Time-dependent scattering theory for ODEs and applications to reaction dynamics

advertisement
Time-dependent scattering theory for ODEs and
applications to reaction dynamics
Daniel Blazevski1 , Rafael de la Llave1
1
1 university station c1200, Department of Mathematics
University of Texas at Austin, Austin, TX 78712
[email protected], [email protected]
Abstract. We develop a time-dependent scattering theory for general vector fields in
Euclidean space. We give conditions that ensure that the wave maps exist, are smooth,
invertible, and depend smoothly on parameters. We then discuss the intertwining relations
and how they can be used to compute stable/unstable manifolds for time-dependent
normally hyperbolic invariant manifolds. The theory is particularly effective for Hamiltonian
mechanics. We also give perturbative calculations of the scattering map that are analogous
to Fermi’s Golden Rule in quantum mechanics. We apply this theory to a problem in
transition state theory concerning the exposure of molecules to a laser pulse for a short
time. We present a method to compute invariant manifolds for the laser-driven HenonHeilies system and give perturbative calculations of the change in the branching ratio.
PACS Numbers: 02.30.Hq, 02.60.Cb, 34.50.Lf, 34.50.Rk, 82.20.Db, 45.50.Pk, 45.10.Hj,
03.65.Nk
Keywords: Scattering theory, transition state theory, chemical reactions, perturbation
theory, Fermi’s Golden Rule, Henon-Heiles, invariant manifolds, numerical analysis,
branching ratio
1. Introduction
Many physical situations are described by an ODE subject to a time dependent perturbation
which is localized in space and time. That is, the trajectories x(t) of the system satisfy
d
x(t) = V(x(t), t)
dt
(1)
where V(x, t) = U(x, t) + P(x, t) and P(x, t), the perturbation, is localized in space and time
( e.g. |P(x, t)| ≤ C exp(−λ|t|) ). All the vector fields in this paper are assumed to be defined
on Rn .
Notice that we do not assume that P is uniformly small. Indeed, in some of the
motivating examples we have in mind P will not depend on a parameter whose value
is taken to be sufficiently small. The examples we have in mind include: exposing molecules
to a laser, an ion passing near a molecule, and similarly the passage of a comet of a a
planet-satellite system. These phenomenon have been studied in chemistry and astronomy
using a variety of methods (e.g. transition state theory [1] and adiabatic perturbation theory
[2, 3, 4]). The goal of this paper is to present a treatment based on scattering theory. Since
the perturbation is localized in time, it is easy to guess that a trajectory of the full system
will behave, both in the past and in the future, like a trajectory of the unperturbed system.
It is then natural to write the future ( or past) asymptotic states as functions of the present
states. This is where the wave maps Ωt±0 come in: Ωt+0 (resp. Ωt−0 ) is a function on the phase
space that takes the present states of a system as an input and outputs the future (resp.
past) states. The composition st0 = Ωt+0 ◦ (Ωt−0 )−1 , called the scattering map, computes the
future states knowing the past states.
2
Classical scattering theory has been considered before, most notably [5, 6, 7, 8],
though only for autonomous Hamiltonian systems. The papers [5, 6] consider Hamiltonians
2
H = |p|2 + V (q) and given conditions on V for the existence of the scattering map. The
conditions are either that V is compactly supported, or that it has sufficient decay at infinity.
These type of assumptions are used in [6, 7, 8] to derive classical analouges of quantum
mechanical scattering theory.
This paper serves three main goals: (1) to give conditions for general (i.e. not necessarily
Hamiltonian) non-autonomous systems for the existence of the scattering map, (2) to indicate
how one can use scattering theory in the numerical computation of invariant manifolds, and
(3) to prove an analogue of Fermi’s Golden Rule, which, as we explain, is of great physical
interest.
In section 2, working in the context of general time-dependent vector fields in Rn , we
give conditions on U and V that ensure the existence, smoothness, and invertibility of the
wave maps Ωt±0 and scattering map st0 . In contrast to [5, 6], where the perturbation is
autonomous and assumed to decay in space, our main existence results (Theorem 1 and
Proposition 1) state, roughly speaking, that the wave and scattering maps exist provided
that the perturbation decays faster in time than the spatial growth rate of the unperturbed
flow.
The method to prove existence uses deformation theory [9] and is analogous to Cook’s
method in quantum mechanics [10, p. 20]. We also give conditions on the vector fields,
Vσ , Uσ and the flow Utt0 that implies that the wave maps are invertible, a property called
asymptotic completeness in quantum mechanical scattering theory. Note that our conditions
do not involve the flow Vtt0 , which is very convenient since this implies we can use scattering
theory to understand Vtt0 knowing only the vector fields and Utt0 .
Using scattering theory, we then show that the existence, smoothness and invertibility
yields intertwining relations for the wave maps Ωt±0 and we show that the dynamics of U and
V are conjugate when viewed as autonomous vector fields on the extended space Rn × R.
Based on the intertwining relations we develop a notion of a time-dependent normally
hyperbolic invariant manifold and discuss how we can use Ωt±0 to compute such manifolds
and their corresponding stable/unstable manifolds.
In Section 3 we give perturbative calculations of the scattering map. Thus, we will
consider vector fields of the form
V(x, t) = U(x, t) + P(x, t)
(2)
In this setting we show that the scattering map is the time- map of a vector field S and
give explicit formulas for S . The result is analogous to Fermi’s Golden Rule in quantum
mechanics as considered in [11, 10, 12, 13].
We also show that, in the case of Hamiltonian systems, S has a corresponding
Hamiltonian S . The physical significance of this result is that this allows us to compute,
3
up to first order in , the total change of any quantity, fast or slow, that arises from the
perturbation. The results and methods of proof are similar to the perturbative calculations
of scattering map for a normally hyperbolic manifold [14].
In Section 4, we apply the theory developed to a problem in transition state theory. The
paper [1] considers the problem of the dynamics of reaction between molecules under the
influence of a laser pulse. In their work, they compute time-dependent invariant manifolds
using normal form theory. We give an alternate method to compute the manifolds using
scattering theory.
We also give perturbative calculations of the quantity called the branching ratio in [1]
which measures the fraction of initial conditions which go into different scattering channels.
In quantum mechanics this is also called the branching fraction (see [15]) but we will stick
with the name “branching ratio” used in [1].
2. Definitions and existence/smoothness results
In this section will work with general time-dependent ODE’s, and consider more specific
cases later. Let Ut and Vt be two time-dependent vector fields on Rn , and let Utt0 and Vtt0 be
the corresponding flows. Thus, Utt0 satisfies the equation
d t
U (x) = Ut (Utt0 (x)),
dt t0
Utt00 (x) = x
(3)
As a consequence of uniqueness of solutions of (3) we have Utt0 ◦ Ust0 = Ust . Of course, Vtt0
satisfies a similar equation.
We define the wave maps by
Ωt±0 = Ωt±0 (U, V ) = lim Utt0 ◦ Vtt0
t→±∞
The intuition is that Ωt+0 (resp. Ωt−0 ) gives the orbital elements in the future (resp. past)
given the present orbital elements. Assuming that Ωt−0 is invertible, the scattering map which
is defined by
−1
st0 = Ωt+0 ◦ Ωt−0
gives the future orbital elements knowing the orbital elements in the past. To make the
intuition of wave maps a bit more concrete notice that for large T > 0 we have
Ut±T
(Ωt±0 (x)) ≈ Vt±T
(x)
0
0
Thus, the wave maps give us a precise way of describing how, for large times, the flow Vtt0
behaves like the flow for Utt0 , though the flows may be quite different for short times (See
Figure 1).
4
Figure 1. Pictorial description of the wave and scattering maps. For a given point x0 let
t0
+
−t −
t
t
x±
0 := Ω± . Then Vt0 (x0 ) converges in the future (resp. past) to Ut0 (x0 ) (resp. Ut0 (x0 ))
+
and the scattering map takes x−
0 as an input and outputs x0 .
Notice that the wave maps depend on the starting time t0 of the non-autonomous
flows. This is what makes the theory different from scattering theory for time independent
vector fields. This will become especially relevant in the next section when discussing the
intertwining relations and conjugacy.
Before we give conditions that ensure the existence, smoothness, invertibility of the
wave maps, we recall the definition the function space C k (BR ), where BR refers the the ball
of radius R in Rn centered at the origin
C k (BR ) = {f : BR → Rn : sup sup kDj f (x)k < ∞}
j
j
x∈BR
j
Here kD f (x)k refers to the norm of the D f (x) as a j−multilinear map, that is the
smallest of all real number M such that
|Dj f (x)(v1 , . . . , vj )| ≤ M |v1 | · · · |vj |
and |vi | simply refers to the Euclidean norm of vi . For a function in C k (BR ) we define its
C k norm as
kf kC k = sup sup kDj f (x)k
j
x∈BR
We now impose conditions on Utt0 and Vtt0 that ensure that the wave maps and scattering
map exist, are smooth, depend smoothly on parameters, and are invertible. The proof we
give is similar to Cook’s method [10] in quantum mechanical scattering theory. Throughout
the paper we will use the language of Cartan calculus, which for our purposes is significantly
more concise than matrix conculations (it handles very well under changes of variables). We
summarize many of the elements in Appendix A. For now, let us recall briefly two basic
definitions from deformation theory that we will use and refer the reader to the Appendix
A for details. Let fa be a family diffeomorphisms, and define the vector field Fa by
d
fa = Fa ◦ fa
da
Fa is called the vector field generating fa . We also use the push-forward of a vector field,
which is defined as follows: Given a vector field V, and a diffeomorphism f : Rn → Rm , the
pushforward of V under f is, denoted f∗ V is the natural way to take a vector field on the
domain Rn and associate, via f , a vector field on Rm . As an equation, the push-forward is
defined by:
(f∗ V)(x) = (Df ◦ f −1 (x))V ◦ f −1 (x) = (Df V) ◦ f −1 (x)
5
For a more thorough treatment of the differential toplogical concepts used see [13, 16, 17].
