Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Integrable hamiltonian systems and spectral theory
.pdf
§5. Examples of Integrable Flows 171
d. Geodesic Flow on the Orthogonal Group (Manakov [8], Mischenko [11])
Arnold studied the geodesic flow on SO(n) under a right-invariant metric,
say
$
tr(˙UTG˙U) dt
where G is a fixed positive definite symmetric matrix and U ∈ SO(n).Ifwe
define the element A =˙UU
Setting L = GA + AG + μG
−1
in the Lie algebra, the Euler equations become
d
(GA + AG)=[A
dt
2
, B = A+ μG with any scalar μ, these equations
2
,G].
can be written in the Lax form
d
L =[B, L].
dt
These formulas were derived by Manakov [8] to construct integrals of this system
from the characteristic polynomial of L.
We should like to consider a 2n-dimensional subsystem of this flow by
setting
G = diag(g
L = x ⊗y − y ⊗x + μG
x
iyj
B =
gi+ g
1,g2
− xjy
j
, ..., gn),
i
+ μG.
2
,
One verifies that this system is compatible and is described by the Hamiltonian
H = −
1
2
i<j
(xiyj− xjyi)
gi+ g
2
.
j
This system is clearly related to those of Section 2 and agrees with (2.9) if we
set
=
ab
cd
j
2
= μg
,βj= μgj=√μαj.
j
α
01
−10
,
Thus this system is integrable and its solutions are expressible in terms of
hyperelliptic functions.
172 Geometry of Quadrics and Spectral Theory
e. Hill’s Equation (McKean and Trubowitz [9, 10])
We consider the Hill equation
2
d
−
y + q(s)y = λy,
ds
where q(s +1)=q(s). McKean and Trubowitz as well as Novikov et al. have
studied the problem of recovering the potential q = q(s) from various spectra.
As a rule one needs two spectra, corresponding to two sets of different boundary
conditions, to determine q(s), whereas there exists a family of potentials giving
rise to one spectrum determined by one set of boundary conditions. Of particular
interest here is the boundary condition of periodicity, say
y(s +1)=±y(s), or y(s +2)=y(s).
The eigenvalues
band spectrum [λ
1λ
0
equation considered on the line −∞ <s<+∞. The other intervals (−∞,λ
(λ
1
,λ
), (λ
2
,λ
), ...are called the instability intervals.
3
4
It is clear that the spectrum λ
0
,λ
<λ
] ∪ [λ
1
1
λ
<λ
2
,λ
] ∪ ... of the continuous spectrum of Hill’s
2
3
cannot be prescribed arbitrarily but is subject
j
3
λ
< ... form the endpoints of the
4
0
to asymptotic restrictions for j →∞. Moreover, McKean and Trubowitz’s work
shows that the position of the λ
is entirely determined by the λ
2j−1
. Of special
2j
interest is the case of a «finite-band» spectrum where all but a finite number of
the eigenvalues are double, e. g.,
<λ
= λ
<... <λ
1
for j N +1.
2j
λ
2j−1
λ
0
In this case the entire spectrum is uniquely determined by λ
2N
;
, λ
, ..., λ
0
2
2N
(5.6)
and
the corresponding potentials have been determined. They form an N-dimensional
real torus which is the real part of the Jacobi variety of the hyperelliptic curve
2n
2
=
w
j=0
(z −λ
).
j
),
It is interesting that this problem is closely related to the above rank-2
perturbations. The following observation is due to E. Trubowitz (Moser [12]).
1
We supply the prime to avoid confusion with the eigenvalues λkof L = L(x, y). Although
the λ
not sufficient to fix q(s).
, λ
, ... are the eigenvalues for the two boundary conditions y(s +1) = ±y(s),theyare
0
1
§5. Examples of Integrable Flows 173
Let q(s) be any periodic potential with the spectrum (5.6), and f
eigenfunctions normalized by
1
2
f
(s) ds =1.
j
0
We consider the spectrum λ
1, ..., N, just depending on the spectrum λ
such that
The determination of the ε
fixed; then there exist positive numbers εj, j =0,
j
N
j=0
2
εjf
(s)=1,
2j
is irrelevant for us; it can be found in McKean and
j
, not on the particular potential,
k
N
εj=1.
j=0
Trubowitz [9].
