix with componentsTXY = ΠXTΠ^∗_Y ∈ L_XY and writeT = (TXY)X,Y∈S. We will be interested in subspaces ofL(H)constructed as direct sums in the following sense. Assume that for each coupleX, Y we are given a closed subspaceR_XY ⊂L_XY. Then we define

R≡(R_XY)X,Y∈S =Pc

X,Y∈SΠ^∗_XR_XYΠY (1.3)

wherePc

means closure of the sum. We say that theR_XY are the components ofR.

For an arbitrary pairX, Y ∈ S we define a closed subspaceT_XY ⊂ L_XY as follows. ChoseZ ∈ S such that X ∪Y ⊂ Z and letϕbe a continuous function with compact support on Z. It is easy to

†L(E,F)andK(E,F)are the spaces of bounded and compact operators respectively between two Banach spacesE,F.

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check that(TXY(ϕ)u)(x) = R

Y ϕ(x−y)u(y)dy defines a continuous operatorH(Y) → H(X). Let T_XY be the norm closure of the set of these operators. This space is independent of the choice ofZand T_XX =C^∗(X)is the groupC^∗-algebra ofX. LetT ≡T_S = (T_XY)X,Y∈Sbe defined as in (1.3). This is clearly a closed self-adjoint subspace ofL(H)but is not an algebra in general.

IfX, Y ∈ S andY ⊂X letπY :X →X/Y be the natural surjection and letCX(Y)∼=Co(X/Y)be theC^∗-algebra of bounded uniformly continuous functions onXof the formϕ◦πY withϕ∈ Co(X/Y).

IfX, Y ∈ SandY 6⊂X letCX(Y) = {0}. Then letCX =Pc

Y CX(Y), this is also aC^∗-algebra of bounded uniformly continuous functions onX. We embedCX ⊂L_Xby identifying a function with the operator onH(X)of multiplication by that function. Then let

C ≡ CS =⊕XCX, (1.4)

this is aC^∗-algebra of operators onH. Moreover, for eachZ∈ Slet

C(Z)≡ CS(Z) =⊕XCX(Z) =⊕X⊃ZCX(Z), (1.5) this is aC^∗-subalgebra ofCand we clearly haveC=Pc

ZC(Z).

Theorem 1.1. The space^†C =T ·T is aC^∗-algebra of operators onHand we have

C =T · C =C ·T (1.6)

For eachZ ∈ Slet

C(Z) =T · C(Z) =C(Z)·T. (1.7)

This is aC^∗-subalgebra ofC and{C(Z)}Z∈Sis a linearly independent family ofC^∗-subalgebras ofC such thatPc

ZC(Z) =C andC(Z^′)C(Z^′′)⊂C(Z^′∩Z^′′)for allZ^′, Z^′′∈ S.

This is the main technical result of our paper. Indeed, by using rather simple techniques involving graded C^∗-algebras and the Mourre method one may deduce from Theorem 1.1 important spectral properties of many-body Hamiltonians. The last assertion of the theorem is an explicit description of the fact thatC is equipped with anS-gradedC^∗-algebra structure.We setC =C_Swhen needed.

The choice ofC may seem arbitrary but in fact is quite natural in our context: not only all the many- body Hamiltonians of interest for us are self-adjoint operators affiliated toC, but alsoC is the smallest C^∗-algebra with this property, cf. Theorem 1.7 for a precise statement.

Remark 1.2. Note thatC_XY =Pc

ZC_XY(Z). In matrix notation we have

C = (C_XY)X,Y∈S where C_XY =CX·T_XY =T_XY · CY

andC(Z) = (C_XY(Z))X,Y∈S where

C_XY(Z) =CX(Z)·T_XY =T_XY · CY(Z) ifZ⊂X∩Y andC_XY(Z) ={0} ifZ 6⊂X∩Y.

We mention that ifZis complemented inXandY thenC_XY(Z)≃ C^∗(Z)⊗K_{X/Z,Y /Z}.

Remark 1.3. IfX ⊃ Y then the space T_XY is a “concrete” realization of the Hilbert C^∗-module introduced by Rieffel in [Ri] which implements the Morita equivalence between the groupC^∗-algebra C^∗(Y)and the crossed productCo(X/Y)⋊X. More precisely,T_XY is equipped with a natural Hilbert C^∗(Y)-module structure such that its imprimitivity algebra is canonically isomorphic withCo(X/Y)⋊X.