Theorem 1 Suppose that the following condition on the flows Utt0 and Vtt0 is satisfied:
The integrals
Z
±∞
I± (t0 ) =
k DUσt0 (Vσ − Uσ ) ◦ Vtσ0 kC k (BR ) dσ < ∞T hen :
(4)
t0
(1) the wave maps
Ωt±0 = lim Utt0 ◦ Vtt0
t→±∞
exist and are in C k (BR ) for every R.
(2) Suppose that k > 1 and that the limits
lim I+ (t0 ) = lim I− (t0 ) = 0
t0 →∞
(5)
t0 →−∞
hold. Then Ωt±0 are invertible, and their inverses are in C k (BR ) for every R > 0
(t ,T )
Proof of (1): First, we show that Ωt+0 exists. Define Ω+0
:= UTt0 ◦ VtT0 . Since the
maps Vtt0 and Utt0 are diffeomophisms we can use deformation theory (c.f. appendix A) to
(t ,T )
construct the vector field generating Ω+0 . Using Proposition 11 from the appendix and
the fact that Ut and Vt generate Utt0 and Vtt0 respectively we have that the generator, OT+ , of
(t ,T )
Ω+0 is given by
OT+ = −(UTt0 )∗ UT + (UTt0 )∗ VT = (UTt0 )∗ (VT − UT )
(6)
The Fundamental Theorem of Calculus implies that
(t ,T +1)
Ω+0
−
(t ,T )
Ω+0
T +1
Z
(t ,σ)
Oσ+ ◦ Ω+0 dσ
=
(7)
T
and hence
∞
X
(t ,N +1)
kΩ+0
−
(t ,N )
Ω+0 kC k (BR )
Z
(t ,σ)
kOσ+ ◦ Ω+0 kC k (BR ) dσ
t0
N >t0
Z
∞
≤
∞
=
(t ,σ)
k(Uσt0 )∗ (Vσ − Uσ ) ◦ Ω+0 kC k (BR ) dσ =
Z
t0
(8)
∞
k DUσt0 (Vσ − Uσ ) ◦ Vtσ0 kC k (BR ) dσ
t0
Since we are assuming the integral converges we conclude that
∞
X
(t ,N +1)
kΩ+0
(t ,N )
− Ω+0
N >t0
6
kC k (BR ) < ∞
(t ,N )
(t ,∞)
and it follows that Ω+0
converges to a function Ω+0 . Moreover, since
Z ∞
(t0 ,∞)
(t0 ,T )
kΩ+
− Ω+ kC k (BR ) ≤
k DUσt0 (Vσ − Uσ ) ◦ Vtσ0 kC k (BR ) ds
T
(t ,∞)
(t ,T )
we conclude that kΩ+0
− Ω+0 kC k (BR ) → 0 as T → ∞ and T ∈ R, which proves the
existence of Ωt+0 .
(t ,T )
t0
◦V−T
, which is generated
Similarly, for Ωt−0 one considers the family of maps Ω−0 = Ut−T
0
t0
−
+
by the vector field OT = −O−T . Using an argument analogous for Ω+ , the existence and
smoothness of Ωt−0 follows provided that
Z t0
Z ∞
−σ
t0
k(DUσt0 (Vσ − Uσ )) ◦ Vtσ0 kC k (BR ) dσ (9)
k(DU−σ (V−σ − U−σ )) ◦ Vt0 kC k (BR ) dσ =
−∞
−t0
is finite.
Proof of (2) Now we show that Ωt±0 are invertible and their inverses are C k . We prove
invertibility for Ωt+0 , and note that proving invertbility for Ωt−0 is similar. We first note observe
it suffices to prove that Ωt+0 is invertible for one value of t0 . Indeed, we have the following
identity
Ωs+ = Uts0 ◦ Ωt+0 ◦ Vst0
(10)
We will discuss (10) in more detail in Section 2.2. For now we notice that it implies that Ωs+
is invertible as long as Ωt+0 is invertible for some t0 , as claimed.
Now we prove that there is one value of t0 for which Ωt+0 is invertible. Notice
Rt
(t ,σ)
(t ,t )
(t ,t)
(t ,s)
that Ω+0 − Ω+0 = s Oσ+ ◦ Ω+0 dσ Since Ω+0 0 (x) = Utt00 Vtt00 (x) = x we have that
R
t
(t ,t)
(t ,σ)
Ω+0 − x = t0 Oσ+ ◦ Ω+0 dσ and hence taking the limit as t → ∞ we have that
Z ∞
(t ,σ)
t0
Ω+ − x =
Oσ+ ◦ Ω+0 dσ
t0
It follows that
kΩt+0
Z
− Id kC k (BR ) ≤
∞
(t ,σ)
k(Uσt0 )∗ (Vσ − Uσ ) ◦ Ω+0 kC k (BR ) dσ
(11)
0
However, assumption (2) is that the integral tends to zero as t0 → ∞ for every R > 0.
Thus given any y ∈ R2n with |y| < R and > 0 we can find a t0 large enough so that
kΩt+0 − Id kC k (B2R ) < Thus Ωt+0 can be made as close to the identity as desired on any given
ball. The next lemma tells us that this implies that Ωt+0 is invertible provided k ≥ 2.
Lemma 1 Suppose that F : Rn → Rn is in C 1 (BR ). There is a 0 > 0 such that if < 0
and
kF − Id kC k (BR ) < Then for any y ∈ B R the equation
2
F (x) = y
has a unique solution x ∈ BR .
7
Proof: The idea is that |F (y) − y| < , that is y is an approximate solution to F (x) = y,
and hence we can use Newton’s method to solve the equation F (x) = y. Thus we consider
G(x) = x − DF (y)−1 (F (x) − y)
since G(x) = x implies F (x) = y. One can show that, for sufficiently small, G satisfies the
hypothesis of the Contraction Mapping Theorem in a ball B(r; y) for some r > 0.
We now apply Lemma 1 to F = Ωt+0 , which satisfies the hypotheses of the lemma
provided t0 is large enough to conclude that Ωt+0 is invertible, and (Ωt±0 )−1 is in C k (BR ) for
every R > 0.
Now let us state several results that explain when the conditions of the Theorem 1 are
satisfied. These results give sufficient conditions to apply the techniques in this paper to
several concrete models, including the models considered in [1].
Corollary 1 Suppose that Ut and Vt are C k (BR ) for every R > 0 and some integer k > 0.
Moreover suppose that Ut − Vt is compactly supported in time, that is, Ut − Vt ≡ 0 for
t∈
/ [T1 , T2 ]. The wave maps Ωt±0 exist, are in C k (BR ) for every R > 0 and are invertible.
R ±∞
Proof: The conditions of Theorem 1 were that I± (t0 ) = t0 k (DUσt0 (Vσ − Uσ )) ◦
Vtσ0 kC k (BR ) dσ < ∞ and limt0 →∞ I+ (t0 ) = limt0 →−∞ I− (t0 ) = 0. In the case that Vσ − Uσ is
compactly supported in time I± (t0 ) are integrals over finite intervals and are also compactly
supported with respect to t0 . Corollary 2 The integrals
±∞
Z
k DUσt0 (Vσ − Uσ ) ◦ Vtσ0 kC k (BR ) dσ
I± (t0 ) =
t0
are finite provided that
(C1)
Z
±∞
I± (t0 ) =
k DUσt0 (Vσ − Uσ ) ◦ Vtσ0 kC 0 (BR ) dσ
t0
are finite and
(C2)
k DUσt0 (Vσ − Uσ ) ◦ Vtσ0 kC k+1 (BR ) ≤ M
where M is independent of σ. From Theorem 1, it follows that under (C1) and (C2) that
the wave maps exist and are in C k (BR ) for every R > 0.
8
Proof: The proof of this is simply an application of Hadamard’s inequality, which asserts
that
kf kC s (BR ) ≤ Cr,t kf kµC r (BR ) kf k1−µ
C t (BR )
t−s
. We simply take s = k, r = 0
for any function f in C t (BR ) where 0 ≤ r < s < t and µ = t−r
and t = k + 1 This inequality was originally proven by Hadamard [18] and later generalized
by Kolmogoroff [19]. A more modern version of the inequality appears in [20] Corollary 3 For n ≥ 2 consider the annuli An = {x ∈ Rn : n − 1 ≤ |x| ≤ n}. Suppose the
estimates hold:
sup kDUσt0 (x)k ≤ Cn eλ(n)|σ−t0 |
An
and
sup |Vσ − Uσ | ≤ C̃n e−µ(n)|σ|
An
Where 0 < λ(n), µ(n), α := supn (λ(n) − µ(n)) < 0 and 0 < Cn C̃n < M for some M > 0.
The integrals
Z ±∞
I± (t0 ) =
k DUσt0 (Vσ − Uσ ) ◦ Vtσ0 kC 0 (BR ) dσ
t0
are finite and hence by Theorem 1 the wave maps exist and in C 0 (BR ) for every R > 0.