We observe that the point x with the coordinates
x
=√εjf2j(s),αj= λ
j
,j=0, 1, ..., N, (5.7)
2j
is restricted to a sphere of radius 1. Moreover,
2
d
x
j
=√εjf
2
ds
This equation can be viewed as constraining the motion x
(s)=qxj− λ2jxj.
2j
j
+ λ
2j
sphere, where q = q(s) plays the role of the normal force,
(s) be the
j
xj=0to a
N
q(s)=
j=0
(λ
2j
x
j
2
(s) − x
2
(s)). (5.8)
j
Thus this problem can be related to Neumann’s constrained motion on the
sphere if we set n = N +1 and use (5.7). The potential q(s) is then obtained
from (5.8).
Now we know that all solutions of the Neumann problem are expressible as
the inverses of hyperelliptic functions, i. e. are quasiperiodic functions associated
to a torus of dimension n − 1=N . Such a torus is characterized by prescribing
the constant values of
εjε
G
= εkf
k
2
2k
+
j
k
2k
− λ
2j
λ
(f2jf
2k
− f2kf
2
)
2j
174 Geometry of Quadrics and Spectral Theory
or alternatively, by prescribing the function
N
G
Φ
(x, y)=
z
k=0
z −λ
k
.
2k
We will show that the periodicity condition q(s +1)=q(s) implies that
N
so that Φ
, ..., λ
λ
2
Φ
vanishes at z = λ
z
.
2N
(x, y)=
z
2k−1
k=1
N
k=0
(k =1, 2, ..., N)and has poles at λ
(z −λ
(z −λ
)
2k−1
, (5.9)
)
2k
Thus the N-gap potentials correspond to just one of the tori of the Neumann
problem of Section 5(b). To prove (5.9) we consider the spectral problems given
by
L = P
where the eigenvalues α
the eigenvalues λ
of L are claimed to be λ
k
(A − y ⊗ y)Px,M= PxAPx,
x
of A are given by αk= λ
k
(k =1, ..., N). By Section 4
2k−1
, k =0, 1, ..., N,and
2k
the underlying hyperelliptic curve is given by
,
0
2
= P(z)=l
w
(0)
(z)a(z)=
N
(z −λk)
1
N
(z −αk).
0
On the other hand, in the theory of McKean and Trubowitz the hyperelliptic
curve is given by
2N
2
=
w
j=0
which shows that the λ
agree with the λ
k
The periodicity condition q(s +1)= q(s) forces that the λ
determined by the λ
, and it is natural to ask for those potentials belonging
2k
to the other tori, i. e. to tori for which the λ
(z −λ
),
j
, as we wanted to show.
2k−1
do not give rise to periodic
2k−1
2k−1
= λkare
potentials. From (5.8) and the above theory it is clear that for general choice
of the λ
the potential is a quasiperiodic function expressible in terms of
2k−1
θ-functions. In fact, such potentials have been considered by Dubrovin, Matveev,
§6. Appendix 175
and Novikov [5]. We restricted ourselves to the finite-band case because it
corresponds to a finite-dimensional mechanical system. Clearly this approach
should be extended to the general case corresponding to the motion of a particle
constrained to an infinite-dimensional sphere, or an isospectral flow in Hilbert
space.
