On the Integrability of the n–Centre Problem

Andreas Knauf^{∗}  Iskander A. Taimanov^{†}

Abstract

It is known that for n ≥ 3 centres and positive energies the ncentre problem of celestial mechanics leads to a flow with a strange repellor and positive topological entropy.

Here we consider the energies above some threshold and show: Whereas for arbitrary g > 1 independent integrals of Gevrey class g exist, no real-analytic (that is, Gevrey class 1) independent integral exists.

1 Introduction

In [BT] the existence of a smoothly integrable geodesic flow on a compact manifold with positive topological entropy was established. Positivity of topological entropy is seen as an indication of complex dynamics, whereas integrability of a Hamiltonian flow is a metaphor for its simplicity. So coexistence of these two properties may not have been expected. In fact, as we show here, such a coexistence takes place in natural physical problems.

To be more specific, in this note we consider the n–centre problem of celestial mechanics in d = 2 and 3 dimensions. We denote by s⃗_{k} ∈ ℝ^{d}, Z_{k} ∈ ℝ \ {0} the location resp. strength of the k–th centre, assuming s⃗_{k} ≠ s⃗_{l} for k ≠ l. Then the Hamiltonian function

Ĥ : T^{∗}M̂ → ℝ , Ĥ (p⃗, q⃗) = \frac{1}{2} p⃗^{2} + V (q⃗),

∗Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstr. 1\frac{1}{2}, D–91054 Erlangen, Germany. e-mail: knauf@mi.uni-erlangen.de

†Institute of Mathematics, 630090 Novosibirsk, Russia. e-mail: taimanov@math.nsc.ru

with potential

V : M̂ → ℝ , V (q⃗) = − ∑_{k=1}^{n} \frac{Z_{k}}{∥q⃗ − s⃗_{k}∥},  (1)

on the cotangent bundle T^{∗}M̂ of configuration space

M̂ := ℝ^{d} \ {s⃗_{1}, …, s⃗_{n}}

generates a – in general incomplete – flow.

Denoting by ω̂ := ∑_{i=1}^{d} dq_{i} ∧ dp_{i}↾_{T^{∗}M̂} the restricted canonical symplectic form, there exists a unique smooth extension

(P, ω, H) of the hamiltonian system (T^{∗}M̂ , ω̂, Ĥ)  (2)

such that the flow Φ : ℝ × P → P of H is complete (see [Kn], Thm. 5.1). Concerning integrability of the flow, the following is known:

• For n = 1 this Hamiltonian system is integrable, with angular momentum

L⃗ : P → ℝ^{3} , L⃗(x) = \{ \begin{matrix}(q⃗ − s⃗_{1}) × p⃗ & , & x = (p⃗, q⃗), q⃗ ≠ s⃗_{1} \\ 0 & , & otherwise \end{matrix}

for dimension d = 3 being a real analytic constant of motion. For Z_{1} > 0 this is called the Kepler problem.

• For n = 2 centres one introduces elliptic prolate coordinates to analytically integrate the flow Φ, see e.g. [Ar] or [Th].

• For n ≥ 3 centres and d = 2 Bolotin showed in [Bo] the nonexistence of an analytic integral of the motion which is non–constant on an energy shell H^{−1}(E), E > 0, see also the discussion in Fomenko [Fo].

• For d = 3 and a collinear configuration of centres the angular momentum w.r.t. that axis is an additional constant of the motion, independent of the number n of centres. However, for d = 3 it was proved by the first author [Kn] for sufficiently large energies E > E_{th} and by Bolotin and Negrini [BN, BN2] for nonnegative energies E ≥ 0 that the topological entropy of the flow, restricted to the set b_{E} of bounded orbits on H^{−1}(E) is positive (and h_{top}(E) = 0 for b_{E} = ∅). Furthermore h_{top}(E) is zero for n = 1 and 2, and h_{top}(E) > 0 if n ≥ 3 and all centres being attracting or not more than two s⃗_{k} being on a line (for collinear configurations with Z_{1}, …, Z_{n} < 0 one has h_{top}(E) = 0 for E > 0).

In the present paper it is proved that

• for d = 2, attracting centres (Z_{k} > 0) and E > 0,

• for d = 3, arbitrary Z_{k} ≠ 0, non-collinear configurations of centres and E > E_{th} where the threshold energy level E_{th} is determined by the data Z_{k}, s⃗_{k}, k = 1, …, n,

the n-centre problem restricted onto the energy level E admits d−1 independent integrals of motion which are smooth (and moreover are of the Gevrey class g for any g > 1) (see Thm. 1). Therefore the restricted problem is smoothly integrable.

On the other hand if the affine span of the (non-collinear) centres equals ℝ^{3} then the restricted problem is not integrable in the real-analytic sense, that is Gevrey class 1 (see Thm. 2).

The article is based on the analysis of the n-centre problem given in the paper by the first author [Kn] where, in particular, it is shown that for high energies there are maps which relate scattering orbits to their asymptotics given by scattering orbits of the Kepler problem which is integrable (the necessary facts extracted from [Kn] are exposed in sections 2–3 of the present paper). Such a relation of the n-centre problem with an integrable problem leads to the smooth integrability of the n-centre problem which is proved here. Here we use a trick for constructing smooth first integrals from discontinuous preserved quantities similar to others used in [Bu, BT]

We would like to notice that in [Bo] the nonexistence of an additional analytic integral of motion for the two-dimensional problem was derived by Bolotin from Kozlov’s theorem [Ko] which reads that the geodesic flow of (real)-analytic metric on a closed oriented surface of genus g > 1 does not admit an additional analytic first integral.

Although one can expect that the integrability of a problem by analytic integrals of motion implies vanishing of the topological entropy it is not proved until now. Therefore the results from [BN, Kn] do not imply the nonexistence of a complete family of analytic first integrals. We prove that by an analysis of the set formed by bounded orbits which supports the restricted flow with positive topological entropy.

