A Comparison of the Georgescu and Vasy Spaces Associated to the N-Body Problems and Applications

Abstract

We provide new insight into the analysis of N-body problems by studying a compactification $M_N$ of ${\mathbb {R}}^{3N}$ that is compatible with the analytic properties of the N-body Hamiltonian $H_N$. We show that our compactification coincides with a compactification introduced by Vasy using blow-ups in order to study the scattering theory of N-body Hamiltonians and with a compactification introduced by Georgescu using $C^*$-algebras. In particular, the compactifications introduced by Georgescu and by Vasy coincide (up to a homeomorphism that is the identity on ${\mathbb {R}}^{3N}$). Our result has applications to the spectral theory of N-body problems and to some related approximation properties. For instance, results about the essential spectrum, the resolvents, and the scattering matrices of $H_N$ (when they exist) may be related to the behavior near $M_N{\setminus } {\mathbb {R}}^{3N}$ (i.e., “at infinity”) of their distribution kernels, which can be efficiently studied using our methods. The compactification $M_N$ is compatible with the action of the permutation group $S_N$, which allows to implement bosonic and fermionic (anti-)symmetry relations. We also indicate how our results lead to a regularity result for the eigenfunctions of $H_N$.

Appendices

Appendix A: Proper Maps

We now provide a characterization of proper maps used in the main body of the paper. Let $f : X \rightarrow Y$ be a continuous map between two Hausdorff spaces. Recall that f is called proper if $f^{-1}(K)$ is compact for every compact subset $K \subset Y$.

Lemma A.1

(Generalizes [53, Prop. 4.32]). Let $f : X \rightarrow Y$ be a continuous map between two Hausdorff spaces with Y locally compact. If f is proper, then f is closed.

In [53, Prop. 4.32], the lemma is stated with the additional requirement that X be locally compact. However, in the proof the locally compactness of X is not needed. We omit the proof since we will apply the lemma only when X is locally compact.

Corollary A.2

Let $f : X \rightarrow Y$ be a continuous injective map between two Hausdorff spaces with Y locally compact. If f is proper, then f is a homeomorphism onto its image.

Proof

The map $f:X\rightarrow f(X)$ is bijective continuous and closed and thus a homeomorphism. $\square$

We shall say that f is locally proper if, for every $y \in Y$, there exists an open neighborhood $V_y$ of y in Y such that the map $f^{-1}(V_y) \rightarrow V_y$ induced by f is proper.

Lemma A.3

Let $f : X \rightarrow Y$ be a continuous map between two Hausdorff spaces with Y locally compact. Then f is proper if, and only if, it is locally proper.

Proof

Clearly, every proper map is locally proper, by definition. Let us assume that f is locally proper and let $K \subset Y$ be a compact subset. For any $y\in K$, we choose the open neighborhood $V_y$ as in the definition of a locally proper map. As Y is locally compact, there is an open neighborhood $W_y$ of y in $V_y$ such that its closure ${\overline{W}}_y$ in Y is a compact subset of $V_y$. The local properness of f together with the choice of $V_y$ implies that $f^{-1}({\overline{W}}_y\cap K)$ is compact. By the compactness of K, we can choose $y_1,\ldots ,y_N$ such that K is covered by $\left( W_{y_j}\right) _{1\le j\le N}$. Then $K=\bigcup _{j=1}^N \left( {\overline{W}}_{y_j}\cap K\right)$. Hence,

$$\begin{aligned} f^{-1}(K)= & {} \bigcup _{j=1}^N f^{-1}({\overline{W}}_{y_j}\cap K) \end{aligned}$$

is also compact. This completes the proof. $\square$

Appendix B: More on Submanifolds of Manifolds with Corners

We discuss here a few other notions of submanifolds and the relation to our concept of weak submanifold. While this is not needed for the proof of the main result, we hope the interested reader will find this material useful.

1.1 B.1 Submanifolds in Melrose’s Sense

We begin with Melrose’s concept of a submanifold in a manifold with corners, following [56, Definition 1.7.3].