Moreover, we also have that limt0 →∞ I+ (t0 ) = limt0 →−∞ I− (t0 ) = 0
Proof: We prove the results for Ωt+0 . We simply break up the integrals I+ into parts
when Vtσ0 lies in An for each n ≥ 2. For example, if we let A1 denote the ball of radius 1 then
we can write I+ as
Z
±∞
I+ (t0 ) =
=
≤
k DUσt0 (Vσ − Uσ ) ◦ Vtσ0 kC 0 (BR ) dσ
t0
∞
XZ
σ
n=1 {σ>t0 :Vt0 ∈An }
∞ Z
X
n=1
k DUσt0 (Vσ − Uσ ) ◦ Vtσ0 kC 0 (BR ) dσ
Me
−ασ
{σ>t0 :Vtσ ∈An }
0
Z
(12)
∞
dσ ≤
M e−ασ eλt0 dσ
t0
It follows that I+ (t0 ) < ∞ and limt0 →∞ I+ (t0 ) = 0.
The importance of this corollary is that it gives conditions that guarantee the existence
of the wave maps that depend only on the vector fields Uσ , Vσ and the flow Utt0 , but not the
flow Vtt0 . Conditions like this are very desirable from the viewpoint of applications since the
goal of scattering theory is to understand better the flow Vtt0 . We now extend Corollary 3 to
obtain conditions that guarantee higher smoothness of the wave maps, which is especially
important since our proof of the invertibility of the wave maps required C 1 smoothness.
9
The idea behind the next proposition is straightforward to understand, though we need
some terminology from differential topology (see [21] for more details). For any smooth
function f : Rn → Rn define T f : R2n → R2n by
T f (x, v) = (x, Df (x)v)
The letter T refers the tangent functor from differential topology. We would like to say that
t0
)
Vt±T
DΩt±0 = lim DΩt±0 ,T = lim D(U±T
0
T →∞
T →∞
exists. Consider the variational equations
d
DUtt0 v = DUt (Utt0 )DUtt0 v
dt
d
DVtt0 v = DVt (Vtt0 )DVtt0 v
dt
(13)
A convenient way of organizing the study of the variational equations (13) is to use the
tangent functor. Indeed, consider the vector field Ut1 defined on Rn × Rn defined by
Ut1 (x, v) = (Ut (x), DUt (x)v)
The corresponding flow Ut1 (t0 , x, η) is given by
Ut1 (t0 , x, v) = (Utt0 (x), DUtt0 (x)v)
and we notice that Ut1 (t0 , x, v) = T Utt0 (x, v)), that is, T Utt0 and T Vtt0 correspond to flows for
a vector field, and hence we can also consider the wave maps for these flows. In fact we have
T Ωt±0 (U, V )(x, v) = (Ωt±0 (x), DΩt±0 (x)v)
= lim (Utt0 ◦ Vtt0 (x), DUtt0 (Vtt0 (x))DVtt0 (x)v) = Ωt±0 (T U, T V )(x, v)
(14)
t→±∞
Thus we conclude that
T Ωt±0 (U, V ) = Ωt±0 (T U, T V )
(15)
The idea for higher regularity is now simple: Corollary 3 gives conditions for the wave maps
to exist and be continuous. Applying Corollary 3 to the flows T U and T V gives conditions
on the vector fields Ut , Vt that guarantee Ωt±0 (T U, T V ) are C 0 . By (15), T Ωt±0 (U, V ) are C 0 ,
which is equivalent to saying that Ωt±0 are C 1 .
Proposition 1 For n ≥ 2 consider the annuli An = {x ∈ Rn : n − 1 ≤ |x| ≤ n}. Suppose
that for some k ≥ 1 we have the following estimates:
k+1
X
i=1
sup kDi Uσt0 (x)k ≤ Cn eλ(n)|σ−t0 |
An
10
(16)
and
k
X
i=1
sup |Di (Vσ − Uσ )| ≤ C̃n e−µ(n)|σ|
(17)
An
Where 0 < λ(n), µ(n), α := supn (λ(n) − µ(n)) < 0 and 0 < Cn C̃n < M for some M > 0.
The wave maps and their inverses exist and are in C k (BR ) for every R > 0.
Proof: For k = 1, we can say that Corollary 3 implies that Ω± (T U, T V ) exists and is in
C (BR ) for every R > 0. By (15) this implies that Ωt±0 (U, V ) are C 1 (BR ) for every R > 0.
For k > 1, we inductively define T k Ωt±0 (U, V ) = T (T k−1 Ωt±0 (U, V )). By using repeatedly
(15) we have that
T k Ωt±0 (U, V ) = Ωt±0 (T k U, T k V )
0
Corollary 3 implies that if conditions (16) and (17) hold, then Ωt±0 (T k U, T k V ) exists and
is C 0 (BR ) for every R > 0, which, by an inductive argument, implies that Ωt±0 (U, V ) are
C k (BR ) for every R > 0. We would like to make a remark about the assumptions of the Proposition 1. A
particular case of these assumptions occurs when one has uniform bounds on all of Rn ,
that is when one has
k+1
X
sup kDi Uσt0 (x)k ≤ Ceλ|σ−t0 |
Rn
i=1
and
k
X
sup |Di (Vσ − Uσ )| ≤ C̃e−µ|σ|
i=1
Rn
with no mention of annuli whatsoever. However, this is not typical, and one commonly has
that the growth rates of U increase as one moves further away from the origin. In this case
the appropriate assumptions to consider are those given in Proposition 1.
Now, we consider the case when the vector fields depend on parameters
Corollary 4 Suppose that Ut and Vt depend on a parameter λ ∈ U , an open subset of
Rm , and let Utt,λ
and Vtt,λ
denote the corresponding flows. Suppose that the conditions are
0
0
satisfied: The integral
Z ±∞
I± (t0 ) =
k DUσt0 ◦ (Vσ − Uσ ) ◦ Vtσ0 kC k (U ×BR ) dσ < ∞
t0
are finite and the following limits hold
lim I+ (t0 ) = 0 lim I− (t0 ) = 0
t0 →∞
t0 →−∞
(note: in this case the norms include both the spatial and the parameter variables) Then, the
wave maps Ωt±0 (λ) are C k with respect to the parameter λ.
The proof, or course, is the same as in Theorem 1. We simply chose to treat this case
separately to make the exposition of Theorem 1 clearer.
11
2.1. Scattering Theory for Classical Hamiltonian Systems
In this section we will consider the case when the vector fields Ut and Vt are Hamiltonian.
Suppose that we are given Hamiltonians H 0 (Q, P, t) and H = H 0 (Q, P, t) + H 1 (Q, P, t) on
R2n and let Ut = J∇Ht0 and Vt = J∇Ht be the corresponding vector fields, where
!
0 Id
J=
−Id 0
The conditions for the existence/smoothness and invertibility of Theorem 1 (i.e. (4)
and (5)) amount to
Z ±∞
k DUσt0 ◦ J∇Ht1 ) ◦ Vtσ0 kC k (Br ) dσ
I± (t0 ) =
t0
is finite and
lim I+ (t0 ) = lim I− (t0 ) = 0
t0 →∞
t0 →−∞
Thus, if we view H 1 as a perturbation of H 0 then the integrability condition requires
that the perturbation decays with time. Moreover, we proved that Ωt±0 is the limit of
t0 ,T
t0
Ω±
= U±T
◦ Vt±T
, which is symplectic in the case of Hamiltonian systems. It follows
0
t0 ,T
that as long as Ω± converges to Ωt±0 in C k (BR ) for every R > 0, then the wave maps are
symplectic.
We also Note that Proposition 1 gives a sufficient condition only in terms of the flow
U and H1 to guarantee that the wave maps exist and are continuous. More precisely,
Proposition 1 asserts the existence of the wave maps provided that the decay rate for H1 is
stronger the the growth rate for U on every annulus.
2.2. Intertwining Relations and Conjugacy
One important fact about the existence of the wave maps is that it implies that the flows
of U and V are conjugate when we view them as autonomous flows in the extended phase
space Rn × R, where we add an extra variable for time. The notion that scattering theory
for classical systems yields a conjugacy between two dynamics was used to study the local
behavior near a hyperbolic fixed point in [11] and was used to prove that certain systems,
for example the Calogero-Moser system, is integrable [22, 23, 24]. For us, we will exploit the
fact that the flows are conjugate to compute invariant manifolds, which we discuss in the
next section.
Let us consider two flows Utt0 and Vtt0 . If we view the flows as defining autonomous flows
on the extended space Rn × R, that is we have autonomous flows Ũ and Ṽ on Rn × R that
solve the ODE’s
12
ẋ = Ut (x) ẋ = Vt (x)
ṫ0 = 1
ṫ0 = 1
(18)
It turns out that the dynamics of Ũ and Ṽ are conjugate, as we show in the following
proposition.
Proposition 2 Suppose that U and V are flows for Ut and Vt , respectively. Moreover,
suppose that the wave maps Ωs± = limt→±∞ Utt0 ◦ Vtt0 exist in the sense that the limits defining
them exist point-wise. Moreover assume that Ωt±0 are invertible.
(1) We have the following intertwining relations
Ωt± ◦ Vtt0 = Utt0 ◦ Ωt±0
(19)
st0 = Utt0 ◦ st ◦ Utt0
(2) The extended flows Ũ and Ṽ are conjugate. More precisely, define the wave maps
on the extended space
Ω̃± (x, t) := lim Ũ−t ◦ Ṽt
t→±∞
These wave maps exist, are as smooth as the wave maps Ωt±0 , are invertible, are moreover we
have the following relation
Ω̃± ◦ Ṽt , = Ũt ◦ Ω̃±
(20)
Proof: The proof of (1) is a consequence of the following calculation
Ωt±
◦
Vtt0
=
=
lim
η→∞
Utt0
Uηt
◦ lim
◦
η→∞
Vtη
Uηt0
◦
◦ Vtt0 = lim Uηt ◦ Vtη0
η→∞
Vtη0
=
Utt0
(21)
◦ Ωt+0
and hence Ωt± ◦ Vtt0 = Utt0 ◦ Ωt±0 as claimed. This identity immediately implies that
st0 = Utt0 ◦ st ◦ Utt0
For the proof of (2) we first show that Ω̃± (x, t0 ) = (Ωt±0 (x), t0 ). Indeed, notice that
0
Ω̃± (x, t0 ) = lim Ũ−t ◦ Ṽt (x, t0 ) = lim Ũ−t (Vtt+t
(x), t + t0 )
0
t→±∞
= lim
t→±∞
t0
(Ut+t
0
◦
t→±∞
t+t0
Vt0 (x), t0 ) =
(Ωt±0 (x), t0 )
(22)
which proves our claim.