§ 6. Appendix
We mention another integrable mechanical system whose integrals can be
interpreted as eigenvalues of a matrix obtained by rank-two perturbation. The
example is the motion of a particle on the sphere |x| =1in R
influence of a potential
U =
1
Ax, x−
2
n
j=1
−2
2
c
x
.
j
j
The corresponding differential equations are
2
c
j
¨x
j
= −U
= −(Ax)j+
x
j
+ λx
. (6.1)
3
x
j
j
By separation of variables in the Hamilton –Jacobi equation Rosochatius [16]
showed that this system is integrable. Here we want to show that this problem
can be described as an isospectral flow of matrices of the form
n
under the
c
L(x, y)=A + ax ⊗ x + r(x ⊗ y −y ⊗ x)+x ⊗
where c/x stands for the vector with components c
. In particular, it follows
j/xj
c
+
⊗ x,
x
x
that the eigenvalues of this matrix are in involution with respect to the standard
symplectic structure. This matrix is again a rank-two perturbation, since
L = A + x ⊗ ξ + η ⊗x
with
ξ = ax + ry +
c
,η= −ry +
x
c
.
x
Without giving the calculation, we give the generalization of Theorem 2 to this
case.
176 Geometry of Quadrics and Spectral Theory
With any fixed diagonal matrix β = diag(β1,β2, ..., βn),define
B = rβ +
where
β
αi− α
i
− β
)
j
2
r
(xiyj− xjyi)+r
j
D
= r
j
k
βj− β
αj− α
c
x
c
k
j
+
2
k
x
j
i
xj+
i
x
c
j
x
x
j
c
k
2
x
.
k
2
k
Then the isospectral deformation
dL
=[B, L]
dt
is equivalent to the Hamiltonian system
*
i
− (δ
ijDj
),
˙x = H
, ˙y = −Hx,
y
with
2H = aβx, x−
We write again
i<j
βi− β
αi− α
H =
+
j
2
r
(xiyj− xjyi)2+ x
j
n
βjG
j
j=1
2
c
j
2
+ x
i
2
x
j
*
2
c
i
2
.
j
2
x
i
and obtain this way n rational functions in involution, and all the above XHare
integrable. In particular, for β = A we obtain
2H = aAx, x−r
2
(|x|2|y|2−x, y2) −x, x
n
j=1
2
c
j
x
2
j
+2
n
j=1
2
c
.
j
Restricting this system to the tangent bundle x, x =1, x, y =0of the sphere
gives the system (6.1) if we set a = −1, r =1. Since also
n
Gj= ax, x/2
2
j=1
belongs to the functions generated by G
, the system (6.1) has the Gjas integrals.
j
The integration of the system can be carried out as above if one uses as
auxiliary matrix
xAPx
.
M = P
References 177
The relevant hyperelliptic curve is
2
w
= P(z)=a2(z)1 −
n
j=1
c
j
z −α
2
j
− a(z)l(z).
Here P(z) is a polynomial of degree 2n − 1, and the curve of genus n −1.But
for c
=0neither αknor the eigenvalues of λkof L are branch points.
j
P. Deift made the interesting observation that the system (6.1) can be derived
from the Neumann system of Section 5(b) if one replaces n by 2n,setsα
(k =1, 2, ..., n) and reduces the system by the n rotations in the
= α
k
x
− x
k
then the free Hamiltonian
becomes
if x
kyk+n
planes. If one sets
k+n
x
k
n
1
(˙r
2
k=1
− x
k+nyk
= rkcos θk,x
n
1
2
2
+ r
k
k=1
2
θ
k
= r
2
(y
2
˙
k
2
k
2
+ y
k
k+n
2
+ αkr
k
˙
= ckis the constant value of these integrals.
θ
k
= rksin θk; yj= xj,
k+n
+ αk(x
)=
2
+ x
k
n
1
2
k=1
2
˙r
+
k
2
k+n
c
r
))
2
k
+ αkr
k
2
k
k+n
=
Constraining this Hamiltonian to the unit tangent bundle gives the system (6.1).
This remark shows that matrices A with multiple eigenvalues are also of
interest.
References
[1] M. Adler, On a trace functional for formal pseudo-differential operators
and the symplectic structure of the Korteweg –de Vries equations,Inv.
Math., 1979, 50 (3) 219–248.
[2] L. Bianchi, Vorlesungen ¨uber Differentialgeometrie, 2, Aufl. Teubner,
Leipzig, Berlin, 1910.
