Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Integrable hamiltonian systems and spectral theory
.pdf
§3.Then-Particle System on the Line with the Inverse Square Potential 51
2
2
are integrals of the motion, which are polynomials in a
, a
1
2
only ν =[n/2] of these are not zero, but the remaining I
2
, ..., a
, I4, ..., I2νare
2
. By (2.2)
n−1
actually independent polynomials.
With
1
uk/2
e
=
a
k
2
the equations (2.4) take the form
1
˙u
(e
=
k
2
where we formally set u
u
0
u
k+1
− e
k−1
)(k =1, 2, ..., n− 1) (2.5)
= −∞, un= −∞. These are the equations which
Kac and v. Moerbeke considered in their discretization of the Korteweg – de Vries
1
equation [8].
The above derivation is, of course, not new; it is quite analogous
to that of Flaschka [4]. But we will use the above representation (2.3) of
the differential equation (2.4) to describe its solutions as rational functions of
exponentials (Section 6) and to investigate the scattering problem related to (2.5)
(Section 7).
Incidentally, the above equations (2.3) do not represent the only deforma-
tions of L preserving the spectrum. On the contrary all B giving rise to such
deformations form an (n − [n/2])-dimensional space [12].
§3. The n-Particle System on the Line with the Inverse
Square Potential
We consider n particles on the line with coordinates x1, x2, ..., xnand
define
U(x)=
(xk− xl)−2,k,l=1, 2, ..., n (3.1)
k<l
as their potential so that the equations of motion are given by
2
x
d
k
2
dt
= −
∂U
∂x
k
=2
(xk− xl)
j=k
−3
(k =1, 2, ..., n). (3.2)
It is remarkable that this system possesses n integrals of the motion which are
polynomials in ˙x
and (xk−xl)−2. This fact can again be derived by considering
k
isospectral deformations of another class of matrices.
1
As I learned from H. Flaschka, this system (2.5) and its relation to the Toda lattice was already
mentioned by M. Henon in a letter of August 28, 1973.
52 Three Integrable Hamiltonian Systems
The quantum-mechanical analog of (3.2) has been studied by Calogero and
Marchioro in a number of papers [2, 3, 11] and Calogero succeeded in determining explicit expressions for the spectrum for this problem. He conjectured from
his work that the classical problem, being the limit of the quantum-theoretical
one, should be integrable. For n =3this was already verified by Marchioro [11]
but his approach does not lend itself to generalization. In order to introduce the
class of matrices adapted to this problem we set
kl
0, for k = l
(x
− xl)−1, for k = l,
k
=
z
and form the matrices
=(z
α
=diag
α
α
) for α =1, 2,
kl
, ..., yn},
1
n
j=1
α
z
kj
for α =2, 3.
(3.3)
Z
Y =diag{y
D
Then we define
so that L = L
L = Y + iZ
∗
is Hermitean and B skew Hermitean.
; B = iD2− iZ2, (3.4)
1
The deformation equations
dL
= BL− LB (3.5)
dt
for this class of matrices can be transformed into the equation of motion (3.2)!
This implies by the argument of the previous section that the coefficients I
of
k
the characteristic polynomial
det(λI − L)=λ
n
+ I1λ
n−1
+ ···+ I
n
are integrals of the differential equations. Moreover, they are rational functions
of the coordinates and in involution.
To relate Eqs. (3.2) and (3.5) to each other observe that (3.5) depends
only on the n − 1 differences x
involves all n coordinates x
. Therefore we rewrite (3.2) in terms of the z
k
− xk(k =1, 2, ..., n− 1), while (3.2)
k+1
kl
§3.Then-Particle System on the Line with the Inverse Square Potential 53
= −˙x
and y
k
Of course this system is highly redundant, since only the n − 1 variables z
k
˙yk= −¨xk= −2
2
˙z
= z
kl
(yk− yl).
kl
n
j=1
3
z
,
kj
(3.6)
k, k+1
are independent, the other being determined by the relations
−1
−1
kr
+ z
−1
if k, l, r distinct, and zkl+ zlk=0. (3.7)
rl
= z
z
kl
But one verifies immediately that these relations are consistent with (3.6): If
they hold for t =0then for all t.
Now we identify (3.6) with the deformation equations (3.5). For this purpose
we have to compute
[B, L]=i[Y, Z
where we used (3.4). The element of [Z
by
hence the corresponding term in [Z
2
(z
zrl− z
kr
r
] − [D2,Z1]+[Z2,Z1](3.8)
2
in the (k, l) position is given
2
zkr),
rl
2
kr
− z
2
rl
)zkl.