This shows that the wave maps Ω̃± on the extended space exist, are smooth as smooth
t0
as Ω± , and are invertible. With this relation we can also deduce (20) as follows
13
0
0
0
◦ Vtt+t
(x), t + t0 )
Ω̃± ◦ Ṽt (x, t0 ) = Ω̃± (Vtt+t
(x), t + t0 ) = (Ωt+t
±
0
0
(23)
Now we use (19) to say that
0
0
0
◦ Vtt+t
(x), t + t0 ) = (Utt+t
◦ Ωt±0 , t + t0 ) = Ũt (Ωt±0 (x), t0 ) = Ũt ◦ Ω̃± (x, t0 )
(Ωt+t
±
0
0
(24)
and hence we conclude that Ω̃± ◦ Ṽt , = Ũt ◦ Ω̃± holds as claimed. 2.3. Time-Dependent Invariant Manifolds
Now that we have established that the existence of the wave maps implies that the flows
U and V are conjugate in the extended space, we can apply this to the theory of invariant
manifolds. The basic idea is that the wave maps give a way to associate invariant manifolds
of U to invariant manifolds of V and vice versa.
Since the non-autonomous flows U and V are conjugate only in the extended space,
we will need to consider an appropriate notion of an invariant manifold, which for the
applications we have in mind will be a time-dependent normally hyperbolic manifold.
Such manifolds appeared in [1] when studying the behavior of reacting molecules under
the influence of a laser-pulse. The definitions we state are very similar to those stated
in [25, 26, 27, 28], but are slightly different since we consider time-dependent invariant
manifolds. The following example motivates the definition we give. This example also
appears in the applications to transition state theory that we discuss later.
2.3.1. Example Let x0 be a hyperbolic fixed point for the autonomous flow Ut corresponding
to the vector field U. Since the flow is autonomous, this means that all eigenvalues of DU(x0 )
have non-zero real part. Note that in the case of non-autonomous flows, conditions for a
point to by a hyperbolic fixed point cannot be given in terms of the vector field (see chapter
3, section 7 of [29] for an example). The fact that the condition on DU(x0 ) imply that x0 is
normally hyperbolic follows immediately from the variational equations.
Now consider the vector field Vt = U + Wt , and let us suppose that the wave maps Ωt±0
corresponding to U and Vt exist. Then the flows Ut and Vtt0 are conjugate in the extended
phase space Rn × R. The natural question arises: what invariant objects do we have for Vtt0
that corresponds to the fixed point, x0 , and it’s corresponding stable/unstable manifolds?
Since we are working in the extended phase space, we consider the line M̃ = {(x0 , t) :
t ∈ R}. Using the fact that the wave maps on the extended phase space, Ω̃± , act trivially
on the time variable, they will take M̃ to
Ñ ± := (Ω̃± )−1 (M̃ ) = {((Ωt±0 )−1 (x0 ), t0 ) : t0 ∈ R}
14
When we project Ñ ± onto Rn we obtain the sets
N ± = Πx Ñ ± = {(Ωt±0 )−1 (x0 ) : t0 ∈ R}
This example clarifies why we consider objects in the extended space: we started off with the
zero-dimensional invariant object {x0 } for Ut and constructed a corresponding 1-dimensional
invariant objects N ± for Vtt0 via the wave maps on the extended space. Moreover, if we define
Nt±0 = {(x0 , t0 ) : (x0 , t0 ) ∈ Ñ ± } then, as we show below these manifolds satisfy
Vtt0 (Nt±0 ) = Nt±
which is a property called time-dependent invariance (note that time-dependent invariance
would not hold if we chose to define Ñ ± := Ω̃± (M̃ ) = {(Ωt±0 (x0 ), t0 ) : t0 ∈ R}). With this example in mind we can now define the invariant manifolds we consider.
Definition 1 Let Ut be a time-dependent vector field on Rn , Utt0 denote the corresponding
flow map. Suppose we have a one-parameter family of manifolds, Mt0 , parameterized by time
with the following property:
Utt0 (Mt0 ) = Mt
The family Mt is called a time-dependent invariant manifold for U .
Moreover, we say that Mt is a time dependent normally hyperbolic invariant manifold
(TDNHIM) for the flow Utt0 provided that Mt is a time dependent invariant manifold and,
for each x0 ∈ Mt0 , we have a splitting
Tx0 Rn = Exs0 (t0 ) ⊕ Tx0 Mt0 ⊕ Exu0 (t0 )
such that the splittings are characterized by
v ∈ Exs0 (t0 ) ⇐⇒ |DUtt0 (x0 )v| ≤ Ce−λ(t−t0 ) |v| for t − t0 ≥ 0
v ∈ Tx0 Mt0 ⇐⇒ |DUtt0 (x0 )v| ≤ Ceµ(t−t0 ) |v| for all t, t0
(25)
v ∈ Exu0 (t0 ) ⇐⇒ |DUtt0 (x0 )v| ≤ Ceβ(t−t0 ) |v| for t − t0 ≤ 0
Where 0 < β < λ, µ. The bundles
E s,u =
[
Exs,u
(t0 )
0
t0 ∈R,x0 ∈Mt0
are called the stable/unstable bundles of Mt .
For us, the main importance of these manifolds is that they possess stable and unstable
manifolds, which is the content of the next proposition.
15
Proposition 3 Suppose that the flow Utt0 is C k (BR ) for every R > 0 and Mt is a timedependent normally hyperbolic invariant manifold for Utt0 . Moreover, suppose that
|Ut (x)| < C
sup
(26)
t∈R,x∈Mt
for some C > 0. Then for any point x0 ∈ M and t0 ∈ R there exist manifolds Wxs0 ,t0
and Wxu0 ,t0 that have the property that
Wxs0 ,t0 = {(x, t0 ) : d(Utt0 (x), Utt0 (x0 )) ≤ Cx e−λ(t−t0 ) , t − t0 ≥ 0}
Wxu0 ,t0 = {(x, t0 ) : d(Utt0 (x), Utt0 (x0 )) ≤ Cx eβ(t−t0 ) , t − t0 ≤ 0}
(27)
The manifolds W s and W u defined by
W s,u =
[
Wxs,u
0 ,t0
(28)
x0 ∈M,t0 ∈R
are called the stable (resp. unstable) manifolds of M and Wxs0 ,t0 (resp Wxu0 ,t0 ) are called the
stable (resp. unstable) fibers of the foliation of W s (resp. W u ). If l is any number that
satisfies
| log λ| | log β|
,
l < min k,
log µ log µ
then M is a C l manifold, Wxs,u
are C k manifolds and the maps (x0 , t0 ) 7→ Wxs,u
are C l−1−j
0 ,t0
0 ,t0
when Wxs,u
are given the C j topologies.
0 ,t0
Proof: The idea is to show that the manifold M̃ = {(x, t0 ) : x ∈ Mt0 } is a NHIM for the
extended flow Ũ . The existence of stable and unstable manifolds will then follow from the
theory developed in [26, 27, 28] or see [25] for a more modern, even infinite dimensional,
version. We note that some versions of the theory of NHIMs assume that the NHIM is
compact. However, in chapter 6 of [27] and in [25] compactness is not needed if one assumes
uniform differentiability.
To see that M̃ is a NHIM for the flow Ũt we compute
!
t+t0
0
Dx Utt+t
(x
)
D
U
(x
)
0
t0 t0
0
0
DŨt (x0 , t0 ) =
(29)
0
1
0
We can compute Dt0 Utt+t
(x0 ) using the chain rule
0
0
Dt0 Utt+t
(x0 ) = Dt Utt0 (x)|t=t+t0 + Dt0 Utt0 (x)|t=t+t0 ,t0 =t0
0
0
0
0
0
= Ut+t0 (Utt+t
(x)) + Dt0 Utt0 (Utt+t
(x)) = Ut+t0 (Utt+t
(x)) − Dx Utt+t
(x0 )Ut0 (x0 )
0
0
0
0
s,u
Thus, if we define Ẽ(x
:= {(v, 0) | v ∈ Exs,u
(t0 )} we have the splitting
0
0 ,t0 )
s
u
T(x0 ,t0 ) Rn × R = Ẽ(x
⊕ Ẽ(x
⊕ T(x0 ,t0 ) M̃
0 ,t0 )
0 ,t0 )
16
(30)
s,u
and (29) implies that Ẽ(x
have the same decay/growth rates of Exs,u
(t0 ).
0
0 ,t0 )
Moreover, notice that T(x0 ,t0 ) M̃ = {(v, τ ) | v ∈ Tx0 Mt0 , τ ∈ R}. Hence, using
assumption (26) and equations (29) and (30) it follows that the same growth rate bounds
on Tx0 M (t0 ) hold for T(x0 ,t0 ) M̃ . It follows that M̃ is NHIM for the extended flow Ũt . By the
s,u
theory developed in [25, 27, 26, 28] we conclude the existence of W(x
, and the regularity
0 ,t0 )
s,u
properties of Mt and W(x0 ,t0 ) .