[3] R. Devaney, Transversal homoclinic orbits in an integrable system,Am.
Math., 1978, 100, 631–648.
[4] L. A. Dikii, Hamiltonian systems connected with the rotation group, Funct.
Anal. and Its Appl., 1972, 6 (4) 83–84.
[5] B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, Nonlinear equations of
Korteweg – de Vries type, finite-zone linear operators, and Abelian varieties,
Russ. Math. Survey, 1976, 31 (1) 59–146.
178 Geometry of Quadrics and Spectral Theory
[6] C. G. Jacobi, Vorlesungen ¨uber Dynamik, In Gesammelte Werke, Supple-
mentband, Berlin, 1884.
[7] T.Kato, Perturbation theory for linear operators, 2nd ed., Springer, 1974,
In particular 244–250.
[8] S. V. Manakov, Remarks on the integrals of the Euler equations of the
n-dimensional heavy top, Func. Anal. and Its Appl., 1976, 10 (4) 93–94.
[9] H. P. McKean, and E. Trubowitz, Hill’s operator and Hyperelliptic function
theory in the presence of infinitely many branch points, Comm. Pure Appl.
Math., 1976, 29 143–226.
[10] H. P. McKean, and E. Trubowitz, Hill’s surfaces and their theta functions,
Bull. Am. Math. Soc., 1979, 84 (6) 1042–1085.
[11] A. S. Mischenko, Integral geodesics of a flow on a Lie group, Funct. Anal.
and Its Appl., 1970, 4 (3) 73–78.
[12] J. Moser, Various aspects of integrable Hamiltonian systems,InProc.CIME
Conference, Bressanone, Italy, June 1978, Prog. Math., 8 Birkh¨auser, 1980.
[13] C. Neumann, Vorlesungen ¨uber Riemanns Theorie der Abelschen Integrale,
2, Auflage. Teubner, Leipzig, 1884.
[14] C. Neumann, De problemate quodam mechanico, quod ad primam integral-
ium ultraellipticorum classem revocatur, Reine und Angew. Math., 1859,
56 46–63.
[15] M. Reid, The complete intersection of two or more quadrics, Thesis, Cam-
bridge Univ. 1972.
[16] E. Rosochatius,¨Uber die Bewegung eines Punktes (Inaugural Dissertation.
Univ. G¨ottingen) Gebr. Unger., Berlin, 1877.
[17] G. Salmon, and W. Fiedler, Analytische Geometrie des Raumes, Teubner,
Leipzig, 1863.
[18] C. L. Siegel, Topics in Complex Function Theory, 2 Wiley-Interscience,
New York, 1971.
[19] P. St¨ackel,¨Uber die Integration der Hamilton – Jacobischen Differential-
gleichung mittelst Separation der Variabeln, Habilitationsschrift, Univ.
Halle-Wittenberg, 1891.
[20] O. Staude, Geometrische Deutung der Additionstheoreme der hyperellip-
tishen Integrale und Functionen 1. Ordnung im System der confokalen
Fl¨achen 2. Grades, Math. Ann., 1883, 82 1–69, 145–176.
k
[21] K. Uhlenbeck, Minimal 2-spheres and tori in S
, Preprint, 1975.
[22] P. van Moerbeke, The spectrum of Jacobi matrices, Inv. Math., 1976, 37
45–81.
Integrable Hamiltonian Systems and Spectral
Theory
§ 1. Introduction
During the last 15 years a large number of publications on integrable Hamiltonian systems, solitons, the Korteweg – de Vries equation has appeared. Integrable Hamiltonian systems are nonlinear differential equations which have
many symmetries and are more or less explicitly solvable (therefore the name).
It turned out that these systems have applications in various fields of physics,
such as fluid mechanics, plasma physics, nonlinear optics, etc. The mathematical
theory revealed deep connections of such systems with differential geometry, the
theory of Lie algebras and algebraic geometry, spectral theory of linear operators
in Hilbert space, but the last word has not been said.