2 Known Smoothness Results

There are three basic types of motion: bounded, scattering and trapped, corresponding to the disjoint subsets b, s, t ⊂ P with

b^{±} := {x ∈ P | q⃗ (±ℝ^{+}, x) is bounded } , b := b^{+} ∩ b^{−},
s^{±} := {x ∈ P | x ∉ b^{±} and H(x) > 0} , s := s^{+} ∩ s^{−}

and t := s^{+}△s^{−}. The orbits in s go to spatial infinity in both time directions, but because of the long range character of the effective potential of strength Z_{∞} := ∑_{k=1}^{n} Z_{k}, one describes these limits by comparison with (regularized) Kepler flow Φ_{∞}^{t} : P_{∞} → P_{∞} generated by the extension of

Ĥ_{∞} : T^{∗}(ℝ^{d} \ \{0\}) → ℝ , Ĥ_{∞}(p⃗, q⃗) := \frac{1}{2}p^{2} − \frac{Z_{∞}}{\left∥{ \atop q}\right∥}  (3)

Identifying the two phase spaces P and P_{∞} outside a region projecting to a ball in configuration space which contains all singularities, the Møller transformations

Ω^{±} : P_{∞,+} := {x ∈ P_{∞} | H_{∞}(x) > 0} → s^{±} , Ω^{±} := lim_{t→±∞} Φ^{−t} ∘ Id ∘ Φ_{∞}^{t} (4)

exist as pointwise limits, and are measure-preserving diffeomorphisms (Thm. 6,3 and 6.5 of [Kn]). Similarly the asymptotic limits of the momentum

p^{±} : s^{±} → ℝ^{d} , p^{±}(x_{0}) := lim_{t→±∞} p⃗ ∘ Φ^{t}(x_{0})  (5)

are smooth. Finally we define time delay τ : s → ℝ of a scattering state x ∈ s by

τ(x) := lim_{R→∞} ∫_{ℝ} (σ(R) ∘ Φ^{t}(x) −  (6)
\frac{1}{2}(σ_{∞}(R) ∘ Φ_{∞}^{t} ∘ Ω_{∗}^{+}(x) + σ_{∞}(R) ∘ Φ_{∞}^{t} ∘ Ω_{∗}^{−}(x)))dt,

where σ(R) : P → {0, 1} and σ_{∞}(R) : P_{∞} → {0, 1} are the characteristic functions σ(R)(x) := θ(R − ∥q⃗(x)∥) and similarly for σ_{∞}(R). That asymptotic difference between the time spent by the orbit and its Kepler limits inside a ball of large radius diverges near b ∪ t. However τ is smooth, as the Møller transformations are.

3 Analyticity Properties

Proposition 1 For a potential V of the form (1) the following maps are real-analytic:

1. the flow Φ : ℝ × P → P

2. the Møller transformations Ω^{±} : P_{∞,+} → s^{±} and asymptotic momenta p^{±} : s^{±} → ℝ^{d}

3. the time delay τ : s → ℝ.

Proof.

1) We first indicate the definition of phase space P in order to show that for the potential (1) the smooth extension (2) actually works in the real-analytic category (in [Kn] more general non-analytic potentials V were considered). We assume d = 3, the case of d = 2 dimensions following by restriction.

For small ε > 0 in the phase space neighbourhood

Û_{l}^{ε} := \{(p⃗, q⃗) ∈ T^{∗}M̂ |∥q⃗ − s⃗_{l}∥ < ε, |p⃗|^{2} > \frac{3}{2} \frac{Z_{l}}{∥q⃗ − s⃗_{l}∥} \} .  (7)

of the lth centre the following real-analytic coordinates are used:

• The restriction of the Hamiltonian function, which splits into

Ĥ(p⃗, q⃗) = Ĥ_{l}(p⃗, q⃗) + W_{l}(q⃗) with Ĥ_{l}(p⃗, q⃗) := \frac{1}{2}p^{2} − \frac{Z_{l}}{∥q⃗ − s⃗_{l}∥} (8)

and

W_{l}(q⃗) = ∑_{i≠l} \frac{−Z_{i}}{∥q⃗ − s⃗_{i}∥}.

• The angular momentum

L̂_{l} : Û_{l}^{ε} → ℝ^{3} , L̂_{l}(p⃗, q⃗) := (q⃗ − s⃗_{l}) × p⃗.  (9)

relative to the position s⃗_{l}.

• The time T̂_{l} : Û_{l}^{ε} → ℝ after which the Kepler orbit generated by Ĥ_{l} is in its pericentre w.r.t. s⃗_{l}

T̂_{l}(p⃗, q⃗) := ∫_{r_{min}(p⃗,q⃗)}^{∥q⃗−s⃗_{l}∥} \frac{r dr}{\sqrt{2r^{2}Ĥ_{l}(p⃗, q⃗) + 2Z_{l}r − L̂_{l}^{2}(p⃗, q⃗)}} · sign((q⃗ − s⃗_{l}) · p⃗).

(10)

The neighbourhood Û_{l}^{ε} ⊂ T^{∗}M̂ is defined in a way which makes the pericentre unique. Its distance from s⃗_{l} equals

r_{min}(p⃗, q⃗) := \{ \begin{matrix}\frac{−Z_{l}+\sqrt{Z_{l}^{2}+2Ĥ_{l}(p⃗,q⃗)L̂_{l}^{2}(p⃗,q⃗)}}{2Ĥ_{l}(p⃗,q⃗)} & , Ĥ_{l} ≠ 0 \\ L̂_{l}^{2}(p⃗, q⃗)/2Z_{l} & , Ĥ_{l} = 0 \end{matrix} .  (11)

• The Runge-Lenz vector relative to s⃗_{l} is given by

F⃗_{l} : Û_{l}^{ε} → ℝ^{3} , F⃗_{l}(p⃗, q⃗) := p⃗ × L̂_{l}(p⃗, q⃗) − Z_{l} \frac{q⃗ − s⃗_{l}}{∥q⃗ − s⃗_{l}∥}.

Since |F⃗_{l}|^{2} > Z_{l}^{2}/4 > 0, the pericentral direction

F̂_{l} : Û_{l}^{ε} → S^{2} , F̂_{l} := F⃗_{l}/|F⃗_{l}|  (12)

is well-defined.

As L̂_{l} · F̂_{l} = 0, we get a real-analytic diffeomorphism onto the image

Ŷ : Û_{l}^{ε} → T^{∗}(ℝ × S^{2}) \ ¯0, (p⃗, q⃗) ↦ (T̂_{l}, L̂_{l}; Ĥ_{l}, F̂_{l})

onto a punctured neighbourhood of the zero section ¯0 of the cotangent bundle. The flow generated by Ĥ_{l} is linearized in these coordinates, and extended on U_{l}^{ε} := Û_{l}^{ε} ∪ (ℝ × S^{2}) by

Y := (T_{l}, L_{l}; H_{l}, F_{l}) : U_{l}^{ε} → T^{∗}(ℝ×S^{2}) , Y(x) := \{ \begin{matrix}(0, 0; h, f) & (h, f) ∈ ℝ × S^{2} \\ Ŷ(x) & otherwise \end{matrix}

to a full neighbourhood of the zero section.