Using the intertwining relations, i.e. Ωt± ◦ Vtt0 = Utt0 ◦ Ωt±0 , we can now use the scattering
theory to relate invariant manifolds of Utt0 to those of Vtt0 and vice versa.
Proposition 4 Let Utt0 and Vtt0 be flows such that the wave maps Ωt±0 exist , are smooth
and are invertible. Furthermore, suppose that Mt is a time-dependent normally hyperbolic
invariant manifold for Utt0 . Suppose that
(1) We have an estimate of the form
sup
|Ut (x)| < C
t∈R,x∈Mt
and (2) the limits
lim (Ωt∓0 )−1
(31)
t0 →±∞
exist.
Define one-parameter families of manifolds N± (t0 ) by
Nt±0 := (Ωt±0 (Mt0 ))−1
Then N ± := Nt0 are time-dependent invariant normally hyperbolic manifolds for V .
Moreover, (Ω̃+ )−1 (resp. (Ω̃− )−1 ) takes the stable/unstable manifolds of Mt to the
stable/unstable of N + (resp. N − ). More precisely we have
(Ωt±0 )−1 (Wxs,u
(Mt0 )) = W s,ut0
0 ,t0
((Ω± )−1 x0 ,t0 )
(Nt±0 )
(32)
Proof: We only consider N + since the proof for N − is similar. We first verify that N + is a
time-dependent invariant. The intertwining relations Ωt± Vtt0 = Utt0 Ωt±0 imply that
Vtt0 (Nt+0 ) = Nt+
(33)
Now we compute the growth/decay rates for DVtt0 on N + . The intertwining relations imply
that, for any x0 in Mt0
DVtt0 ((Ωt+0 )−1 (x0 )) = D(Ωt+ )−1 (Utt0 (x0 ))DUtt0 (x0 )DΩt+0 ((Ωt+0 )−1 (x0 ))
17
(34)
We want to control the limit as t → ∞. The third factor on the right side of (34) only
depends on t0 and is thus easy to control. We are assuming that we have estimates on the
second term, however the first term D(Ωt+ )−1 (Utt0 (x0 )) is a bit subtle to estimate. However,
in the proof that Ωt+0 is invertible, we saw that limt0 →∞ kΩt+0 − Id kC 1 (BR ) = 0.
Thus, kD(Ωt+0 )−1 (x0 )k is bounded as a function t0 > 0. However, we are assuming that
the corresponding limit
lim (Ωt+0 )−1 = lim Vtt0 ◦ Utt0
t0 →∞
t0 →∞
exists, though it generally will not be the identity, and hence kD(Ωt+0 )−1 (x0 )k is bounded
for all t0 . Thus, if vts0 ∈ Exs0 is a vector in the stable bundle for M then we we define
v̄ts0 := (DΩt+0 ((Ωt+0 )−1 (x0 )))−1 vts0 and notice that
|DVtt0 ((Ωt+0 )−1 (x0 ))v̄ts0 | = |D(Ωt+ )−1 (Utt0 (x0 ))DUtt0 (x0 )DΩt+0 ((Ωt+0 )−1 (x0 ))v̄ts0 |
= |D(Ωt+ )−1 (Utt0 (x0 ))DUtt0 (x0 )vts0 | ≤ M Ce−λ(t−t0 ) |v̄ts0 |
(35)
Thus, Ēts0 := (DΩt+0 ((Ωt+0 )−1 (x0 )))−1 (Ets0 ) is the stable bundle for N + . Similarly, we
define the unstable and center bundle for N + and show that the growth rates are the same
as for M , which proves that N + is in fact a time-dependent normally hyperbolic invariant
manifold.
The proof that the stable/unstable manifolds of M get mapped to stable/unstable
manifolds of N is a similar argument using the intertwining relations and using the fact
that the (four) limits
lim (Ωt±0 )−1 and lim (Ωt∓0 )−1
t0 →±∞
t0 →±∞
exist.
Remark. In Proposition 4, we made the assumption that limt0 →±∞ (Ωt∓0 )−1 exist. In
the proof of Proposition 4 we saw that limt0 →±∞ (Ωt±0 )−1 = Id. Using this and the fact
that st0 = Ωt+0 ◦ (Ωt−0 )−1 we have limt0 →±∞ (Ωt∓0 )−1 = limt0 →±∞ st0 , which may not exist in
general, and hence we make it an assumption. Without this assumption, we still have the
manifolds N ± , which are time-dependent invariant. However, to check that they are normally
hyperbolic and to check that Ωt±0 takes the stable/unstable manifolds of M to those of N ±
we needed these limits to exist.
3. Perturbative Calculations
In this section we will study perturbations of the flows, so that our flows will depend on
two parameters t and . Let Ut be a time-dependent vector field that also depends on a
parameter and let Utt,
denotes the corresponding flow. We will compare the dynamics for
0
18
= 0 and 6= 0 by considering the wave maps
t0 ,0
0
Ω,t
◦ Utt,
± = lim Ut
0
t→±∞
t0 −1
0
and scattering map st0 = Ω,t
+ ◦ (Ω− ) . The main content of this section are Theorems 2
and 3 where perturbative calculations of st0 are given.
Following the ideas of deformation theory (see [30] and [9]) we will write these families
in terms of their generators with respect to t and . The flow Utt,
and vector field Ut are
0
related by
d t,
U = Ut ◦ Utt,
0
dt t0
In this section we will also consider the vector field F defined by
d t,
U = F ◦ Utt,
0
d t0
In the literature, both Ut and F are referred to as the generator of Utt,
and to distinguish
0
between the two usages, we will refer to Ut and F as the t-generator and -generator,
respectively.
Theorem 2 Suppose that the following conditions hold: There exists an interval I
containing zero and an integer k ≥ 1 such that
Z ±∞
I± (t0 ) =
k DUσt0 ,0 (Uσ − Uσ0 ) ◦ Utσ0 , kC k (I×BR ) dσ < ∞
t0
for every R > 0. Here I is an interval in which the parameter lies in. Also assume that
and the following hold
lim I+ (t0 ) = lim I− (t0 ) = 0
t0 →∞
t0 →−∞
By Theorem 1 the wave and scattering maps exist, are C k , are diffeomorphisms, depend
smoothly on . If the limit
Z T1
d t0 ,,T1
t0 ,
t0
U dσ
S := lim
(Ω+
◦ Uσ )∗
T1 ,T2 →∞ −T
d σ
2
converges in C 1 (BR ) for every R > 0, then St0 is the vector field generating st0 .
We anticipate that in the case the vector fields are Hamiltonian, the symplectic geometry
will allow us to simplify Theorem 2. This is carried out in Section 3.1.
Note that an immediate corollary of Theorem 2 is that, if the conditions of Theorem 2
hold, then we have the following perturbative expansion of the scattering map
st0 = Id + S0t0 + O(2 )
19
where
S0t0
Z
=
T2
(Uσt0 ,0 )∗
lim
T1 ,T2 →∞
d U |=0 dσ
d σ
−T1
t0 ,0 ±T,
Proof: As in the proof of Theorem 1 we consider Ωt±0 ,,T := U±T
Ut0 . The method
t0 ,,T
t0 ,
proof of Theorem 1 and Corollary 4 imply that Ω±
converges to Ω± uniformly in BR × I
for any R > 0. This time, however, we will compute the vector field generating Ωt±0 ,,T with
t0 ,T
respect to not T . That is, we want to compute the vector fields X±,
that satisfy
d t0 ,,T
t0 ,T
Ω
= X±,
◦ Ωt±0 ,,T
d ±
To do this, we use Proposition 11 (c.f. Appendix A), which tells us how to compute generators
of compositions of diffeomorphisms. If Ftt,
is the -generator of Utt,
then Proposition 9
0
0
implies that
t0 ,T
t0 ,0
X±,
= (U±t
)∗ Ft±t,
0
The next proposition explains how to compute Ftt,
0
Proposition 5 If Utt,
denotes the time-t flow for Ut , then the generator Ftt,
is given by
0
0
Z t
d t,
t,
U dσ
Ft0 =
(Uσ )∗
d σ
t0
Proof: First note that we have the following variational equations (in what follows · and
D refer to differentiation with respect to and x respectively)
d t,
U̇t0 = DUt ◦ Utt,
◦ U̇tt,
+ U̇t ◦ Utt,
0
0
0
dt
d
DUtt,
= DUt ◦ Utt,
◦ DUtt,
0
0
0
dt
(36)
We consider (36) as equations for U̇tt,
and DUtt,
. These are supplemented with the
0
0
t0 ,
t0 ,
initial conditions U̇t0 = 0 and DUt0 = Id, which come from taking derivatives of the
equation Utt00 , = x with respect to and x.
We see that the first equation of (36) is linear nonhomogeneous, while the second is the
corresponding homogeneous equation. Thus, using the fact that (DUtt,
)−1 = DUtt0 , Utt,
we
0
0
can use the variation of parameters formula to say that
U̇tt,
0
Z
=
Z
t
DUtt,
DUσt0 ,
0
◦
Utσ,
U̇σ
0
◦
Utσ,
dσ
0
=
Z
t0
t
=
(Utt,
) ((Uσt0 , )∗ U̇σ )dσ
0 ∗
◦
Utt,
0
t0
Z
=
DUtt,
(U t0 , σ )∗ U̇σ dσ
0
t0
t
t,
t0 ,
(Ut0 ◦ Uσ )∗ U̇σ dσ ◦ Utt,
0
t0
20
t
(37)
Rt
It follows that the -generator of Utt,
is t0 (Uσt, )∗ U̇σ dσ.