(z
r
r
(z
2
kr
2
rl
2,Z1]kl
zrl− z
] − [D2,Z1] is
2,Z1
zkr) −
To simplify this expression we use the identities (3.7) as follows: The
summands of the sum above can be factored
kl, r
kr
2
= z
Q
zrl− z
2
zkr− (z
rl
2
kr
2
− z
)zkl=(zkr− zrl)P
rl
kl, r
with
=(zkrzrl− (zkr+ zrl)zkl).
P
kl, r
If all k, l, r are distinct, this takes the form
−1
−1
P
= zkrzrlzkl{z
kl, r
− z
kl
−1
− z
kr
} =0
rl
on account of (3.7). For k = l one gets obviously
= −z
2
, if r = k or r = l.
kl
P
kl, r
,
54 Three Integrable Hamiltonian Systems
Thus
which shows that [Z
2,Z1
n
Q
=0 for k = l
kl, r
r=1
] −[D2,Z1] is a diagonal matrix. If one computes the
diagonal elements one finds
[Z
] − [D2,Z1]=−2D3,
2,Z1
with the notation of (3.3). Thus with (3.8) the equations (3.5) take the form
dL
= i[Y, Z
dt
] − 2D3,
2
and, in components,
= −2z
˙y
k
˙z
=(yk− yl)z
kl
3
,
kj
2
,
kl
in agreement with (3.6).
This establishes the existence of the integrals, as well as their rational
character. In Section 4, in which we study the scattering problem for this system,
we will find without further calculation that these integrals are in involution
1
.
§ 4. Asymptotic Behavior, Marchioro’s Conjecture
The n-particle system of the preceding section has a very simple behavior.
Since the particles exert a repelling force on each other they fly apart as t →±∞
and ultimately behave like force particles. From this it is clear that the limits
˙xk(±t)= ˙xk(±∞) exist. As a matter of fact, these limit velocities or
lim
t→∞
their symmetric functions can be assigned as integrals to the orbits to which they
belong. Thus the existence of integrals is no surprise for a system like (3.2).
However, the existence of rational integrals is remarkable, and it implies that
(+∞)= ˙x
˙x
k
(−∞),k=1, 2, ..., n, (4.1)
n+1−k
so that the particles simply exchange their velocity. Moreover, the above velocities are distinct and agree with the negative of the eigenvalues of the matrix (3.4)
1
Extending this method, M. Adler, a student at New York University, found n rational integrals
for U =
{α(xk− xl)−2+ β(xk− xl)2}.
k<l
§4. Asymptotic Behavior, Marchioro’s Conjecture 55
belonging to the orbit considered. This way we will prove the fact that matrices
of the form (3.4) always have simple eigenvalues. One may ask for the phase
shifts δ
defined by
k
x
k
(t) − x
(−t) − 2˙xk(∞)t → δ
n−k+1
k
for t → +∞. It is easily verified that δ1= δ2=0for n =2, and one may
conjecture that δ
=0for any n>2, but this we have not been able to establish1.
k
The relations (4.1) have been established by Marchioro [11] for the case
n =3and were conjectured by him for arbitrary n. For the quantum-mechanical
problem they were established by Calogero [2].
To prove the above assertion we observe that we may label the particles
according to the order
x
1<x2
<... <xn.
Indeed, since the Hamiltonian of (3.2) is given by
n
1
=
2
k=1
2
y
+
k
(xk− xl)−2, (4.2)
k<l
the minimal existence of the particles is bounded away from zero for any solution.
Moreover, the velocities −y
=˙xkare bounded for all t for every orbit.
k
Our next goal is to show that
lim
yk(±t)=yk(±∞)(4.3)
t→∞
exists and that
y
(+∞) >y2(+∞) >... >yn(+∞). (4.4)
1
From
1
(¨x
− ¨x1)=
n
2
and the boundedness of ˙x
+∞
(xn− xj)−3+
j<n
we conclude, by integration that
k
(xj− x1)−3> 0(4.4)
j>1
(xk− xl)−3dt < ∞ for k>l=1and for l<k= n. (4.5)
−∞
1
Note added in proof. Meanwhile we have been able to verify that indeed δk=0for nq.