In a final step the hamiltonian flow Φ : ℝ × P → P generated the continuous extension H : P → ℝ of the Hamiltonian function Ĥ : T^{∗}M̂ → ℝ is linearized by slightly changing the coordinates Y using the (real-analytic but incomplete) flow generated by Ĥ. See [Kn] for details.

In summary, we obtain a real-analytic extension (2) of the Hamiltonian system (T^{∗}M̂ , ω̂, Ĥ). Thus the Hamiltonian vector field X_{H} (defined by i_{X_{H}}ω = dH) is real-analytic, too, and thus the flow Φ : ℝ × P → P is known to be real-analytic (see, e.g. [Ho]), proving assertion 1.

2) This, however is insufficient to show real-analyticity of the Møller transformations and asymptotic momenta. It is known that even for smooth potentials these maps may be very non-smooth, see [Si].

There exists an energy-dependent virial radius R_{vir} > 0, with

\frac{d}{dt} ⟨q⃗, p⃗⟩ > \frac{E}{2} > 0 if \left∥{ \atop q}\right∥ ≥ R_{vir}(E) and E := H(p⃗, q⃗).  (13)

If we assume q_{0} := ∥q⃗_{0}∥ ≥ R_{vir}(E) and ⟨q⃗_{0}, p⃗_{0}⟩ ≥ 0, then

∥q⃗(t)∥ ≥ q_{0} · ⟨t⟩_{λ} for all t ≥ 0, with λ := \sqrt{E/2}/q_{0}  (14)

and

⟨t⟩_{λ} := \sqrt{1 + (λt)^{2}} , ⟨t⟩ := ⟨t⟩_{1}.

In particular a trajectory leaving the ball of radius R_{vir}(E) cannot reenter this ball in the future but must go to spatial infinity.

Analyticity estimates for the trajectory are then derived from the integral equation with initial conditions x_{0} = (p⃗_{0}, q⃗_{0})

q⃗(t, x_{0}) = q⃗_{0} + tp⃗_{0} − ∫_{0}^{t} ∫_{0}^{s} ∇V (q⃗(τ, x_{0})) dτ ds,

using the decay property of the potential

|∂^{β}V (q⃗)| ≤ |β|! (\frac{C}{∥q⃗∥})^{|β|+1} (β ∈ ℕ_{0}^{3})

outside the virial radius, and (14).

Setting

∥w⃗∥_{λ} := sup_{t≥0} \frac{∥w⃗(t)∥}{⟨t⟩_{λ}}

we obtain for γ := (α, β), ∂_{x_{0}}^{γ} := ∂_{p_{0}}^{α}∂_{q_{0}}^{β}

∥∂_{x_{0}}^{γ}q⃗(·, x_{0})∥_{λ} ≤ α! q_{0}^{−|β|+δ_{|γ|,1}}E^{− \frac{1}{2} |α|−1+δ_{|γ|,1}}.  (15)

This allows us to perform the (locally uniform w.r.t. x_{0}) time limit in

∂_{x_{0}}^{γ}(p⃗(t, x_{0}) − p⃗_{0}) = −  (16)

∑_{N=1}^{g} ∑_{|γ^{(i)}|>0^{γ^{(1)}+...+γ^{(N)}=γ}} ∫_{0}^{t} D^{N}∇V (q⃗(τ, x_{0})) (∂_{x_{0}}^{γ^{(1)}} q⃗(τ, x_{0}), …, ∂_{x_{0}}^{γ^{(N)}} q⃗(τ, x_{0})) dτ.

with g := |γ| ≥ 1, and to conclude that p^{+} is real-analytic at x_{0}. We can substitute the assumption x_{0} ∈ s^{+} for the stronger assumptions q_{0} := ∥q_{0}∥ ≥ R_{vir}(E), ⟨q⃗_{0}, p⃗_{0}⟩ ≥ 0, as initial conditions meeting the first lead to data meeting the second one after some t. The same holds for p^{−} by reversibility (p^{−}(p⃗_{0}, q⃗_{0}) = −p^{+}(−p⃗_{0}, q⃗_{0})).

The proof of real-analyticity of the Møller transforms is based on the integral equation

∂_{x_{0}}^{γ}r⃗(t) = ∫_{t}^{∞} ∫_{s}^{∞} ∂_{x_{0}}^{γ}∇ (\frac{−Z_{∞}}{∥Q⃗(τ; x_{0})∥} − V (q⃗(τ, x_{0}))) dτ ds

for r⃗(t; x_{0}) := q⃗(t, x_{0})−Q⃗(t; x_{0}) with Kepler hyperbola (P⃗ (t; x_{0}), Q⃗(t; x_{0})) = Φ_{∞}^{t}(X_{0}). Inspecting the proof of smoothness for the Møller transforms in [Kn], one sees that the estimates for ∂_{x_{0}}^{γ}r⃗ can be dominated by |γ|!C^{|γ|}.

3) For a scattering state x ∈ s time delay equals

τ(x) = \frac{1}{2} lim_{R→∞} ∫_{ℝ} σ^{+} ∘ Φ^{t}(x) · (σ_{∞}(R) ∘ Ω_{∗}^{+} − σ_{∞}(R) ∘ Ω_{∗}^{−}) (Φ^{t}(x))dt
+ \frac{1}{2} lim_{R→∞} ∫_{ℝ} σ^{−} ∘ Φ^{t}(x) · (σ_{∞}(R) ∘ Ω_{∗}^{−} − σ_{∞}(R) ∘ Ω_{∗}^{+}) (Φ^{t}(x))dt,

with σ^{±}(p⃗, q⃗) := θ(±q⃗·p⃗), see [KK] and Ω_{∗}^{±} := (Ω^{±})^{−1}. Using the intertwining property Ω_{∗}^{±} ∘ Φ^{t} = Φ_{∞}^{t} ∘ Ω_{∗}^{±} this can be written entirely in terms of Møller transformations and the Kepler flow:

τ(x) = \frac{1}{2} lim_{R→∞} ∫_{ℝ} σ^{+} ∘ Φ_{∞}^{t}(x) · (σ_{∞}(R) ∘ Φ_{∞}^{t} ∘ Ω_{∗}^{+}(x) − σ_{∞}(R) ∘ Φ_{∞}^{t} ∘ Ω_{∗}^{−}(x)) dt
− \frac{1}{2} lim_{R→∞} ∫_{ℝ} σ^{−} ∘ Φ_{∞}^{t}(x) · (σ_{∞}(R) ∘ Φ_{∞}^{t} ∘ Ω_{∗}^{+}(x) − σ_{∞}(R) ∘ Φ_{∞}^{t} ∘ Ω_{∗}^{−}(x)) dt.

