Exhaustive families of representations of $C^*$-algebras associated to $N$-body Hamiltonians with asymptotically homogeneous interactions

TL;DR

This paper studies the spectral properties of $N$-body Hamiltonians with asymptotically homogeneous interactions using algebraic methods, extending previous frameworks and providing Fredholm criteria for associated operators.

Contribution

It introduces a comprehensive algebraic framework for analyzing $N$-body Hamiltonians with asymptotically homogeneous interactions, extending prior work and addressing open questions.

Findings

Spectral analysis of $N$-body Hamiltonians with asymptotic homogeneity.

Extension of algebraic methods to broader classes of operators.

Fredholm conditions for elliptic operators in this context.

Abstract

We continue the analysis of algebras introduced by Georgescu, Nistor and their coauthors, in order to study $N$-body type Hamiltonians with interactions. More precisely, let $Y$ be a linear subspace of a finite dimensional Euclidean space $X$, and $v_Y$ be a continuous function on $X/Y$ that has uniform homogeneous radial limits at infinity. We consider, in this paper, Hamiltonians of the form $H = - \Delta + \sum_{Y \in S} v_Y$, where the subspaces $Y$ belong to some given family S of subspaces. We prove results on the spectral theory of the Hamiltonian when $S$ is any family of subspaces and extend those results to other operators affiliated to a larger algebra of pseudo-differential operators associated to the action of $X$ introduced by Connes. In addition, we exhibit Fredholm conditions for such elliptic operators. We also note that the algebras we consider answer a question of Melrose and Singer.

Equations

$$H=-\Delta +\sum _{Y\in \mathcal{S}}{v}_Y,$$

$${\mathcal{E}}_{\mathcal{S}}(X)=⟨C(\overline{X/Y}),Y\in \mathcal{S}⟩\subset C_b^u(X).$$

$${\mathcal{S}}_i^n=\{(x_1,\dots ,x_n)\in X^n;x_i=0\}$$

$${\mathcal{S}}_{ij}^n=\{(x_1,\dots ,x_n)\in X^n;x_i=x_j\}$$

$${\mathcal{E}}_{\mathcal{S}}(X)⋊X,$$

$$C_{\varphi }u(x):={\int }_X\varphi (y)u(x-y)dy$$

$$K(x,y)=f(x)\varphi (y-x),(\text{resp. }K(x,y)=f(y)\varphi (y-x)).$$

$$K(x,y)=\varphi (y-x)(f(x)-f(y)).$$

$$(H+i{)}^{-1}=(-\Delta +i{)}^{-1}{\left(1+V(-\Delta +i{)}^{-1}\right)}^{-1},$$

$$0\to {\mathcal{E}}_{\mathcal{S}}(X)⋊X\to \Psi \mathrm{DO}({\mathcal{E}}_{\mathcal{S}}(X),X)\stackrel{{\sigma }_0}{⟶}C({\text{SS}}_X,{\mathcal{E}}_{\mathcal{S}}(X))\to 0.$$

$${\tau }_{\alpha }(P)=\underset{r\to +\infty }{\lim }{T}_{ra}^{\ast }P{T}_{ra},\text{mbox}if\alpha =\hat{a}\in {\text{SS}}_X,$$

$$\tau_{\alpha}(f)(x)=\left\{\begin{array}[]{ll}f(x)&\text{ if }Y\supset\alpha\,,\\ f(\pi_{Y}(\alpha))&\text{ else. }\end{array}\right.$$

$${\tau }_{\alpha }:{\mathcal{E}}_{\mathcal{S}}(X)\to {\mathcal{E}}_{{\mathcal{S}}_{\alpha }}(X).$$

$${\mathrm{Spec}}_{\mathrm{ess}}(P)={\cup }_{\alpha \in {\text{SS}}_X}\mathrm{Spec}({\tau }_{\alpha }(P)).$$

$${\mathrm{Spec}}_{\mathrm{ess}}(H)={\overline{\cup }}_{\alpha \in {\text{SS}}_X}\mathrm{Spec}({\tau }_{\alpha }(H))$$