It is not our aim to give a survey of tins fascinating subject but rather
describe several integrable Hamiltonian systems of classical mechanics, such
as the geodesic flow on an n-dimensional ellipsoid going back to Jacobi, and
their close connection to the inverse spectral theory of the one-dimensional
Schr¨odinger equation.
What is the inverse spectral problem? The usual problem of spectral theory
asks for the spectrum of an operator, e.g. the Schr¨odinger operator
which can be defined as a selfadjoint operator in a dense domain of
2
(−∞, +∞),ifq(x) is a continuous bounded function. The relevant the-
L
ory for this singular eigenvalue problem was developed by H. Weyl in 1910 [30]
based on the earlier work by Hilbert and Hellinger. This operator plays a central
role in quantum theory where it is usually assumed that the function q(x) tends
to zero at some rate. In this case one has a continuous spectrum in (0, +∞) and
some point eigenvalues on the negative axis. The inverse spectral problem asks
for the potentials q(x) giving rise to a given spectrum. Since the knowledge of
the spectrum does not suffice to recover q(x) one usually prescribes in addition
1
Integrable Hamiltonian Systems and Spectral Theory. Fermi Lectures, Pisa 1981. Lezioni
Fermiane, Acad. Nat. dei Lincei, Pisa 1981.
d
−
dx
1
2
+ q(x)(1.1)
180 Integrable Hamiltonian Systems and Spectral Theory
the phase shift; this leads to scattering theory which we will not pursue here
(see [5], [7]). Instead we assume the potential q(x) to be periodic or almost
periodic, which is connected with a different type of spectrum. For example,
if q(x) is a periodic function one has no point eigenvalues and the continuous
spectrum generally consists of infinitely many intervals on the real axis, the
so-called band spectrum. In special cases, such as for the constant potential or
for the elliptic p-function (the Lam´e equation), does one have only finitely many
intervals, one of which extending to infinity. In this case one speaks of a finite
band spectrum.
In the inverse spectral theory one prescribes the spectrum, for example, a
sequence of disjoint intervals and asks for the corresponding potential. It is a
priori not clear whether any set of intervals is admissible. Indeed a necessary
condition is that the spectrum extends to +∞. If one insists on periodic potentials
then the position of the intervals have to satisfy various conditions; in the case
of a finite band spectrum only one end-point of the intervals can be prescribed
(see McKean – van Moerbeke [17]). However, if the intervals are prescribed
arbitrarily, i. e. finitely many disjoint intervals and one half interval extending to
infinity, it is possible to construct almost periodic potentials q = q(x),forwhich
the above operator (1.1) has as its spectrum the given set of intervals.
Secondly, it is in general not true that the potential is uniquely prescribed
by the spectrum. Indeed q(x) and its translate q(x + t) clearly give rise to the
same spectrum. But generally the potentials belonging to a given spectrum form
an infinite dimensional manifold. The inverse spectrum problem consists then
in the characterization of closed sets on the real axis which are admissible as
spectra and the determination of all potentials giving rise to this spectrum.
We will describe the complete solutions of this problem in the case of a
finite band spectrum, where finitely many intervals I
half-infinite interval I
, extending to +∞ are prescribed. If the I0, I1, ..., Igare
g
, I1, I2, ..., I
0
g−1
and a
disjoint, then the set of corresponding potentials forms a g-dimensional torus T
and the q(x) are given as hyperelliptic functions. The torus Tgis the real part
of the Jacobi variety belonging to the Riemann surface which is obtained by
slitting two copies of the complex plane along the spectrum I
, I2, ..., Igand
0
glueing the two copies across in the familiar manner. These beautiful connections
between spectral theory and complex analysis were discovered by McKean and
van Moerbeke [17], and Novikov et al. [6] independently.
In these lectures we will present a different derivation of these results
by reducing the determination of the finite band potentials to the study of the
geodesics on an ellipsoid. It was known that the geodesics on an ellipsoid can
g
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]