With Assertion 2 and the analog of formula (10) this implies analyticity of τ. □

4 Gevrey Integrals of Motion

We now show the existence of independent constants of motion for all energies E > E_{th}. In [Kn] many estimates are shown to hold true above a threshold energy E_{th} ≥ 0:

In [Kn] many estimates are shown to hold true above a threshold energy E_{th} ≥ 0:

• For d = 2 and attracting centres (Z_{k} > 0) E_{th} = 0.

• For d = 3, arbitrary Z_{k} ≠ 0 and non-collinear configurations of the s⃗_{k} ∈ ℝ^{3} the existence of such a threshold is proven.

In particular the set b_{E} of bounded orbits of energy E > E_{th} is shown to be of measure zero (and has a Cantor set structure for n ≥ 3).

According to the standard definition (see, e.g. [AM], Def. 5.2.20) functions f_{0}, …, f_{k} : P̃ → ℝ on a symplectic manifold (P̃, ω̃) are called independent if the set of singular points of F := f_{0} × … × f_{k} : P̃ → ℝ^{k} has measure zero.

To simplify discussion, we set

P̃ := H^{−1}([E_{1}, E_{2}])

for an arbitrary energy interval, E_{th} ≤ E_{1} ≤ E_{2} < ∞, and f_{0}^{g} ≡ f_{0} := H|P̃. Then for a parameter g > 1 we define for k = 1, 2

f_{k}^{g} : P̃ → ℝ , f_{k}^{g}(x) := \{ \begin{matrix}p_{k}^{+}(x) exp (−e^{C(g)⟨τ(x)⟩}) & , & x ∈ s \\ 0 & , & x ∉ s, \end{matrix}  (17)

where C(g) = \frac{C_{2}}{g−1} with C_{2} to be defined below. Putting things together we get a map F^{g} : P̃ → ℝ^{3}.

The notation is chosen so that F^{g} belongs to the Gevrey class of index g. Now we collect some information about Gevrey functions, which were

introduced in [Ge].

Definition 1 For g ≥ 1 and an open set Ω ⊂ ℝ^{n} a function f ∈ C^{∞}(Ω) is called of Gevrey class g if for every compact K ⊂ Ω there exist A_{K}, C_{K} > 0 with

max_{x∈K}|∂^{α}f(x)| ≤ A_{K} C_{K}^{|α|} (α!)^{g} (α ∈ ℕ_{0}^{n}).

Here |α| = α_{1} + … + α_{n} and α! = α_{1}! · . . . · α_{n}! if α = (α_{1}, …, α_{n}). The vector space of these functions is denoted by G_{g}(Ω).

Then G_{1}(Ω) is the space of real-analytic functions, and G_{g′}(Ω) ⊃ G_{g}(Ω) for g′ > g. G_{g}(Ω) is stable w.r.t. partial derivatives, compositions, and the implicit function theorem holds within the class. For g > 1 Gevrey partitions of unity exist.

Example 1 For g > 1

f^{g}(x) := \{ \begin{matrix}exp(−x^{−1/(g−1)}) & , & x > 0 \\ 0 & , & x ≤ 0, \end{matrix}

is a function in G_{g}(ℝ) but not in G_{g′}(ℝ) for any g′ < g, see e.g. [Ju].

For a real-analytic manifold we can define the spaces of Gevrey functions, as the defining family of bounds is preserved by coordinate changes.

Theorem 1 For all g > 1 the functions f_{0}^{g}, f_{1}^{g}, f_{2}^{g} are independent, in involution and of Gevrey class g.

Proof. In Thm. 12.8 of [Kn] it was shown that the set b_{E} of bounded orbits of energy E is of Liouville measure zero for all E > E_{th}. As the set t_{E} := {x ∈ H^{−1}(E) | x ∈ T } of trapped orbits is always of Liouville measure zero, on P̃ the set P̃ \ s is of measure zero. On P̃ ∩ s the functions f_{1}^{g} and f_{2}^{g} are real-analytic, using Proposition 1, whereas they are zero on P̃\s.

We study their decay near P̃ \ s in order to prove that they are in G^{g}(P̃). All orbits in P̃ \ s enter the interaction zone, so we need only orbits in

P̃ ∩ s entering the interaction zone. W.l.o.g. we assume R_{vir} to be constant on [E_{1}, E_{2}]. Then the region

Ĩ:= {x ∈ P̃ | ∥q(x)∥ ≤ R_{vir}}

projecting to the interaction zone is compact. The restriction of the realanalytic flow Φ : ℝ × P → ℝ to a domain of the form [−ε, ε] × Ĩ thus has partial derivatives |∂^{α}Φ(t, x)| ≤ C̃_{1}C̃_{2}^{|α|}α!. We conclude, using Corollary 1 that for arbitrary t ∈ ℝ such that Φ(t, x) ∈ Ĩ, too,

|∂^{α}Φ(t, x)| ≤ C_{1} exp(C_{2}|α| ⟨t⟩) α!.  (18)

Next we analyse time delay (6) for orbits entering Ĩ. By compactness of Ĩ there exists a R̃ ≥ R_{vir} such that all Kepler orbits {Φ_{∞}^{t} ∘ Ω_{∗}^{±}(x) | t ∈ ℝ}

s⃗_{n} s⃗_{1} R_{vir} R

Figure 1: Scattering trajectory in configuration space (solid line), with associated Kepler hyperbolae (broken lines)

through points x ∈ Ĩ enter a configuration space region of radius R̃. In (6) we only consider radii R ≥ R̃ and, denoting by τ_{R}(x) the integral (6), we split this quantity in

τ_{R}(x) = τ_{R}^{+}(x) + τ_{R}^{−}(x) + τ^{0}(x).