In Section 4 we shall see that for arbitraryX, Y ∈ S the spaceT_XY has a canonical structure of Hilbert (Co(X/(X∩Y))⋊X,Co(Y /(X∩Y))⋊Y)imprimitivity bimodule. This fact is technically important for the proof of our main results but plays no role in this introduction.

†IfE,F,Gare Banach spaces and(e, f)7→efis a bilinear mapE × F → Gand ifE⊂ E, F⊂ Fare linear subspaces then EFis the linear subspace ofGgenerated by the elementsefwithe∈E, f∈FandE·Fis its closure.

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Remark 1.4. A simple extension of our formalism allows one to treat particles with arbitrary spin.

Indeed, if E is a complex Hilbert then the last part of Theorem 1.1 remains true if C is replaced by C^E =C ⊗K(E)and theC(Z)byC(Z)⊗K(E). IfEis the spin space then it is finite dimensional and one obtainsC^Eexactly as above by replacing theH(X)byH(X)⊗E =L²(X;E). Then in our later results one may consider instead of scalar kinetic energy functionshself-adjoint operator valued functionsh:X^∗ →L(E). For example, we may take as one particle kinetic energy operators the Pauli or Dirac Hamiltonians.

The preceding definition ofC is quite efficient for theoretical purposes but much less for practical ques- tions: for example, it is not obvious how to decide if a self-adjoint operator is affiliated to it. Our next result is an “intrinsic” characterization ofC_XY(Z)which is relatively easy to check. SinceC is constructed in terms of theC_XY(Z), we get simple affiliation criteria.

Forx∈X andk ∈ X^∗(dual group) we define unitary operators inH(X)by(Uxu)(x^′) =u(x^′+x) and(Vku)(x) = k(x)u(x). These correspond to the momentum and position observablesP ≡PXand Q ≡ QX of the system. IfX, Y ∈ S then one can associate to an elementz ∈ X∩Y a translation operator in H(X) and a second one in H(Y). We shall however denote both of them by Uz since which of them is really involved in some relation will always be obvious from the context. IfX and Y are subgroups of a lca groupG(equipped with the topologies induced byG) then we have canonical surjectionsG^∗ →X^∗andG^∗→Y^∗defined by restriction of characters. So a characterk∈G^∗defines an operator of multiplication byk|XonH(X)and an operator of multiplication byk|Y onH(Y). Both will be denotedVk. In our context the lca groupX+Y is well defined (but generally does not belong to S) and we may takeG=X+Y, cf. Remark 6.3. Below we denoteZ^⊥the polar set ofZ ⊂XinX^∗. Theorem 1.5. IfZ ⊂X∩Y thenC_XY(Z)is the set ofT ∈L_XY satisfyingU_z^∗T Uz=Tifz∈Zand such that

(i) k(Ux−1)Tk →0ifx→0inXandkT(Uy−1)k →0ify→0inY,

(ii) kV_k^∗T Vk−Tk →0ifk→0in(X+Y)^∗andk(Vk−1)Tk →0ifk→0inZ^⊥.

Theorem 1.5 becomes simpler and can be improved in the context of Example 3 page 4. So let us assume that S consists of finite dimensional subspaces of a real prehilbert space. Then eachX is equipped with an Euclidean structure and this allows to identifyX^∗ =X such thatVk becomes the operator of multiplication by the functionx7→e^ihx|kiwhere the scalar producthx|kiis well defined for anyx, kin the ambient prehilbert space. ForX⊃Y we identifyX/Y =XªY, the orthogonal ofY inX.

Corollary 1.6. Under the conditions of Example 3 page 4 the spaceC_XY(Z)is the set ofT ∈ L_XY satisfying the next two conditions:

(i) U_z^∗T Uz=Tforz∈ZandkV_z^∗T Vz−Tk →0ifz→0inZ,

(ii) kT(Uy−1)k →0ify→0inY andkT(Vk−1)k →0ifk→0inY /Z.

Condition 2 may be replaced with:

(iii) k(Ux−1)Tk →0ifx→0inXandk(Vk−1)Tk →0ifk→0inX/Z.