0
Thus, we obtain that
Z ±T
±T,
t0 ,T
t0 ,0
t0 ,
(Ut0 ◦ Uσ )∗ U̇σ dσ
X±, = (U±T )∗
t0
Z
±T
t0 ,0
(U±T
=
◦
Ut±T,
0
Uσt0 , )∗ U̇σ dσ
◦
Z
(38)
±T
=
(Ωt±0 ,,T
◦
Uσt0 , )∗ U̇σ dσ
t0
t0
Thus, the -generator St0 ,T of st0 ,T is given by
t0 ,T
t0 ,T
t0 ,T
t0 ,T
St0 ,T = X+,
− (Ωt+0 ,,T )∗ ((Ωt−0 ,,T )−1 )∗ X−,
= X+,
− (Ωt+0 ,,T ◦ (Ωt−0 ,,T )−1 )∗ X−,
Z −T
Z T
t0 ,,T
t0 ,,T
t0 ,,T −1
t0 ,,T
t0 ,
t0 ,
=
(Ω+
◦ Uσ )∗ U̇σ dσ − (Ω+
◦ (Ω− ) )∗
(Ω−
◦ Uσ )∗ U̇σ dσ
t0
T
Z
=
t0
(Ωt+0 ,,T
◦
Uσt0 , )∗ U̇σ dσ
Z
t0
+
(Ωt+0 ,,T
◦
Uσt0 , )∗ U̇σ dσ
Z
=
−T
t0
T
(Ωt+0 ,,T ◦ Uσt0 , )∗ U̇σ dσ
−T
(39)
Our assumptions in Theorem 2 are that St0 ,T converges, as T → ∞, to a function that is in
C k (BR ) for every R > 0. Let us call the limiting function St0 ,∞ .
Now we will show show that St0 ,∞ is in fact the vector field generating st0 , that is, we
need to show that
d t0
s = St0 ,∞ ◦ st0
d
t0 ,T
t0
Since we know that s → s uniformly in we can say that
d
d t0
d t0 ,T
s =
lim st0 ,T = lim
s
T →∞ d
d
d T →∞
= lim St0 ,T ◦ st0 ,T = St0 ,∞ ◦ st0
(40)
T →∞
It follows that
St0
Z
=
T1
lim
T1 ,T2 →∞
(Ωt+0 ,,T1 ◦ Uσt0 , )∗ U̇σ dσ
(41)
−T2
3.1. The Hamiltonian case
In the case that the flow Utt,
corresponds to the time-t evolution of a Hamiltonian system
0
with Hamiltonian H (Q, P, t) we show that the scattering map st0 also has a Hamiltonian
structure. To be more precise, we refine the results of the previous section to show that st0
is the time- map of a Hamiltonian St0 , which we now compute. Before we state the next
21
theorem, let us introduce some terminology. If ft is a 2 parameter family of exact symplectic
diffeomorphisms, then as is shown in Appendix A, the -generator of ft satisfies
Ft = J∇Ft
for some function Ft , which we call the Hamiltonian -generating ft .
Theorem 3 Suppose that the following condition holds: There exists an interval I
containing zero and an integer k ≥ 1 such that
Z ±∞
k(DUσt0 ,0 J∇Hσ1 )(Utσ,
)kC k (I×Br ) dσ < ∞
I± (t0 ) =
0
t0
for every R > 0. Here I is an interval in which the parameter lies in. Also assume that
and the following limits hold
lim I+ (t0 ) = lim I− (t0 ) = 0
t0 →∞
t0 →−∞
By Theorem 1 the wave and scattering maps exist, are C k , are diffeomorphisms, depend
smoothly on . If the limit
Z T2 d
t0
S = lim
H ◦ UTσ, ◦ UtT,0
dσ
(42)
0
T1 ,T2 →∞
d
−T1
converges in C 1 (BR ) for every R > 0, then St0 is the Hamiltonian -generating st0
The significance of Theorem 3 is that it allows us to compute st0 . More precisely, Theorem
3 implies that st0 is the time- map of the Hamiltonian S given in (42). Hence, if F is any
observable, it follows that
F ◦ st0 = F + {S0 , F } + O(2 )
where
S0t0
Z
=
T2
lim
T1 ,T2 →∞
−T1
d
H |=0 ◦ Utσ,0
dσ
0
d
In particular, one can take F to be a coordinate function qi or pi , and hence we see that the
above formula allows us to compute st up to first order in knowing only the unperturbed
flow and the Hamiltonian.
These first order calculations are often called Melnikov theory. We note that many
expositions require extra assumptions such as the limiting behavior of the orbits [31]
The fact that Theorem 3 allows us to compute the effect of the scattering map on any
observable is quite significant. Indeed, one can use averaging [32] to understand how the
action, a slow variable, of a nearly integrable system changes in time. However, averaging
22
does not yield estimates on how the fast variables, i.e. the angle variables, change. Theorem
3 gives a way to compute the effect of an external force has on any quantity, including the
fast ones.
t0 ,
◦ U−T
◦ Ut−T,0
and
Proof: We first compute the Hamiltonian for st0 ,T = UTt0 ,0 ◦ UtT,
0
0
t0 ,T
take the limit as T → ∞. Since s is exact symplectic, we can use deformation theory to
construct the Hamiltonian St0 ,T -generating st0 ,T (c.f. Appendix A), i.e. St0 ,T satisfies
d t0 ,T
s = J∇St0 ,T ◦ st0 ,T
d
Using Proposition 11 from Appendix A, we have that
T,
St0 ,T = F−T
◦ UtT,0
0
T,
T,
is the Hamiltonian generating U−T
, which we can compute via the following
where F−T
Proposition.
Proposition 6 If Utt,
denotes the time-t flow for Ut , then the Hamiltonian Ftt,
-generating
0
0
t,
Ut0 is given by
Z t
d
,t
Ft0 =
H ◦ Utσ, dσ
d
t0
Proof: We computed the vector field FtT,
generating UtT,
, and hence we only need to
0
0
compute the contraction of this vector field with the symplectic form ω, which need not be
the canonical symplectic form, and obtain
Z
t
t
(Uσt, )∗ iU̇σ ωdσ
t
t0
Z 0t
Z t
t,
=
(Uσ )∗ dḢσ dσ = d
Ḣσ ◦ Utσ, dσ
iFtt, ω =
0
Z
i(Uσt, )∗ U̇σ ωdσ =
t0
(43)
t0
It follows that
St0 ,T
Z
T
=
−T
d H
d σ
dσ
◦ UTσ, ◦ UtT,0
0
(44)
Our assumptions guarantee that St0 ,T converges, as T → ∞ to a function St0 ,∞ in
C k (BR ) for every R > 0. The proof that St0 ,∞ is the Hamiltonian generating st0 is similar
the proof that St0 ,∞ is the vector field generating st0 in Theorem 2
23
3.1.1. Comparison to Fermi’s Golden Rule in quantum mechanics In this section we relate
the perturbative calculations given in Theorems 2 and 3 to Fermi’s Golden Rule in quantum
mechanics [33, 12] . For a given Hamiltonian H, i.e. a self-adjoint operator in this context,
we can consider the corresponding Schrödinger equation
i∂t U (t) = H(t)U (t)
U (0) = U0
(45)
Suppose that we have a Hamiltonians H0 and H = H0 + H1 and let U and V denote
their respective solution operators. We can also define the wave operators in this setting in
the exact same manner as before
t0 t
Vt0
Ωt±0 = lim U±t
t0 →±∞
And the scattering operator is defined as
st0 = Ω+ ◦ (Ω− )−1 =
lim
T1 ,T2 →∞
T1
UTt01 V−T
U −T2
2 t0
(46)
In the classical setting, the physical interpretation of st0 is that its input is the asymptotic
past of an orbit, and the output is the asymptotic future. The analogy in quantum mechanics
is that the quantity
Z
t0
u+ |s |u− = u+ (x)st0 u− (x)dx
Is the probability that the system is asymptotically in state x+ in the future if it was
asymptotically in state x− in the past.
In similar spirit as in [33], we can now formulate Fermi’s Golden Rule and see that it
is very similar to the perturbative calculations. Let us suppose that H = H0 + H1 + O(2 )
and consider the scattering operator
st0 =
lim
T1 ,T2 →∞
T1
UTt01 V−T
U −T2
2 t0
(47)
The variation of parameters formula says that
Z t
t
t
t
Vt0 = Ut0 − Ut0
Ust0 iH1 (s)Uts0 ds + O(2 )
t0
and hence substituting this expression for Vtt0 into (47)we can compute the scattering operator
as
Z
T1
st0 = st00 + lim
T1 ,T2 →∞
UTt01
−T2
iH1 Utt0 ds UTt02 + O(2 )
This formula is Fermi’s Golden Rule. This formula gives us a perturbative way to compute
the probability of a system having a certain asymptotic past and future. As in Theorems 2
and 3, the result is that we integrate the perturbation along the unperturbed trajectory.
24
4. Application to transition state theory
In this section we show how to use scattering theory to understand better a problem arising
in transition state theory [34]. This theory was developed to determine the rate of chemical
reactions. The theory has also been used to develop a general qualitative and quantitative
understanding of how chemical reactions take place. For example, invariant manifold theory
has been used to separate regions consisting of initial conditions resulting in a reaction.
In [1], they consider two reactant molecules, R1 and R2 that collide with each other to
form a meta-stable complex. After some time, the meta-stable complex can dissolve either
back to original state R1 + R2 or to one of two products P1 + P2 and P10 + P20 . The goal is
to understand how exposing the molecules to a laser shortly before and after the collision
affects the outcome of the reaction.