Definition B.1

A subset S of a manifold with corners M of dimension n is a submanifold (in the sense of manifolds with corners) if, for every $p \in S$, there exists $0 \le k \le n$ and a (corner) chart $\phi :U\rightarrow \Omega \subset {\mathbb {R}}_k^n := [0,\infty )^k \times {\mathbb {R}}^{n-k}$, numbers $n'\le n$ and $k'\le n'$, and a matrix $G\in \mathop {\mathrm {GL}}(n,{\mathbb {R}})$ such that

• (1) $p \in U$

• (2) $G \left( {\mathbb {R}}^{n'}_{k'}\times \{0\}\right) \subset {\mathbb {R}}^n_k.$

• (3) $\phi (S \cap U)\ =\ G \left( {\mathbb {R}}^{n'}_{k'}\times \{0\}\right) \cap \Omega .$

Obviously, every submanifold in the sense of manifolds with corners is a weak submanifold, see Definition 2.10. In [56], all submanifolds are submanifolds in the sense of Definition 2.10, see e.g., Lemma 2.11. In Remarks 2.12 (a), we explained that any weak submanifold of a manifold with corners inherits an atlas, and thus this also applies to submanifolds in the above sense. However, it can be shown [50] that many submanifolds in our article are not submanifolds in the sense of manifolds with corners, but only weak submanifolds, as defined in Definition 2.10. In Example B.3, we provide an example of a weak submanifold of ${\mathbb {R}}^2_1$ that is not one in the sense of Definition B.1.

Example B.2

(Diagonal). Let N be a manifold with corners. Then $M:=N\times N$ is also a manifold with corners. Consider the diagonal $\Delta _N:=\{(p,p)\in M\mid p\in N\}$. Then $\Delta _N$ is a submanifold of M in the sense of manifolds with corners.

The following provides examples of weak submanifolds that are not submanifolds in the sense of Definition B.1.

Examples B.3

• (1) The function $f : {\mathbb {R}}^2_1 := [0, \infty ) \times {\mathbb {R}}\rightarrow {\mathbb {R}}^2_1$, $f(x,y):=(x+y^2,y)$, is an injective immersion. It is a homeomorphism onto its image $S:=f\bigl ({\mathbb {R}}^2_1\bigr )$. However, it can be easily seen that S is not a submanifold of ${\mathbb {R}}^2_1$ in the sense of manifolds with corners. On the other hand, S is a submanifold of ${\mathbb {R}}^2$ in the sense of manifolds with corners.

• (2) The function $f(x):=(x,x^2)$ defines a injective immersion ${\mathbb {R}}^1_1\rightarrow {\mathbb {R}}^2_2$. It is a homeomorphism onto its image $S:=f\bigl ({\mathbb {R}}^1_1\bigr )$. However, S is not a submanifold ${\mathbb {R}}^2_2$ in the sense of manifolds with corners. On the other hand, S is a submanifold of ${\mathbb {R}}^2_1$ and of ${\mathbb {R}}^2$ in the sense of manifolds with corners.

In our article, injective immersions that are homeomorphisms onto their images play an important role. Recall the following classical fact for manifolds N and M without boundary and without corners:

$$\begin{aligned} (*) {\left\{ \begin{array}{ll} \;\text {If }f:N\rightarrow M\text { is an injective immersions, then }f(N)\text { is a submanifold}\\ \;\text {if, and only if, }f\text { maps }N\text { homeomorphically to } f(N). \end{array}\right. } \end{aligned}$$

Examples B.3 show that $(*)$ does no longer hold if M and N are manifolds with corners and if we understand the word “submanifold” in the sense of Definition B.1. On the other, we proved in Proposition 2.13 that $(*)$ holds for manifolds with corners, if we replace “a submanifold” by “a weak submanifold.”

1.2 B.2 Other Classes of Submanifolds

For comparison and completeness, we recall now the definitions of some further classes of submanifolds. The reason the reader might be interested in these concepts is that the concept of a submanifold in the sense of Definition B.1 seems to be too unspecific and the concept of a p-submanifold (Definition 2.15) seems to be sometimes too restrictive. A first alternative is the concept of a “wib-submanifold”, where “wib” stands for a submanifold without an interior boundary.