Here

τ_{R}^{±}(x) := ∫_{ℝ} (σ^{+} ∘ Φ_{t}(x) · σ(R)(1 − σ(R_{vir})) ∘ Φ^{t}(x)
−\frac{1}{2}σ_{∞}(R) ∘ Φ_{∞}^{t} ∘ Ω_{∗}^{±}(x)) dt  (19)

and

τ^{0}(x) := ∫_{ℝ} σ(R_{vir}) ∘ Φ^{t}(x) dt.  (20)

Whereas the limits τ^{±}(x) := lim_{R→∞} τ_{R}^{±}(x) of (19) meet Gevrey class 1 estimates uniform on Ĩ∩ s, τ^{0}(x) defined in (20), the time spent by the orbit in the interaction zone, is not uniformly bounded on Ĩ∩ s for n ≥ 2 centres, see Figure 1.

We need only consider scattering states in a slightly smaller domain Ĩ_{ε} := {x ∈ P̃ | ∥q(x)∥ ≤ R_{vir} − ε}. By (13) the configuration space trajectories with initial conditions in Ĩ_{ε} ∩s intersect the boundary of the interaction zone with angles uniformly bounded away from zero. Therefore (18) shows that for x ∈ Ĩ_{ε} ∩ s

|∂^{α}τ^{0}(x)| ≤ C_{1} exp(C_{2}|α| ⟨τ(x)⟩) α!,

and τ = τ^{+} + τ^{−} + τ^{0} satisfies the same kind of estimate (with enlarged constants).

A similar kind of reasoning applies to the asymptotic momenta p^{±}:

|∂^{α}p_{k}^{+}(x)| ≤ C_{1} exp(C_{2}|α| ⟨τ(x)⟩) α!

for x ∈ Ĩ_{ε} ∩ s.

¿From this one can conclude that f_{1}^{g} and f_{2}^{g} are of Gevrey class g for C(g) large enough, see Proposition 3.

The functions p^{±} and τ are Φ_{t}–invariant. Therefore f_{1}^{g} and f_{2}^{g} are Φ_{t}– invariant, too. As f_{0}^{g} = H \ P̃ generates the flow Φ_{t} on P̃, the Poisson brackets {f_{0}^{g}, f_{1}^{g}} and {f_{0}^{g}, f_{2}^{g}} vanish.

By definition p^{±}(x) = lim_{t→±∞} p ∘ Φ_{t}(x), and the decay estimates (16) imply that

{ lim_{t→∞} p_{1} ∘ Φ_{t}(x), lim_{t→∞} p_{2} ∘ Φ_{t}(x)} = lim_{t→∞}{p_{1}, p_{2}} ∘ Φ_{t}(x) = 0.

Thus the Poisson bracket {f_{1}^{g}, f_{2}^{g}} vanishes, too.

To see that the functions f_{k}^{g} are independent, it suffices to show that the (analytic) restriction

F^{g}|_{P̃∩s}

has a measure zero set of singular points. As the energy range [E_{1}, E_{2}] under consideration consists of regular values of H, we may consider instead the independence of f_{1}^{g} and f_{2}^{g}, restricted to H^{−1}(E) ∩ s, for E ∈ [E_{1}, E_{2}].

Independence of f_{1}^{g} and f_{2}^{g}, restricted to an energy surface H^{−1}(E) with E ∈ [E_{1}, E_{2}] follows from their real-analyticity on H^{−1}(E) ∩ s and the independence of p_{1}^{+} and p_{2}^{+}, remarking that for ⟨q⃗, p⃗⟩ → ∞, p^{+}((p⃗, q⃗)) → p⃗. □

Proposition 2 Consider real-analytic maps

f : Ω_{f} → ℝ^{d} , g : Ω_{g} → Ω_{f}

with open sets Ω_{f}, Ω_{d} ⊂ ℝ^{d}, meeting

|∂^{α}f_{i}| ≤ M_{f} C_{f}^{|α|} α! , |∂^{α}g_{i}| ≤ M_{g} C_{g}^{|α|} α! (i = 1 , …, d, α ∈ ℕ_{0}^{d}).

Then

|∂^{α}(f ∘ g)_{i}| ≤ M_{f} C_{fg}^{|α|} α! with C_{fg} := C_{g}(1 + C_{f}dM_{g})d.

Proof. The chain rule has the form

∂^{α}f ∘ g = ∑_{k=1}^{n} ∑_{A_{1},...,A_{k}} D^{k}f(∂^{α_{A_{1}}}g, …, ∂^{α_{A_{k}}} g)

with n = |α|, (A_{1}, …, A_{k}) running through the k–partitions of {1, …, n} and α_{A_{l}} ∈ ℕ_{0}^{d} is the multiindex of size |α_{A_{l}}| = |A_{l}| corresponding to A_{l} ⊂ {1, …, n} in a, say lexicographic, ordering of the n partial derivatives. Therefore with λ := C_{f}dM_{g}

|∂^{α}(f ∘ g)| ≤ ∑_{k=1}^{n} M_{f}C_{f}^{k}d^{k}k! ∑_{A_{1},...,A_{k}} ∏_{l=1}^{k} M_{g}C_{g}^{|A_{l}|}α_{A_{l}}!

= M_{f}C_{g}^{n} ∑_{k=1}^{n}(C_{f}dM_{g})^{k}k! ∑_{A_{1},...,A_{k}} ∏_{l=1}^{k} α_{A_{l}}!

≤ M_{f}C_{g}^{n} ∑_{k=1}^{n} λ^{k}k! ∑_{A_{1},...,A_{k}} ∏_{l=1}^{k} |A_{l}|!

= M_{f}C_{g}^{n}n! ∑_{k=1}^{n} λ^{k}\binom{n−1}{k−1}
= M_{f}C_{g}^{n}λ(1 + λ)^{n−1}n!,

since k! ∑_{A_{1},...,A_{k}} ∏_{l=1}^{k} |A_{l}|! = n!\binom{n−1}{k−1}. As in d dimensions

|α|! ≤ d^{|α|} α! (α ∈ ℕ_{0}^{d}),

the estimate follows. □

This estimate is iterated in order to estimate the flow for long times:

Corollary 1 Assume that for k = 1, …, t the maps

f^{(k)} : Ω^{(k)} → f^{(k)}(Ω^{(k)}) ⊂ Ω^{(k+1)} on Ω^{(k)} ⊂ ℝ^{d}

admit the uniform estimates |∂^{α}f_{i}^{(k)}| ≤ MC̃^{|α|} α! with M ≥ 1.