The system they consider is the laser driven Henon-Heiles system given by the
Hamiltonian
1
1
1
(48)
H = (p2x + p2x ) + (x2 + y 2 ) + x2 y − y 3 + E1 (t) exp(−αx2 − βy 2 )
2
2
3
∂
where E1 = − ∂t
A(t) where A(t) is given by
(
ωt
−A0 cos2 2N
, if |t| < Nωπ
A(t) =
0,
otherwise
The key features of the the Hamiltonian H0 = 12 (p2x + p2x ) + 21 (x2 + y 2 ) + x2 y − 13 y 3 is that
the Henon-Heiles potential U (x, y) = 21 (x2 + y 2 ) + x2 y − 31 y 3 has three saddle points which
give rise to three asymptotic channels in phase space. These channels correspond to each of
the three possible outcomes of the reaction. The driving term, H1 = E1 (t) exp(−αx2 − βy 2 )
corresponds to adding a laser to the system, which one hopes can be used to manipulate the
result of the interaction of the two molecules.
In [1], they use time-dependent normal-form theory to carry out numerical computations
of invariant manifolds for the system. These invariant manifolds separate the reactive and
non-reactive regions of phase space.
We now explain how to use the scattering theory developed in Section 2 to construct the
invariant manifolds computed in their paper. Moreover, we suggest an alternative method
to compute numerically the invariant manifolds.
First let us have a closer look at the system without the presence of a laser field, that
is the system given by H0 = 12 (p2x + p2x ) + 21 (x2 + y 2 ) + x2 y − 31 y 3 . This system has three
√
√
saddle points, x1 = ( 3/2, −1/2), x2 = (− 3/2, −1/2), and x3 = (0, 1). These points define
three asymptotic channels, one of which is interpreted as defining a reactant region, while
the other two are interpreted as defining two distinct product regions.
These saddle points have stable (resp. unstable) invariant manifolds Wxsi (resp. Wxui ).
These manifolds are characterized by having the following asymptotic properties
25
Wxsi = {x | d(Ut0 (x), xi ) ≤ Cx e−λt , t ≥ 0}
(49)
Wxui = {x | d(Ut0 (x), xi ) ≤ Cx eµt , t ≤ 0}
(50)
Here λ and µ is the expansion/contraction rate of the saddle points. In the presence of a
laser field, these manifolds become time-dependent normally hyperbolic invariant manifolds.
The next proposition allows us to compute an associated TDNHIM and the associated
stable/unstable manifolds for the full system.
Proposition 7 Let x0 be a fixed point of the system without the laser-driving term. Then
Nt±0 := (Ωt±0 )−1 (x0 ) are time-dependent normally hyperbolic invariant manifolds for the
system with the presence of a laser-pulse. Moreover, we have that
s,u
({x0 })) = W s,ut0
(Ωt±0 )−1 (W(x
0 ,t0 )
((Ω± )−1 (x0 ),t0 )
(Nt±0 )
Note that this follows immediately from Proposition 4. The conditions that the limits
lim (Ωt∓0 )−1
t0 →±∞
exist follows from the fact that the laser-pulse term is compactly supported in time. Indeed
in this case
t0
lim (Ωt∓0 )−1 = V±T
◦ Ut±T
0
t0 →±∞
where the support of H1 lies in [−T, T ].
Proposition 7 could be used as a basis as an alternate approach to computing the
stable/unstable manifolds computed in [1]. All that is necessary is to compute the
stable/unstable manifolds of the system in the absence of a laser field, numerically compute
Ω± , and finally compute the images of the original stable/unstable manifolds under (Ω± )−1 .
4.1. The branching ratio
An important quantity in transition state theory is the branching ratio, also commonly
called the branching fraction (see [15]). Recall that for the Henon-Heiles system, there are
two possible products, P1 + P2 and P10 + P20 , that the meta-stable complex can dissolve into.
Informally speaking, the branching ratio is the ratio of the probabilities that the outcome of
the reaction is one product versus another.
For the Henon-Heiles system without the laser term, the measure of the asymptotic
channels for each of the two products is infinite. However, it is common to consider only a
subset of the phase space. Indeed, in [1] restrict themselves to a energy surface and then
take a surface of section and obtain a two-dimensional planar subset of phase space. Since
the region is planar, the probability that a given reaction takes place can be computed by
taking the Lebesgue measure of the corresponding asymptotic channel intersected with the
26
Figure 2. The unperturbed asymptotic channel c0 is the right-half plane, the blue (color
may not appear on the printed version) dashed line is the boundary of the the perturbed
asymptotic channel c , the sample space B is the unit ball, the net volume of the green
(or the shaded region in the printed version) region is |c ∩ B| − |c0 ∩ B| and is given by
integrating the flux of J∇X + over ∂(B ∩ c0 ) ∩ B
planar region, which is finite. In [1] it is noticed that there is a change in the branching ratio
when the laser-term is included.
We now give perturbative calculations of the change in the branching ratio that sheds
light on the calculations given in [1]. The calculations we give can be thought of as a version
of Fermi’s Golden Rule for the branching ratio. Indeed, the results and techniques are similar
to the perturbative calculations for the scattering map given in Section 3.
We will consider general Hamiltonian systems. Thus we let H (Q, P, t) be a Hamiltonian
and let c01 and c02 be two asymptotic channels for H0 , that is c10 and c20 are regions of phase
space that are invariant for the unperturbed system. Since the asymptotic channels will
generally have infinite measure, we take a sample space B of initial data, which we take to
be an open subset of phase space. Using the intertwining relations, we deduce that
0 ,
cti,±
:= (Ωt±0 )−1 (c0i )
are asymptotic channels for the full system. As expected, the asymptotic channel depends on
the starting time t0 . For example, for the laser-driven Henon-Heiles system the asymptotic
channels are different for starting times t0 before, during, or after the laser-pulse. The
branching ratio is defined as
|c1,± ∩ B|
bt±0 , (c1 , c2 , U ) = |c2,± ∩ B|
where |A| denotes the Lebesgue measure of a set A. We now give perturbative
0 ,
calculations for |cti,±
∩ B|.
Figure 2 helps to understand the computation. In this case the channel c0 is the righthalf plane and the sample space B is the unit ball and the blue (color may not show up
on the printed version) dashed line is the boundary of the perturbed asymptotic channel c .
The following formula computes the change in the in the volume due to the presence of a
laser-field
Z
t0 ,
0
|ci,± ∩ B| = |ci ∩ B| + J∇Xt0,±
(x) · n(x)dA + O(2 )
0
∂(B∩c0i )∩B
Here Xt,±
is the Hamiltonian -generating (Ωt±0 , )−1 with respect to and n is the outward
0
pointing normal. Thus the change in volume is computed by integrating the flux of J∇Xt,±
0
27
over ∂(B ∩ c0i ) ∩ B. In a similar fashion as the perturbative calculations given in Theorem
3 we can compute Xt,±
by first computing Xt,+
, the Hamiltonian -generating
0
0 ,T
t0 , ±T,0
(Ωt±0 ,T )−1 = U±T
Ut0
and taking the limit as T → ∞. Using Proposition 11 we have that
Xt,±
= −Ft,±T
◦ Ut±T,
0
0
0 ,T
Where FtT
is the Hamiltonian -generating UtT0 , and by Proposition 6 is given by Ft,T
=
0
0
Rt
R ±∞ d
σ,
,±
σ,
Ḣ ◦ Ut dσ, and hence we conclude that Xt0 = t0 d H ◦ Ut0 and hence,
t0
Xt0,±
0
±∞
Z
=
t0
d
H |=0 ◦ Utσ,0
0
d
We summarize this in the following Proposition.
Proposition 8 Let H (Q, P, t) be a Hamiltonian with corresponding flow Utt0 and suppose
0
that the wave maps Ω,t
exist, are smooth and invertible. Let c01 , c02 be two asymptotic
±
0 ,
channels for H0 and U an open subset of phase space. Then cti,±
:= (Ωt±0 )−1 (c0i ) are asymptotic
channels for H and
Z
t0 ,
0
J∇Xt0,±
|ci,± ∩ U | = |ci ∩ U | + (x) · n(x)dA + O(2 )
0
∂(U ∩c0i )∩U
where
Xt0,±
0
Z
±∞
=
t0
d
(H ) |=0 ◦ Utσ,0
0
d
Appendix A. Appendix on Deformation Theory
In this section we recall some facts from deformation theory that we used throughout the
paper. The treatment here follows [9] and we refer the reader to [30] for more details on the
subject. Let M be a manifold. Suppose we have a family of diffeomorphisms fa : M → M ,
where a lies in some (possibly infinite) interval. In the context of proving Theorem 1 the
variable a corresponded to the time variable t, whereas in the the context of Theorem 3 we
used the perturbation parameter . Hence, we use a as a neutral symbol for the parameter of
the family in this section to avoid confusion in our two different uses of deformation theory
in this paper.
Given a family of diffeomorphisms fa we can obtain a vector field Fa defined by
d
fa = Fa ◦ fa
da
28
(A.1)
Conversely, given a C 1 vector field Fa and some initial conditions f0 , we can determine
uniquely fa by the theory of ODE’s.
Hence, it is equivalent to speak of the family fa or of (Fa , f0 ). Note that we can think
of Fa as an “infinitesimal diffeomorphism”. Many functional equations that are non-linear
and non-local when formulated in terms of fa become linear and local when formulated in
terms of Fa .