Definition B.4

A submanifold $S\subset M$ is called a wib-submanifold or a submanifold without interior boundary if it can be defined locally in suitable charts as the kernel of a linear function. More precisely: $S\subset M$ is a wib-submanifold if, for every $x \in S$, there exists a (corner) chart $\phi :U\rightarrow \Omega \subset {\mathbb {R}}^n_k$, and a linear subspace L of ${\mathbb {R}}^n$, such that

• (1) $x \in U$ and

• (2) $\phi (S \cap U)\ =\ L \cap \Omega .$

If $G\in \mathop {\mathrm {GL}}(n,{\mathbb {R}})$ is as in Definition B.1, then we necessarily have $L = G \left( {\mathbb {R}}^{n'}\times \{0\}\right)$. If $x\in S\cap U$, then $n':=\dim (L)$ is the dimension of S in x defined above. Obviously all p-submanifolds are wib-manifolds, which can be easily seen by defining the L in the definition above as the linear extension of $L_I$ in Definition 2.15.

Remark B.5

In the above definition, we explicitly required S to be a submanifold. To justify this requirement, we will give an example of a closed subset $S\subset M$ that is not a submanifold, but fulfills all other requirements of the definition of a wib-submanifold. Indeed, let

$$\begin{aligned} K \ :=\ \{(x_1,x_2,x_3)\in {\mathbb {R}}^3\mid x_1\ge 0,\; x_2\ge 0,\;x_1\le x_3,\;x_2\le x_3\}, \end{aligned}$$

which is a cone over a square. The map $f:{\mathbb {R}}^3\rightarrow {\mathbb {R}}^4$, $f(x_1,x_2,x_3)= (x_1,x_2,x_3-x_1,x_3-x_2)$ has the property $f^{-1}({\mathbb {R}}^4_4)=K$. Then for $\phi ={\text {id}}$, $x=0$, and $L:=f({\mathbb {R}}^3)$ all requirements of the definition are satisfied, but $S:=f(K)$ is not a submanifold of ${\mathbb {R}}^4_4$. It it were a submanifold, then its dimension would have to be 3, and then any boundary point of S is in at most 3 closed boundary hyperfaces. But $0\in S$ is in 4 closed boundary hyperfaces of S.

Remark B.6

Note that Melrose also introduces the notions d-submanifold [56, Def. 1.7.4] and b-submanifold [56, Def. 1.12.9], whose definitions will not be recalled here. They satisfy

$$\begin{aligned}&S \text { is a p-submanifold} \;\Longrightarrow \;S \text { is a d-submanifold} \Longrightarrow S \text { is a b-submanifold{}} \\&\quad \Longrightarrow S \text { is a submanifold}\;\Longrightarrow \;S \text { is a weak submanifold}. \end{aligned}$$

However, there are wib-manifolds that are not b-submanifolds, such as Melrose’s example of the submanifold $\{x_3=x_1+x_2\}\in {\mathbb {R}}^3_3$. There are d-manifolds that are no wib-manifolds, for instance, ${\mathbb {R}}^1_1=[0,\infty )\subset {\mathbb {R}}$ or any surface with boundary in ${\mathbb {R}}^3$. However, all p-submanifolds introduced below are d-submanifolds and wib-submanifolds. Melrose shows that the diagonal $\Delta _N$ is a b-submanifold of $N\times N$, but in general not a d-submanifold. It follows that $\Delta _N$ is not a p-submanifold.

Remark B.7

Let us remark that the concept of a tame submanifold considered in [2, Sect. 2.3] is a concept of a submanifold in an essentially different sense, it is actually a more restrictive notion of submanifold than the ones encountered in this paper. All notions of submanifolds discussed so far involve properties that may or may not hold for a subset N of a manifold with corners M. In contrast to this, tame submanifolds in [2, Sect. 2.3] are submanifolds of a Lie manifold (M, A), where M is a manifold with corners and A is a Lie algebroid on M with some compatibility conditions. Whether a subset N of M is a tame submanifold of (M, A) or not depends also on the Lie algebroid A. In any case, a tame submanifold will have a tubular neighborhood in the strongest sense. Similar remarks apply to the $A({\mathcal {G}})$-tame submanifolds considered in [66].