$$(\forall a\in A)\mathrm{Spec}(a)={\cup }_{i\in I}\mathrm{Spec}({\varphi }_i(a))$$

$$\begin{diagram}$$

$$(\forall Y\in \mathcal{S},\alpha \not\subset Y,\forall u\in C(\overline{X/Y}))\chi (u)=u({\pi }_Y(\alpha )).$$

$$\ker ({\tau }_{\alpha }⋊X)=(\ker {\tau }_{\alpha })⋊X\subset \ker \pi .$$

Full text

An new approach in the study of Hamiltonians of $N$-body type with interactions that are asymptotically homogeneous at infinity on a finite dimensional Euclidean space $X$ was initiated by Georgescu and Nistor [3, 5, 6].

For any finite real vector space $Z$, we let $\overline{Z}$ denote its spherical compactification. A function in $C(\overline{Z})$ is thus a continuous function on $Z$ that has uniform radial limits at infinity. Let $\SS_{Z}$ be the set of half-lines in $Z$, that is $\SS_{Z}:=\{\,\hat{a},\ a\in Z,a\neq 0\,\}$ where $\hat{a}:=\{ra,\,r>0\}$. We identify $\SS_{Z}=\overline{Z}\smallsetminus Z$ .

For any subspace $Y\subset X$, $\pi_{Y}:X\to X/Y$ denotes the canonical projection. Let

[TABLE]

where $v_{Y}\in C(\overline{X/Y})$ is seen as a bounded continuous function on $X$ via the projection $\pi_{Y}:X\to X/Y$. The sum is over all subspaces $Y\subset X$, $Y\in\mathcal{S}$ and is assumed to be uniformly convergent. One of the main results of [5, 9] describe the essential spectrum of $H$ extending the celebrated HVZ theorem [13]. The goal of this paper is to explain how these results can be extended to any family of subspaces that contains $\{0\}$ and to more general operators using $C^{*}$-algebras techniques.

Let $\mathcal{S}$ be a family of subspaces of $X$ with $0\in\mathcal{S}$. We define the commutative sub-$C^{*}$-algebra $\mathcal{E}_{\mathcal{S}}(X)$ of the commutative $C^{*}$-algebra $C_{b}^{u}(X)$ of bounded uniformly continous functions on $X$ by

[TABLE]

The algebras $\mathcal{E}_{\mathcal{S}}(X)$ give an answer to a question of Melrose and Singer [8].

[TABLE]

Then the spectrum $\Omega_{\mathcal{S}^{n}}$ of $\mathcal{E}_{\mathcal{S}^{n}}(X^{n})$ is a compactification of $X^{n}$ satisfying the following properties :

1. (1)

$\Omega_{\mathcal{S}^{1}}$* is the spherical compactification $\overline{X}$,* 2. (2)

The action of the symmetric group $\mathfrak{S}_{n}$ on $X^{n}$ extends continuously to $\Omega_{\mathcal{S}_{n}}$, 3. (3)

The projections $p^{n,k}_{I}\colon X^{n}\to X^{k}$, $p^{n,k}_{I}(x_{1},\ldots,x_{n})=(x_{i_{1}},\ldots,x_{i_{k}})$ extend continuously to $p^{n,k}_{I}\colon\Omega_{\mathcal{S}_{n}}\to\Omega_{\mathcal{S}_{k}}$, 4. (4)

The difference maps $\delta_{ij}(x_{1},\ldots,x_{n})=x_{i}-x_{j}$ from $X^{n}$ to $X$ extend continuously to the compactifications.

Actually, the spectrum $\Omega_{\mathcal{S}^{n}}$ have very strong connection with the space built by Vasy in [15] and generalized by Kottke in the last section of [7].