1.3 Elementary Hamiltonians

Our purpose in this subsection is to show thatC is aC^∗-algebra of Hamiltonians in a rather precise sense, according to the terminology used in [GI1, GI2]: we show thatC is theC^∗-algebra generated by a simple class of Hamiltonians which have a natural quantum field theoretic interpretation. Since our desire is only to motivate our construction, in this subsection we shall make two simplifying assumptions: S is finite and ifX, Y ∈ SwithX ⊃Y, thenY is complemented inX.

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For each coupleX, Y ∈ S such thatX ⊃ Y we chose a closed subgroupX/Y of X such thatX = (X/Y)⊕Y. Moreover, we equipX/Y with the quotient Haar measure which gives us a factorization H(X) = H(X/Y)⊗ H(Y). Then we defineΦXY ⊂ L_XY as the closed linear subspace consisting of “creation operators” associated to states fromH(X/Y), i.e. operatorsa^∗(θ) : H(Y)→ H(X)with θ∈ H(X/Y)which act asu7→θ⊗u. We setΦY X = Φ^∗_XY ⊂L_{Y X}, this is the space of “annihilation operators” a(θ) = a^∗(θ)^∗ defined by H(X/Y). This definesΦXY when X, Y are comparable, i.e.

X ⊃ Y or X ⊂Y, which we abbreviate byX ∼ Y. IfX 6∼ Y then we takeΦXY = 0. Note that ΦXX =C1X, where1X is the identity operator onH(X), becauseH(O) =C.

The spaceΦXY forX ⊃Y clearly depends on the choice of the complementX/Y. On the other hand, according to Definition 4.7 and Proposition 4.19, we have

C^∗(X)·ΦXY = ΦXY · C^∗(Y) =T_XY ifX ∼Y. (1.8) This seems to us a rather remarkable feature because not onlyT_XY is independent ofX/Y but is also well defined even ifY is not complemented inX.

Now we define Φ = (ΦXY)X,Y∈S ⊂ L(H). This is a closed self-adjoint linear space of bounded operators onH. A symmetric elementφ ∈ Φwill be calledfield operator, this is the analog of a field operator in the present context. Giving such aφis equivalent to giving a familyθ = (θXY)X⊃Y of elementsθXY ∈ H(X/Y), the components of the operatorφ≡φ(θ)being given by:φXY =a^∗(θXY) ifX ⊃Y, thenφXY = a(θY X)ifX ⊂Y, and finallyφXY = 0ifX 6∼Y. Note thatΦXX =C1X

becauseH(O) =C. Ifu= (uX)X∈Sthen we have hu|φui=P

X⊃Y2ℜhθXY ⊗uY|uXi.

Astandard kinetic energy operatoris an operator onHof the formK=⊕XhX(P)wherehX:X^∗→R is continuous andlimk→∞|hX(k)|=∞. The operators of the formK+φ, whereKis a standard kinetic energy operator andφ∈Φis a field operator, will be calledPauli-Fierz Hamiltonians.

The proof of the next theorem may be found in the Appendix.

Theorem 1.7. Assume thatSis finite and thatY is complemented inX ifX ⊃Y. ThenC coincides with theC^∗-algebra generated by the Pauli-Fierz Hamiltonians.

Remark 1.8. It is interesting and important to note that C is generated by a class of Hamiltonians involving only an elementary class of field type interactions. However, as we shall see in§1.5, the class of Hamiltonians affiliated toC is very large and coversN-body systems interacting between themselves (i.e. for varyingN) with field type interactions. In particular, theN-body type interactions are generated by pure field interactions and this thanks to the semilattice structure ofS.

1.4 Essential spectrum of operators affiliated toC

The main assertion of Theorem 1.1 is thatC is anS-gradedC^∗-algebra. The class ofC^∗-algebras graded by finite semilattices has been introduced and their role in the spectral theory ofN-body systems has been pointed out in [BG1, BG2]. Then the theory has been extended to infinite semilattices in [DaG2].

A much deeper study of this class ofC^∗-algebras is the subject of the thesis [Ma1] of Athina Mageira (see also [Ma2, Ma3]) whose results allowed us to consider a semilatticeS of arbitrary abelian groups (and this is important in certain applications that we do not mention in this paper). We mention that her results cover non-abelian groups and the assumption (iv) (on non-compact quotients) is not necessary in her construction. This could open the way to interesting extensions of our formalism.