56 Three Integrable Hamiltonian Systems
Considering the other differential equations one concludes with a simple induction argument (which we forego) that (4.5) holds for all pairs k>l.This,in
turn implies from (3.2) that the limits lim
˙xk(±t) exist, proving (4.3). Because
t→∞
of the ordering of the particles we have obviously
˙x
(+∞) ˙x2(+∞) ... ˙xn(+∞),
1
(−∞) ˙x2(−∞) ... ˙xn(−∞).
˙x
1
(4.6)
To prove (4.4) we proceed as follows: Consider first ϕ(t)=x
which, by (4.4
), satisfies
1
¨ϕ 2(x
2
− x1)−3> 0. (4.7)
n
− x1> 0
n
Thus ˙ϕ is monotone increasing and ˙ϕ(+∞) 0, by (4.6). Were ˙ϕ(+∞)=0
then ˙ϕ(t) < 0 and thus ϕ bounded. But then the righthand side of (4.7) would
be bounded away from zero, hence ϕ unbounded. This contradiction shows that
˙x
(+∞) < ˙xn(+∞).
1
Thus, in the first row of (4.6) we do not have equality in all places, i. e.,
there exists an s with
˙x
s
From this we will show now ˙x
(+∞) < ˙x
(+∞) < ˙xs(+∞) and ˙x
1
(+∞). (4.8)
s+1
(+∞) < ˙xn(+∞)
s+1
which implies readily that all velocity are different. It suffices to show
(+∞) < ˙xs(+∞), the other case being symmetric to it.
˙x
1
From (4.8) we conclude that x
2
1
d
(xs− x1)=
2
2
dt
2(x
Thus ψ = x
− x1+ At−1with some positive constant A satisfies
s
(xs− xj)−3− O(t−3)+
j<s
s
¨
ψ 4(x
− xs= O(t−1) for j>sand therefore
j
(xj− x1)−3
j>1
− x1)−3− O(t−3).
−3
− x1)
s
for t>t
0
and is bounded from below. Thus˙ψ is increasing and˙ψ(+∞) 0.Asbefore
we conclude that the assumption˙ψ(∞)=0leads to a contradiction. Since
˙
ψ(t) <˙ψ(∞)=0(t>t
) implies ψ to be bounded for t>t0hence¨ψ would
0
§5. The Periodic Case — Sutherland’s Equation 57
be bounded away from zero, and so ψ unbounded. Thus˙ψ(∞) > 0 as we
wanted to show.
Since y
= −˙xkwe have established (4.4). This implies obviously
k
(x
− xl)−1= O(t−1) for t → +∞,k= l
k
so that we can see that the matrix L(t) has a limit L(∞) which is a diagonal
matrix. Since the eigenvalues λ
of L(t) are independent of t we have
k
y
(∞)=λ
k
k
if we make the convention to order these like
λ
n<λn−1
<... <λ1.
for t →−∞the matrix L also approaches a diagonal matrix with the same
eigenvalues in the diagonal, but, because of (4.6) in reversed order. Thus
˙x
(+∞)=−yk(+∞)=−λk;˙x
k
n+1−k
(−∞)=−y
n+1−k
(−∞)=−λk,
and (4.1) is proven.
Finally, we observe that the integrals I
are in involution. For x
derivatives to σ
converges to {σ
(y), the symmetric functions of y. Thus the Poisson bracket
k
k,σl
− x
k
G
kl
→∞these integrals Ikconverge with their
k−1
n
∂(Ik,Il)
=
∂(xr,yr)
r=1
} =0. Thus, along any solution of our system Gkl→ 0
as t →∞. On the other hand, as is well known, G
hence G
=0for all x, y.