Proposition 9 If fa , ga : M → M are diffeomorphisms and Fa , Ga are their generators,
then we have
(A 1) If we define ha = ga ◦ fa , then its generator Ha is given by
Ha = Ga + (ga )∗ Fa
where (ga )∗ is the push-forward:
(ga )∗ Fa = Dga ◦ ga−1 Fa ◦ ga−1 = (Dga Fa ) ◦ ga−1
(A 2) If ga ◦ fa = Id then
Ga + (ga )∗ Fa = 0
The proof of this proposition simply involves recalling the definition of Ha . With this,
we can now state the following proposition, that asserts that the existence of a Hamiltonian
Fa that generates fa . We used this proposition several times to prove Theorems 1 and 2.
We will now consider the case when fa are exact symplectic and show that they have
an associated Hamiltonian.
Proposition 10 Let fa be a smooth family of exact symplectic diffeomorphisms. We have
If f0 is exact symplectic, then fa is exact symplectic if and only if there exists a family
of functions Fa : M → R such that:
iFa ω = dFa
Proof:
By the definition of fa being exact symplectic, we know that there exists a family of
primitive functions P fa which satisfy
fa∗ (α) = α + P fa
and hence we have that
d
d fa
P
da
=
d ∗
f (α)
da a
The right side of the equation is the Lie derivative of α with respect to the vector field Fa .
Thus we can use Cartan’s formula to say that
29
d
d fa
P
da
= fa∗ (iFa dα + diFa α)
And hence, by rearranging terms we have that
d fa
∗
∗
f (iFa ω) = d
P − f IFa α = dψa
da
where ψa =
d
P fa
da
− f ∗ IFa α. Since fa is a diffeomorphism, we can say that
d fa
iFa ω = d (fa )∗
− i Fa α
P
da
∗
where, recall that (fa )∗ = (fa−1 ) . Hence we can take
d fa
Fa = (fa )∗
P
− iFa α
da
Now we derive a formula for the Hamiltonian generating ha = ga ◦ fa in terms of Fa and
Ga .
With this we can now state the following proposition that tells us how to compute the
Hamiltonian of the composition and inverses of families of exact symplectic diffeomorphisms
was needed in both Theorems 1 and 3
Proposition 11 If fa : M → M and ga : M → M are exact symplectic diffeomorphisms
generated by their Hamiltonians Fa : M → R and Ga :→ R respectively, then we have:
(1) If we define ha = ga ◦ fa , its Hamiltonian Ha is given by
Ha = Ga + Fa ◦ ga−1
(2) If ga ◦ fa = Id then
Ga + Fa ◦ ga−1 = 0
Proof: By Prop. 10 we have that
iHa ω = iGa +(ga )∗ Fa ω = Ga + i(ga )∗ Fa ω
Now we compute that
(i(ga )∗ Fa ω)(v) = ω(Dga ◦ ga−1 Fa ◦ ga−1 , v)
Since ga−1 is symplectic we can furthermore say that
ω(Dga ◦ ga−1 Fa ◦ ga−1 , v) = ω(Dga−1 Dga ◦ ga−1 Fa ◦ ga−1 , Dga−1 v)
= ω(Fa ◦ ga−1 , Dga−1 v)
30
(A.2)
However, by the definition of Fa we have that
ω(Fa ◦ ga−1 , Dga−1 v) = dFa ◦ ga−1 · Dga−1 v = d(Fa ga−1 ) · v
Thus we have proven that
i(ga )∗ Fa ω = d(Fa ga−1 )
which thus completes the proof of the proposition. Acknowledgements
We thank Profs. T. Bartsch, F. Borondo, M. Capinski, A. Delshams, M. Gidea and T.
M.-Seara for several conversations on the topics of this paper and for encouragement. The
work of the authors has been partially supported by NSF grant DMS 0901389 and by Texas
Coordinating Board ARP 0223
References
[1] Shinnosuke Kawai, Andre D. Bandrauk, Charles Jaffe, Thomas Bartsch, Jesus Palacian, and T. Uzer.
Transition state theory for laser-driven reactions. The Journal of Chemical Physics, 126(16):164306,
2007.
[2] Dmitry Treschev and Oleg Zubelevich. Introduction to the perturbation theory of Hamiltonian systems.
Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010.
[3] Antonio Gamboa Suárez, Daniel Hestroffer, and David Farrelly. Formation of the extreme Kuiperbelt binary 2001 QW322 through adiabatic switching of orbital elements. Celestial Mech. Dynam.
Astronom., 106(3):245–259, 2010.
[4] C. Jaffe, S.D. Ross, M.W. Lo, J. Marsden, D. Farrelly, and T. Uzer. Statistical theory of asteroid escape
rates. Phys Rev Lett, 89(1):011101, 2002.
[5] W. Hunziker. The S-matrix in classical mechanics. Comm. Math. Phys., 8(4):282–299, 1968.
[6] H. Narnhofer and W. Thirring. Canonical scattering transformation in classical mechanics. Phys. Rev.
A (3), 23(4):1688–1697, 1981.
[7] A. Knauf. Qualitative aspects of classical potential scattering. Regul. Chaotic Dyn., 4(1):3–22, 1999.
[8] Vladimir Buslaev and Alexander Pushnitski. The scattering matrix and associated formulas in
Hamiltonian mechanics. Comm. Math. Phys., 293(2):563–588, 2010.
[9] R. de la Llave, J. M. Marco, and R. Moriyón. Canonical perturbation theory of Anosov systems and
regularity results for the Livšic cohomology equation. Ann. of Math. (2), 123(3):537–611, 1986.
[10] Michael Reed and Barry Simon. Methods of modern mathematical physics. III. Academic Press
[Harcourt Brace Jovanovich Publishers], New York, 1979. Scattering theory.
[11] Edward Nelson. Topics in dynamics. I: Flows. Mathematical Notes. Princeton University Press,
Princeton, N.J., 1969.
[12] Walter Thirring. A course in mathematical physics. Vol. 3. Springer-Verlag, New York, 1981. Quantum
mechanics of atoms and molecules, Translated from the German by Evans M. Harrell, Lecture Notes
in Physics, 141.
[13] Walter Thirring. Classical mathematical physics. Springer-Verlag, New York, third edition, 1997.
Dynamical systems and field theories, Translated from the German by Evans M. Harrell, II.
31
[14] Amadeu Delshams, Rafael de la Llave, and Tere M. Seara. Geometric properties of the scattering map
of a normally hyperbolic invariant manifold. Adv. Math., 217(3):1096–1153, 2008.
[15] Iupac compendium of chemical terminology – the gold book, 2009.
[16] Ralph Abraham and Jerrold E. Marsden. Foundations of mechanics. Benjamin/Cummings Publishing
Co. Inc. Advanced Book Program, Reading, Mass., 1978. Second edition, revised and enlarged, With
the assistance of Tudor Raţiu and Richard Cushman.
[17] Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. Gravitation. W. H. Freeman and
Co., San Francisco, Calif., 1973.
[18] J. Hadamard. Sur le module maximum d’une fonction et de ses derives. Bull. Soc. Math. France,
42:68–72, 1898.
[19] A. Kolmogoroff. On inequalities between the upper bounds of the successive derivatives of an arbitrary
function on an infinite interval. Amer. Math. Soc. Translation, 1949(4):19, 1949.
[20] R. de la Llave and R. Obaya. Regularity of the composition operator in spaces of Hölder functions.
Discrete Contin. Dynam. Systems, 5(1):157–184, 1999.
[21] Ralph Abraham and Joel Robbin. Transversal mappings and flows. An appendix by Al Kelley. W. A.
Benjamin, Inc., New York-Amsterdam, 1967.
[22] Eugene Gutkin. Integrable Hamiltonians with exponential potential. Phys. D, 16(3):398–404, 1985.
[23] Eugene Gutkin. Asymptotics of trajectories for cone potentials. Phys. D, 17(2):235–242, 1985.
[24] Andrea Hubacher. Classical scattering theory in one dimension. Comm. Math. Phys., 123(3):353–375,
1989.
[25] Peter W. Bates, Kening Lu, and Chongchun Zeng. Approximately invariant manifolds and global
dynamics of spike states. Invent. Math., 174(2):355–433, 2008.
[26] Neil Fenichel. Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J.,
21:193–226, 1971/1972.
[27] M. W. Hirsch, C. C. Pugh, and M. Shub. Invariant manifolds. Lecture Notes in Mathematics, Vol.
583. Springer-Verlag, Berlin, 1977.
[28] Yakov B. Pesin. Lectures on partial hyperbolicity and stable ergodicity. Zurich Lectures in Advanced
Mathematics. European Mathematical Society (EMS), Zürich, 2004.
[29] Jack K. Hale. Ordinary differential equations. Robert E. Krieger Publishing Co. Inc., Huntington,
N.Y., second edition, 1980.
[30] R. Thom and H. Levine. Singularities of differentiable mappings. In C.T. Wall, editor, Proceedings of
Liverpool Singularities— Symposium. I (1 969/1970)., pages 1–89. Springer-Verlag, Berlin, 1971.
[31] Philip J. Holmes and Jerrold E. Marsden. Melnikov’s method and Arnol0 d diffusion for perturbations
of integrable Hamiltonian systems. J. Math. Phys., 23(4):669–675, 1982.
[32] V. I. Arnol0 d. Mathematical methods of classical mechanics, volume 60 of Graduate Texts in
Mathematics. Springer-Verlag, New York, second edition, 1989. Translated from the Russian by
K. Vogtmann and A. Weinstein.
[33] Roger G. Newton. Scattering theory of waves and particles. Dover Publications Inc., Mineola, NY,
2002. Reprint of the 1982 second edition [Springer, New York; MR0666397 (84f:81001)], with list of
errata prepared for this edition by the author.
[34] Holger Waalkens, Roman Schubert, and Stephen Wiggins. Wigner’s dynamical transition state theory
in phase space: classical and quantum. Nonlinearity, 21(1):R1–R118, 2008.
32
Download