In§5.1 we recall some basic facts concerning gradedC^∗-algebras. Our main tool for the spectral analysis of the self-adjoint operators affiliated toC is Theorem 5.2. For example, it is easy to derive from it the

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abstract HVZ type description of the essential spectrum given in Theorem 5.3. Here we give a concrete application in the present framework, more general results may be found in Sections 5 and 7.

For eachX ∈ Swe define a closed subspace ofHby H≥X =L

Y⊃XH(Y). (1.9)

This is associated to the semilatticeS≥X={Y ∈ S |Y ⊃X}in the same way asHis associated toS.

LetC_≥Xbe theC^∗-subalgebra ofC given by C_≥X =Pc

Y⊃XC(Y)∼=¡Pc

Y⊃XC_EF(Y)¢

E∩F⊃X (1.10)

and note thatC_≥X lives on the subspaceH≥XofH. Moreover,C andC_≥Xare nondegenerate algebras of operators on the Hilbert spacesHandH≥Xrespectively. It can be shown that there is a unique linear continuous projectionP_≥X :C →C_≥X such thatP_≥X(T) = 0ifT ∈C(Y)withY 6⊃X and that this projection is a morphism, cf. Theorem 5.2.

LetH be a self-adjoint operator on a Hilbert spaceHaffiliated to aC^∗-algebra of operatorsA onH.

Then ϕ(H) ∈ A for all ϕ ∈ Co(R). IfA is the closed linear span of the elementsϕ(H)A with ϕ∈ Co(R)andA∈A, we say thatH isstrictly affiliated toA.

Assume that the semilatticeShas a smallest elementminS. ThenX ∈ Sis an atom if the only element ofSstrictly included inXisminS. LetP(S)be the set of atoms ofS. We say thatSisatomicif each of its elements not equal tominScontains an atom. It is clear that if the zero groupObelongs toSthen Ois the smallest element ofSandC(O) =K(H).

Theorem 1.9. IfH is a self-adjoint operator onHstrictly affiliated toC then for eachX ∈ S there is a unique self-adjoint operatorH≥X ≡ P_≥X(H)onH≥X such thatP_≥X(ϕ(H)) =ϕ(H≥X)for all ϕ∈ Co(R). The operatorH≥X is strictly affiliated toC_≥X. IfO∈ SandS is atomic then the essential spectrum ofHis given by

Spess(H) =S

X∈P(S)Sp(H≥X). (1.11)

1.5 Hamiltonians affiliated toC

We shall give now examples of self-adjoint operators strictly affiliated toC. The argument is relatively straightforward thanks to Theorem 1.5 but the fact thatSis allowed to be infinite brings some additional difficulties. We are interested in Hamiltonians of the formH =K+IwhereK is the kinetic energy operator of the system andIis the interaction term. FormallyHis a matrix of operators(HXY)X,Y∈S, the operatorHXY is defined on a subspace ofH(Y)and has values inH(X), and we haveH_XY^∗ =HY X

(again formally). ThenHXY =KXY+IXY and our assumptions will be thatKis diagonal, soKXY = 0 ifX 6= Y andKXX ≡ KX. The interactions will be of the formIXY = P

Z⊂X∩YIXY(Z), this expresses the N-body structures of the various systems (with various N, of course). ThenHXX = KX+IXX will be a generalizedN-body type Hamiltonian (IXX may depend on the momentum). The non-diagonal operatorsHXY =IXY define the interaction between the systemsXandY (these operators too may depend on the momentum of the systemsX, Y). We give now a rigorous construction of such Hamiltonians.

(a)For eachX we choose a kinetic energy operatorKX =hX(P)for the system havingX as configuration space. The functionhX : X^∗ → Rmust be continuous and such that|hX(x)| → ∞ifk→ ∞.

We emphasize the fact that there are no relations between the kinetic energiesKXof the systems corre- sponding to differentX. IfSis infinite, we requirelimXinfk|hX(k)|=∞, more explicitly:

for each realEthere is a finite setT ⊂ Ssuch thatinfk|hX(k)|> EifX /∈ T.

Hilbert C*-modules and spectral analysis of many-body systems

HAL Id: hal-00285199

https://hal.archives-ouvertes.fr/hal-00285199

Hilbert C*-modules and spectral analysis of many-body systems

Mondher Damak, Vladimir Georgescu

To cite this version:

Hilbert C

-modules and spectral analysis of many-body systems

Contents

1 Introduction and main results

**Hilbert C*-modules and spectral analysis of many-body systems**