The additive group $X$ acts by translation on $C_{b}^{u}(X)$ and the subalgebra $\mathcal{E}_{\mathcal{S}}(X)$ is invariant. So a crossed product $C^{*}$-algebra is obtained

[TABLE]

which can be regarded as an algebra of operators on $L^{2}(X)$. Thanks to the assumption $0\in\mathcal{S}$, the algebra $C_{0}(X)$ belongs $\mathcal{E}_{\mathcal{S}}(X)$. Hence $C_{0}(X)\rtimes X$ is contained in $\mathcal{E}_{\mathcal{S}}(X)\rtimes X$. It follows from the definition of crossed products algebras that the $C^{*}$-algebra $\mathcal{E}_{\mathcal{S}}(X)\rtimes X$ is generated by two kinds of operators : multiplication operators $m_{f}$ associated to functions $f\in\mathcal{E}_{\mathcal{S}}(X)$, and convolution operators

[TABLE]

with $\phi\in C_{c}(X)$, a continuous compactly supported function. An immediate computation shows that $m_{f}c_{\phi}$ (resp. $c_{\phi}m_{f}$) is a kernel operator with kernel

[TABLE]

(ii) For $f\in C(\overline{X})$ and $\phi\in C_{c}(X)$ the commutator $[m_{f},c_{\phi}]$ is compact.*

The point $(i)$ is a consequence of equation (4) because the kernel $K$ has compact support when $f$ does and the result follows by density. Again, thanks to equation (4), one sees that the commutator is a kernel operator with kernel

[TABLE]

Hence, in view of $\phi\in C_{c}(X)$, the support of $K$ is contained in a band around the diagonal. The distance between the border of the band and the diagonal is bounded. Moreover, $K$ goes to [math] at infinity because $f$ has radial limits. So the commutator is a limit of Hilbert-Schmidt operators, and hence is compact.

Recall that a self-adjoint operator $P$ on $L^{2}(X)$ is said to be affiliated to a $C^{*}$-algrebra $A$ of bounded operators on $L^{2}(X)$ if for some (and hence any) function $h\in C_{0}(\mathbb{R})$ then $h(P)$ belongs to $A$. For example,, it follows from the identity

[TABLE]

that $H$ is affiliated to $\mathcal{E}_{\mathcal{S}}(X)\rtimes X$. More generally, for any $C^{*}$-algebra $A$, a morphism $h\colon C_{0}(\mathbb{R})\to A$ is called an operator affiliated to $A$. Following Connes [2] and Baaj [1] we introduce the $C^{*}$-algebra of non positive order pseudo-differential operators $\Psi\mathrm{DO}(\mathcal{E}_{\mathcal{S}}(X),X)$ together with the symbol map exact sequence

[TABLE]

Positive order pseudo-differential operators are examples of operators affiliated to the algebra of non positive order pseudo-differential operators $\Psi\mathrm{DO}(\mathcal{E}_{\mathcal{S}}(X),X)$.

Let $\alpha\in\mathbb{S}_{X}$. For each $x\in X$, we let $(T_{x}f)(y)=f(y-x)$ denote the translation on $L^{2}(X)$. For any operator $P$ on $L^{2}(X)$, we let

[TABLE]

whenever the strong limit exists.

[TABLE]

We define $\mathcal{S}_{\alpha}=\{Y\in\mathcal{S}\,;\,\alpha\subset Y\}$. It follows from the previous lemma that on $\mathcal{E}_{\mathcal{S}}(X)$, $\tau_{\alpha}$ is the projection on the subalgebra $\mathcal{E}_{\mathcal{S}_{\alpha}}(X)$,

[TABLE]

1. (1)

Let $P$ be a self-adjoint operator affiliated to $\Psi\mathrm{DO}(\mathcal{E}_{\mathcal{S}}(X),X)$ and $\alpha=\hat{a}\in\SS_{X}$. Then the limit $\tau_{\alpha}(P):=\lim_{r\to+\infty}T_{ra}^{*}PT_{ra}$ exists and

[TABLE] 2. (2)

Let $P\in\Psi\mathrm{DO}(\mathcal{E}_{\mathcal{S}}(X),X)$. Then $P$ is a Fredholm operator if and only if $P$ is elliptic (i.e. $\sigma_{0}(P)$ is invertible) and for all $\alpha\in\SS_{X}$, $\tau_{\alpha}(P)$ is invertible.