Then their iterates T^{(k)} := f^{(k)} ∘ T^{(k−1)} with T^{(0)} := Id_{Ω_{1}} are estimated by

|∂^{α}T_{i}^{(k)}| ≤ M exp(C |α| k) α!  (α ∈ ℕ_{0}^{d}, k = 1, …, d)  (21)

with C := ln(d) + MdC̃.

Proof. (21) holds for k = 1, and is assumed to hold for some k < t. Then by Proposition 2

|∂^{α}T_{i}^{(k+1)}| ≤ M(exp(Ck)(1 + MdC̃)d)^{|α|} α!
≤ M exp(C(k + 1))^{|α|} α!,

proving (21). □

Proposition 3 For the choice C(g) := \frac{C_{2}}{g−1} in (17) the functions f_{k}^{g} are of Gevrey class g.

Proof. As |∂^{α} ⟨τ(x)⟩ | ≤ C_{1} exp(C_{2}|α| ⟨τ(x)⟩) α!,

|∂^{α} exp(C(g) ⟨τ(x)⟩)| ≤ exp (C(g) ⟨τ(x)⟩ ) ∑_{k=1}^{|α|} C(g)^{k} ∑_{A_{1},...,A_{k}} ∏_{l=1}^{k} ∂^{α_{A_{l}}} ⟨τ⟩

≤ exp ((C(g) + C_{2}n) ⟨τ(x)⟩ ) (∑_{k=1}^{n} C(g)^{k}C_{1}^{k}\binom{n−1}{k−1}) α!
= exp ((C(g) + C_{2}n) ⟨τ(x)⟩ )(1 + C(g)C_{1})^{n} α!

with n := |α|.

The ⟨τ⟩ dependent part of |∂^{α}f_{k}^{g}| is of the form

exp (−e^{C(g)⟨τ⟩}) e^{C_{2}n⟨τ⟩} ≤ x^{x},

as it has its maximum for ⟨τ⟩ with

x := e^{C(g)⟨τ⟩} = \frac{C_{2}n}{C(g)}.

Thus the choice C(g) := \frac{C_{2}}{g−1} leads to the proof. □

5 Nonexistence of Analytic Integrals of Motion

We start with the following simple observation for a hamiltonian system (P, ω, H):

Proposition 4 Let γ ⊂ Σ_{E} := H^{−1}(E) be a periodic orbit of the Hamiltonian flow Φ_{t} generated by the Hamiltonian function H which is isolated on the energy surface Σ_{E}. Then there is no additional integral of motion which is functionally independent of H on γ.

Proof. Let us assume that there is an additional integral of motion J : P → ℝ. By localizing J around γ if necessary, the Hamiltonian flow Ψ_{s} generated by J exists for all times s ∈ ℝ. Since these integrals are in involution, their flows commute:

Φ_{t}Ψ_{s} = Ψ_{s}Φ_{t} (t, s ∈ ℝ).

Let T be the period of γ and x ∈ γ. We have

Φ_{T} Ψ_{s}(x) = Ψ_{s}Φ_{T} (x) = Ψ_{s}(x) (s ∈ ℝ).

This implies that Ψ_{s} maps (within a given energy surface Σ_{E}) periodic orbits of the flow Φ_{t} into periodic orbits of this flow. But γ ⊂ Σ_{E} is an isolated periodic orbit and therefore Ψ_{s}(x) ∈ γ for all s ∈ ℝ. This implies that H and J are functionally dependent on γ. □

Remark 1 Note that single isolated periodic orbits need not form an obstruction to the existence of an additional analytic integral of motion J : P → ℝ, independent of H in the sense defined in Sect. 4. If the periodic orbit γ is hyperbolic, then J must be constant on its stable and unstable manifold, but still on a neighbourhood of γ ⊂ Σ_{E} the singular set of J|_{Σ_{E}} may only consist of γ which is of measure zero.

Easy examples for this are given by motion in a smooth potential V : ℝ^{2} → ℝ which are rotationally symmetric. Then angular momentum is an independent analytic integral of the motion, but circular hyperbolic orbits may exist for some energies.

In the present context the energy surface Σ_{E} (E > E_{th}) for the two–centre problem contains exactly one bounded orbit γ. γ is periodic and collides with the two centres. As commented in the Introduction, the two–centre problem is analytically integrable.

Let us return to the n-centre problem. By Thm. 12.8 from [Kn] the set of bounded orbits b_{E} on the energy level E > E_{th} consists in hyperbolic trajectories, and for n ≥ 3 is locally homeomorphic to a product of a Cantor set and the interval. Moreover its Liouville measure vanishes. We now show that b_{E} can form an obstruction to the existence of independent analytic integrals:

Theorem 2 If the affine span of the (non-collinear) centres s⃗_{1}, …, s⃗_{n} equals ℝ^{3}, then for E > E′ ≥ E_{th} the n–centre problem does not admit a pair of independent analytic integrals I_{1}, I_{2} : Σ_{E} → ℝ of motion on the energy surface Σ_{E} := H^{−1}(E).

Proof. Assume that I_{1}, I_{2} : Σ_{E} → ℝ are analytic integrals of motion. Then on the five–dimensional manifold Σ_{E} we consider the singular set

S := {x ∈ Σ_{E} | rank(DI(x)) < 2}

with I := (I_{1}, I_{2}).

By Prop. 4 the periodic orbits within b_{E} belong to the (closed) set S (as in Prop. 4 we consider functions J : P → ℝ, we use appropriate smooth extensions J of I_{k} : Σ_{E} → ℝ).

In [Kn] it was shown that for energies E > E_{th} the periodic orbits are dense within b_{E}. Thus b_{E} ⊂ S, too.

According to [vdD] (see also [Ta] where a sketch for more general subanalytic sets appears) S admits an analytic simplicial decomposition which is locally finite and whose simplices are semianalytic. As b_{E} ⊂ S is compact, there exist (disjoint) simplices Δ_{1}, …, Δ_{m} with

b_{E} ⊂ ⋃_{i=1}^{m} Δ_{i}.  (22)

Now we assume I_{1}, I_{2} to be indendent integrals, contrary to the statement of the theorem. Then K := max_{i} dim(Δ_{i}) < dim(Σ_{E}) = 5. As the set S is Φ_{t}–invariant, we consider the transversal intersections

Δ̃_{i} := Δ_{i} ∩ H_{E}  (23)

of the simplices with the Poincaré surface H_{E} ⊂ Σ_{E} (defined in Sect. 10 of [Kn]). We have K̃ := max_{i} dim(Δ̃_{i}) = K − 1 < dim(H_{E}) = 4. The intersection Λ_{E} := b_{E} ∩ H_{E} of the bounded orbits with the Poincaré surface has the form

Λ_{E} = Λ_{E}^{+} ∩ Λ_{E}^{−},

Λ_{E}^{±} ⊂ H_{E} being the (un–) stable manifolds, consisting of two–dimensional leaves which intersect transversally.