kl
= Ik(x, y)(k =1, 2, ..., n)
k
= {I
}
k,Il
are integrals themselves,
kl
§ 5. The Periodic Case — Sutherland’s Equation
If one wants to study the problems of the previous two sections on the circle
it is natural to use the identity
+∞
(x − kπ)−2=sin−2x
k=−∞
58 Three Integrable Hamiltonian Systems
as motivation to introduce the potential
U(x)=
1
2
α2sin−2(α(xk− xl)) (α>0) (5.1)
k=l(n)
where the summation is taken over all distinct pairs k, l (mod n). The coordinates x
of the particles may be defined for all integers k such that
k
x
= xl(mod (π/α)) if and only if k = l (mod n),
k
so that is suffices to consider x
for k =1, 2, ...,n. The differential equations
k
take the form
2
d
x
k
∂U
dt
= −
2
∂x
=2α
k
which is the classical analog of Sutherland’s equation [14]. With y
3
j=k(n)
cot α(xk− xj)sin−2(α(xk− xj)) (5.2)
k
= −˙xkthe
Hamiltonian is
=
1
2
k (mod n)
2
y
k
showing that, on an energy surface
particles remains bounded away from zero and the velocities |y
2
α
+
2
sin−2(α(xk− xl))
k=l(n)
=constthe minimal distance of the
| bounded away
k
from ∞. Thus the energy surface is compact and most solutions of (5.2) turn
out to be quasi-periodic. This will be a consequence of well known facts [1]
about integrable Hamiltonian systems if we show that (5.2) has n independent
integrals which are in involution.
The construction of these integrals follows the pattern of Section 3. We set
= αcot α(xk− xl), if k = l(n),
z
kl
z
=0, if k = l(n)
kl
and rewrite the system (5.2) in the form
= U
˙y
k
˙z
=(α2+ z
kl
x
= −2
k
2
)(yk− yl) for k = l(n).
kl
zkj(α2+ z
j=k(n)
2
)
kj
(5.3)
Here the last line follows from the differential equation for cot x.
§5. The Periodic Case — Sutherland’s Equation 59
To put these differential equations in the form (3.5) we introduce the n by n
matrices
=(zkl); Z2=(z
Z
1
2
kl
+ α2)
where k, l =1, 2, ..., n. With
D
D
=diag
2
=diag
3
j (mod n)
j (mod n)
Y =diag{y
(z
zkj(z
k
2
+ α2);
kj
2
kj
}.
+ α2);
we set
L = Y + iZ; B = iD
− iZ2. (5.4)
2
Then it is a straightforward, though surprising, calculation that (5.3) can be
written in the form
dL
= BL− LB. (5.5)
dt
In fact for α → 0 the formal identities go over into those of Section 3, except
for the boundary conditions.
Thus it follows that the coefficients I
det(λI −L)=λn+ I1λ
, I2, ..., I
1
n−1
+ ···+ I
n
n
are independent integrals of the motion. We will not verify here that they are
involution
1
, but observe that they are rational functions of ykand e
iα(xk−xl)
.
To verify (5.5) one has to use the addition theorem for cot x which gives,
for k, l, r distinct modulo n:
− α
2
rl
z
z
krzrl
=
kl
zkr+ z
hence, for k = l (mod n)
0 if r = k, l(n)
P
= zkrzrl− α2− (zkr+ zrl)zkl=
kl, r
1
This could be done by replacing α by iα and using the same argument as in the previous section.
− z
2
2
− α
kl
if r = k, l(n).
60 Three Integrable Hamiltonian Systems
This implies for
=(zkr− zrl)P
Q
kl, r
kl, r
= −zkr(z
2
+ α2)+zrl(z
rl
2
kr
+ α2) − (z
kr
2
2
− z
)z
kl
rl
that
Q
r
so that the matrix with the elements
trix −2D
.
3
kl, r
⎧
0 if k = l (mod n)
⎨
⎩
− 2
r
(z
2
kr
+ α2)z
Q
r
if k = l(mod n)
kr
agrees with the diagonal ma-
kl, r
=
Now we compute the commutator
− D2,Z1]=
[Z
2
r
Q
kl, r
= −2D
3
and, thus, from (5.4)
[B, L]=i[Y, Z
] − [D2,Z1]+[Z2,Z1]=i[Y, Z2] − 2D3.
2
From this identity one reads off that (5.5) agrees with the equation (5.3). This
makes the statement about the I
being integrals of the motion again obvious.
k
§ 6. Rational Character of the Solution of (2.4)
We return to the equations (2.4) or (2.5) and investigate their solutions
using the fact that these differential equations describe isospectral deformation
of Jacobi matrices. We begin with introducing a set of variables r
manifold of Jacobi matrices (2.1) for which the spectrum is fixed. This is the
analog of the inverse spectrum problem.
Let
R(λ)=(λI − L)
−1
and e1be the vector with components (1, 0, ..., 0). We introduce the rational
function
f(λ)=(R(λ)e
which has simple poles at λ = λ
so that
f(λ)=
with a positive residue which we denote by r
k
n
λ − λ
k=1
1,e1
2
r
k
)
.
k
k
on the
2
k
,
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]