This extends theorems of [5, 9] in the following sense : only operators affiliated to $\mathcal{E}_{\mathcal{S}}(X)\rtimes X$ are considered there, and the relation is

[TABLE]

in [5]. In [9] only finite semi-lattice $\mathcal{S}$ are considered. The equation (6) means that the family $(\tau_{\alpha})$ is a faithful family of morphism of $\mathcal{E}_{\mathcal{S}}(X)\rtimes X$. The stronger result of [9] is obtained by showing that the family $(\tau_{\alpha}\rtimes X)_{\alpha\in\SS_{X}}$ is actually an exhaustive family of representations of $\mathcal{E}_{\mathcal{S}}(X)\rtimes X$, when $\mathcal{S}$ is a finite semi-lattice. In the framework of admissible locally compact group, decomposition of essential spectrum involving exhaustive families can be found in [10] [11]. In fact, by [12, Proposition 3.12], exhaustive families are also strictly spectral families in the following sense.

1. (1)

A family $(\phi_{i})_{i\in I}$ of morphisms of a $C^{*}$-algebra $A$ is said to be exhaustive if any primitive ideal contains at least $\ker\phi_{i}$ for some $i\in I$. 2. (2)

A family $(\phi_{i})_{i\in I}$ of morphisms of a unital $C^{*}$-algebra $A$ is said to be strictly spectral if

[TABLE]

Let us prove this result. Let $\pi$ be an irreducible representation of $\mathcal{E}_{\mathcal{S}}(X)\rtimes X/\mathcal{K}(X)$. It extends to an irreducible representation of $\mathcal{E}_{\mathcal{S}}(X)\rtimes X$ as well as to their multipliers algebras $\mathcal{M}(\mathcal{E}_{\mathcal{S}}(X)\rtimes X/\mathcal{K}(X))$ and $\mathcal{M}(\mathcal{E}_{\mathcal{S}}(X)\rtimes X)$. By proposition 2(i), one obtains the following commutative diagram:

[TABLE]

In fact it is enough to show that any $f\in C(\overline{X})$ commutes with any element of $\mathcal{E}_{\mathcal{S}}(X)\rtimes X$ modulo a compact operator. But the result is true on the generators by Proposition 2(ii), so the lemma follows by density.

By the Schur Lemma, we deduce that $\pi\circ\phi$ is a character of $C(\SS_{X})$. Hence there exists some $\alpha\in\SS_{X}$ such that $\pi|_{C(\overline{X})}=\chi_{\alpha}I$, where $\chi_{\alpha}$ is the character of $C(\overline{X})$ given by the evaluation at $\alpha\in\SS_{X}$.

[TABLE]

In particular, for $Y=0$, we see that $\chi_{|C(\overline{X})}=\chi_{\alpha}$. Reciprocally it follows from [5, Lemma 6.7] that if $\chi_{|C(\overline{X})}=\chi_{\alpha}$ then relation (8) is true. On the other hand, the characters of $\mathcal{E}_{\mathcal{S}}(X)/(\ker\chi_{\alpha})\mathcal{E}_{\mathcal{S}}(X)$ are precisely the characters $\chi$ of $\mathcal{E}_{\mathcal{S}}(X)$ such that $\chi_{C(\overline{X})}=\chi_{\alpha}$. So $\ker\tau_{\alpha}=(\ker\chi_{\alpha})\mathcal{E}_{\mathcal{S}}(X)$ as claimed. ∎

Now if $\pi_{|C(\overline{X})}=\chi_{\alpha}$, one has $\ker\pi\supset(\ker\chi_{\alpha})\mathcal{E}_{\mathcal{S}}(X)=\ker\tau_{\alpha}$. Finally,

[TABLE]

It follows that $(\tau_{\alpha}\rtimes X)_{\alpha\in\SS_{X}}$ is an exhaustive family of morphisms.

See also [4, Example 6.35] for others physical interesting operators.

Acknowledgments

The authors thank Victor Nistor for useful discussions.