It is known from Hasselblatt [Ha] that the Hölder regularity of (the distributions of) Λ_{E}^{±} can be controlled by the so-called bunching constant. For the case of the n-centre problem Prop. 11.2 of [Kn] controls the expansion resp. contraction rates of the Poincaré map on Λ_{E}^{±}, which differ from a constant times E by O(E^{0}). Thus applying [Ha] we have C^{2−ε} regularity of Λ_{E}^{±} for all large energies E > E_{th}.

Now as K̃ < 4, for each Δ̃_{i} in (23) at least one of the intersections

Δ_{i}^{±} := Δ̃_{i} ∩ Λ_{E}^{±}

must be of dimension ≤ 1. By reversibility of the flow we assume w.l.o.g. that dim(Δ_{1}^{−}) ≤ 1 and derive a contradiction.

As Λ_{E} is a Cantor set (see Thm. 12.8 of [Kn]), we assume w.l.o.g. that Δ_{1}^{−} contains a sequence (x^{(i)})_{i∈ℕ} of points x^{(i)} ∈ Λ_{E} converging to x ∈ Δ_{1}^{−}. By going to a subsequence, if necessary, we assume that the unit vectors v^{(i)} ∈ T_{x}H_{E} with exp_{x}(t^{(i)}v^{(i)}) = x^{(i)} have a limit v := lim_{i→∞} v^{(i)} (which is in fact independent of the choice of Riemannian metric used to define the exponential map).

In [Kn] starting from the alphabet S := {1, …, n} the space of symbol sequences

X := \{\underline{k} ∈ S^{ℤ} | ∀ i ∈ ℤ : k_{i+1} ≠ k_{i}\}

is equipped with the metric

d(\underline{k}, \underline{l}) := ∑_{i∈ℤ}2^{−|i|} · (1 − δ_{k_{i},l_{i}}), (\underline{k}, \underline{l} ∈ X).

Then denoting the shift by

σ : X → X , σ(\underline{k})_{i} := k_{i+1} (i ∈ ℤ),

there exists a Hölder homeomorphism

F_{E} : X → Λ_{E}  (24)

conjugating σ and the restriction of the Poincaré map to Λ_{E}, see Lemma 12.2 of [Kn].

The Poincaré surface is the disjoint union

H_{E} = ⋃_{k≠l^{k,l=1}}^{n} H_{E}^{k,l},

the H_{E}^{k,l} being open regions in the intersection of a five-dimensional affine space and Σ_{E}. Starting with

V_{E}(k_{0}, k_{1}) := W_{E}(k_{0}, k_{1}) := H_{E}^{k_{0},k_{1}},

in [Kn] for (k_{−m}, …, k_{0}) admissible (that is k_{l} ≠ k_{l+1}) the nested subsets

W_{E}(k_{−m}, …, k_{0}) := W_{E}(k_{−1}, k_{0}) ∩ P_{E}(W_{E}(k_{−m}, …, k_{−1})),  (25)

resp. for (k_{0}, …, k_{m}) admissible

V_{E}(k_{0}, …, k_{m}) := V_{E}(k_{0}, k_{1}) ∩ P_{E}^{−1}(V_{E}(k_{1}, …, k_{m})),  (26)

were defined, using the Poincaré map P_{E}.

By going to subsequences we can assume that the points x^{(i)} ∈ Λ_{E}^{−} correspond to symbol sequences \underline{k}^{(i)} ∈ X which are related to the symbol sequence \underline{k} := F_{E}^{−1}(x) ∈ X of x by

k_{j}^{(i)} = k_{j} (i ∈ ℕ; j > χ(i))

but k_{χ(i)}^{(i)} ≠ k_{χ(i)}, where χ(i) → −∞.

By construction the vector v is tangent to the one-dimensional manifold Δ_{1}^{−} at x.

We now show the existence of a second sequence (y^{(i)})_{i∈ℕ} of points y^{(i)} ∈ Λ_{E} converging to x, but with the following property: There exists an α ∈ (0, 1) such that writing the points in the form

y^{(i)} = exp_{x}(s^{(i)}w^{(i)})

with units vectors w^{(i)} ∈ T_{x}H_{E},

d(w^{(i)}, span(v)) ≥ |s^{(i)}|^{α+1}.  (27)

Namely as the number n of centres is ≥ 4 (which follows from our assumption on the positions s⃗_{1}, …, s⃗_{n}) we find symbol sequences \underline{l}^{(i)} ∈ X with

l_{j}^{(i)} = k_{j} (i ∈ ℕ; j < χ(i))

but l_{χ(i)}^{(i)} ≠ k_{χ(i)}, and

affine span (s⃗_{k_{χ(i)−1}}, s⃗_{k_{χ(i)}}, s⃗_{k_{χ(i)}^{(i)}}, s⃗_{l_{χ(i)}^{(i)}}) = ℝ^{3}.  (28)

Then the y^{(i)} := F_{E}(\underline{l}^{(i)}) converge to

lim_{i→∞} y^{(i)} = F_{E} ( lim_{i→∞} \underline{l}^{(i)}) = F_{E}(\underline{k}) = x.

Next we consider the geometric situation at the (early) time χ(i). More precisely we set

˜\underline{k} := σ^{χ(i)}(\underline{k}) , ˜ \underline{k}^{(i)} := σ^{χ(i)}(\underline{k}^{(i)}) and ˜ \underline{l}^{(i)} := σ^{χ(i)}(\underline{l}^{(i)}).

Thus

(k̃_{1}, k̃_{2}) = (k̃_{1}^{(i)}, k̃_{2}^{(i)}) = (l̃_{1}^{(i)}, l̃_{2}^{(i)}) = (k_{χ(i)+1}, k_{χ(i)+2})

but k̃_{0} ≠ k̃_{0}^{(i)} ≠ l̃_{0}^{(i)} ≠ k̃_{0}.

By the conjugacy property of (24) the identities x̃ := F_{E}(˜\underline{k}) = P_{E}^{χ(i)}(x), x̃^{(i)} := F_{E} (˜\underline{k}^{(i)}) = P_{E}^{χ(i)}(x^{(i)}) and ỹ:= F_{E} (˜\underline{l}^{(i)}) = P_{E}^{χ(i)} (y^{(i)}) are true.

All of these points are contained in the local stable manifold

V_{E}(k̃_{1}, k̃_{2}, k̃_{3}, …) ⊂ V_{E}(k̃_{1}, k̃_{2}, k̃_{3}) ⊂ H_{E}^{k̃_{1},k̃_{2}},

but at the same time in the following disjoint sets

x̃ ∈ W_{E}(k̃_{0}, k̃_{1}, k̃_{2}) , x̃^{(i)} ∈ W_{E}(k̃_{0}^{(i)}, k̃_{1}, k̃_{2}) , ỹ^{(i)} ∈ W_{E}(l̃_{0}^{(i)}, k̃_{1}, k̃_{2}).

In H_{E}^{k̃_{1},k̃_{2}} ⊂ H_{E} the minimal angle between vectors from (points in)

W_{E}(k̃_{0}, k̃_{1}, k̃_{2}) ∩ V_{E}(k̃_{1}, k̃_{2}, k̃_{3}) to W_{E}(k̃_{0}^{(i)}, k̃_{1}, k̃_{2}) ∩ V_{E}(k̃_{1}, k̃_{2}, k̃_{3})

and vectors from

W_{E}(k̃_{0}, k̃_{1}, k̃_{2}) ∩ V_{E}(k̃_{1}, k̃_{2}, k̃_{3}) to W_{E}(l̃_{0}^{(i)}, k̃_{1}, k̃_{2}) ∩ V_{E}(k̃_{1}, k̃_{2}, k̃_{3})

is bounded away from zero by some Δ > 0 (see (26) and (25)). This follows from (28). As there are only finitely many (at most n, to be more precise) choices for each of the indices k̃_{0}, k̃_{1}, k̃_{2}, k̃_{0}^{(i)} and l̃_{0}^{(i)}, this bound is uniform.

So the angle between the unit vectors ṽ^{(i)} and w̃^{(i)} defined by

x̃^{(i)} = expx̃(s̃^{(i)}ṽ^{(i)}) , ỹ^{(i)} = expx̃(t̃^{(i)}w̃^{(i)})

is bounded below by Δ. But by Prop. 11.2 of [Kn] the contraction rates of these vectors w.r.t. the Poincaré P_{E} differ at most by the order O(1/E).

By estimate (11.2) of [Kn] we conclude that after iterating −χ(i) > 0 times these vectors still have an angle bounded away from zero by

Δ (1 − \frac{C}{E} )^{|χ(i)|}

whereas their length is reduced at least by a factor (cE)^{|χ(i)|}.

Thus by enlarging the energy E, we can choose an arbitrarily small α > 0 in (27).

But (27) is incompatible with y^{(i)} belonging to the one–dimensional submanifold Δ_{1}^{−}.

As we can apply the same argument to all x ∈ Λ_{E}, we have derived a contradiction to the finite covering assumption (22). □

The second author (I.A.T.) was supported by RFBR (grant 03-01-00403) and Max-Planck-Institute on Mathematics in Bonn.

References

[AM] Abraham, R., Marsden, J.E.: Foundations of Mechanics. Reading: Benjamin 1978

[Ar] Arnol’d, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics 60. Berlin: Springer 1989

[Bo] Bolotin, S.V.: Nonintegrability of the n-center problem for n > 2. Vestnik Mosk. Gos. Univ., ser. I, math. mekh. No.3, 65–68 (1984)

[BN] Bolotin, S.V.; Negrini, P.: Regularization and topological entropy for the spatial n-center problem. Ergodic Theory and Dynamical Systems 21, 383–399 (2001).

[BN2] Bolotin, S.V.; Negrini, P.: Global regularization for the n-center problem on a manifold. Discrete and Continuous Dynamical Systems - Series A8, 873–892 (2002)

[BT] Bolsinov, A.V.; Taimanov, I.A.: Integrable geodesic flows with positive topological entropy. Invent. Math. 140, 639–650 (2000)

[Bu] Butler, L.T.: New Examples of Integrable Geodesic Flows. Asian J. Math. 4, 515–526 (2000)

[vdD] van den Dries, L.: Tame Topology and O-minimal Structures. London Math. Society, Lecture Note Series 248, Cambridge University Press, Cambridge, 1998.

[Fo] Fomenko, A.T.: Integrability and Nonintegrability in Geometry and Mechanics. Dordrecht: Kluwer 1988

[Ge] Gevrey, M.: Sur la nature analytique des solutions des équations aux dérivées partielles. Ann. Scient. Éc. Norm. Sup. 35 (1918), 129–189; In: Œuvres de Maurice Gevrey. CNRS 1970

[Ha] Hasselblatt, B.: Regularity of the Anosov splitting II. Ergodic Theory and Dynamical Systems 17, 169–172 (1997)

[Ho] Horn, J.; Wittich, H. Gewöhnliche Differentialgleichungen. Berlin: de Gruyter 1960

[Ju] Jung, K.: Adiabatic Invariance and the Regularity of Perturbations. Nonlinearity 8, 891–900 (1995)

[KK] Klein, M.; Knauf, A.: Classical Planar Scattering by Coulombic Potentials. Lecture Notes in Physics m 13. Berlin: Springer 1992

[Kn] Knauf, A.: The n-Centre Problem of Celestial Mechanics. J. Europ. Math. Soc. 4, 1–114 (2002)

[Ko] Kozlov, V. V.: Topological Obstructions to the Integrability of Natural Mechanical Systems. Soviet Math. Dokl. 20, 1413–1415 (1979)

[Si] Simon, B.: Wave Operators for Classical Particle Scattering. Commun. Math. Phys. 23, 37–49 (1971)

[Ta] Taimanov, I.A.: Topological obstructions to the integrability of geodesic flows on nonsimply connected manifolds. Math. USSR-Izv. 30, 403–409 (1988)

[Th] Thirring, W.: Lehrbuch der Mathematischen Physik 1. 2nd Ed.; Wien: Springer 1988