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§8. Hill’s Equation 121
This is easily verified by taking the differential of this relation with respect to q,
(L − μ
Taking the inner product with ϕ
1
ϕj(L − μj)δϕjdx +
0
)δϕj+ ϕjδq = δμjϕj.
j
we get
j
1
2
ϕ
δq dx = δμj.
j
0
Noting that the first term vanishes as L is selfadjoint, we have the stated relation.
Therefore we have for the translation the differential equation
1
dμ
j
dt
δμ
=
j
δqdqdt
0
Using the differential equation we replace ϕ
dμ
j
= −ϕ
dt
Clearly, ϕ
(x) is a multiple of the solution y2(x, μj) which also vanishes
j
dx =
j2
1
2
ϕ
(x)q(x) dx = −2
j
0
1
0
=(ϕ
(0))2− (ϕ
j
q by ϕ
j
1
ϕ
ϕjqdx.
j
0
+ μjϕjand obtain
j
(1))2.
j
at x =0. One computes
ϕ
(x)=(˙y2(1,μj)y
j
(1,μj))
2
−1/2
y2(x, μj)
where the dot indicates λ-derivative. Thus
(ϕ
(0))2− (ϕ
j
If we use the relation y
we have y
=0we find
2
(1))2=˙y2(1,μj)−1(y
j
y
− y
1
2
Therefore we have the identity
1
+ y
2
for x =1, λ = μ
(y
. Hence, with Δ=y1+ y
j
4 − Δ2=(y1− y
(1,μj)−1− y
2
y2=1and the fact that for λ = μj,andx =1
1
y
=1.
y
1
2
)2+(y1− y
2
)2=4y1y
2
2
)2=(y
2−1
2
− y
=4
)
2
(1,μj)).
2
122 Various Aspects of Integrable Hamiltonian Systems
and
dμ
j
2
(0) − ϕ
= ϕ
dt
Since y
the λ
(1,λ) has as zeroes the μk(k =1, 2, ...) and (4 −Δ2(λ)) as zeroes
2
, λ1, ... we obtain the differential equation (8) to a constant. We forego
0
j
2
(1) = ˙y2(1,μj)
j
$
−1
4 − Δ2(μj).
the determination of the constant, which is found from the asymptotic behavior
for large λ.
In fact, Trubowitz uses these differential equations in the infinite dimen-
sional case in the form
dμ
dt
k
2π2
= k
μj− μ
j=k
j2π
−1#
k
2
Δ2(μk) − 4
which is valid in the general case and uses them to solve the inverse spectral
problem in the periodic case. The point is that the right-hand side admits a
uniform Lipschitz estimate to allow global solution of the differential equation.
However, we restrict ourselves to the finite gap case. For example, if N =1
the differential equation for μ = μ
dμ
=2
dt
which gives the elliptic p-function plus a constant. Formula (6
becomes
1
#
(λ0− μ)(λ1− μ)(λ2− μ)
) shows that the
corresponding potential is also an elliptic function with real period 1. This is the
same equation.
f) A mechanical description of the differential equations (8) or (9). McK-
ean and Trubowitz derived an interesting identity for the eigenfunctions ϕ
belonging to the eigenvalues λ2j. They satisfy the relation
∞
j=0
The ε
depend on the λ0, λ1, ...only and are positive if λ2j− λ
j
are equal to zero if λ
2
j
2j
2j−λ2j−1
j
y
=0. Thus in our case (7) of finitely many gaps
Δ(λ
(1,λ2j)
1
)
2j
. (10)
2j−1
> 0,and
this identity reduces to
N
j=1
2
εjϕ
(x)=1. (10)
2j
2j
Setting
§8. Hill’s Equation 123
=√εjϕ2j(j =0, 1, ..., N)
x
j
we see that the x =(x
, ..., xN) are restricted to the unit sphere. We wish
0,x1
to derive the differential equations on the unit sphere which correspond to the
translation q(x) → q(x + t). For this purpose we use the differential equations
for ϕ
The fact that |x|
x, ¨x + | ˙x|
which give
2j
2
d
xj=√εjϕ
2
dt
2
=1is invariant under the flow implies that x, ˙x =0and
2
=0. Thus, taking the inner production with x we obtain from the
=(g −λ2j)√εjϕ2j=(q −λ2j)xj. (11)
2j
differential equation:
N
−|˙x|
2
= q|x|2−
j=0
λ2jx
2
j
or
N
q =
j=0
2
λ2jx
−|˙x|2. (12)
j
We notice that this system (11), (12) is precisely the integrable system of
the particle moving on the sphere under the influence of a quadratic potential
discussed in a previous section. Here q plays the role of the normal force and
N
λ2jxjof the quadratic potential. We can make use of our information about
j=0
the mechanical problem to gain information about the spectral problem.
For example, (12) yields the identity
N
q(x)=
(λ2jεjϕ
j=0
2
2j
− εjϕ
2j2
)
for the potential. In McKean –Trubowitz (p. 223) one finds another formula for
of the form
q(x)=2
N
j=0
λ2jεjϕ
2
+ λ0−
2j
N
j=1
(λ2j− λ
). (13)
2j−1
124 Various Aspects of Integrable Hamiltonian Systems
Taking the difference of the two expressions we find
N
εjϕ
j=0
Now, the left-hand side is |˙x|
2j2
N
+
j=0
λ2jεjϕ
2
+
2
= −λ0+
2j
N
λ2jx
j=0
N
(λ2j− λ
j=1
2
, i. e. twice the total energy for
j
2j−1
).
the mechanical problem, which is clearly a constant. Its value is given by the
right-hand side.
N
g) Identification of the torus T
. Thus we see that the differential
equation of translation q(x) → q(x + t) agree with the mechanical problem
=(q −λ2j)x
¨x
j
j
q as in (12). But clearly not all solutions give rise to periodic potentials. Which
solutions correspond to the N -gap potentials described by (7)? They form an
N-dimensional torus which we wish to identify.
For this purpose we use the previously discussed integrals
(ϕ2νϕ
− ϕ2μϕ
2μ
λ2ν− λ
2μ
2
= ενϕ
2
+
ενε
2ν
μ=ν
μ
ν
= x
2
ν
F
+
μ=ν
(xνyμ− xμyν)
λ2ν− λ
2μ
2
)
2ν
and the corresponding function
N
F
ν
=
Φ
z
ν=0
λ2ν− z
.
Since the integrals are in involution it is clear that the manifolds F
= cνare tori,
ν
provided they are regular, compact and connected. Therefore it is to be expected
that the desired manifold corresponding to the N -gap potential is described in
the algebraic form F
instead of giving the values of the constants c
= cν(ν =0, 1, ..., N). This is indeed the case. But
ν
we describe the zeroes of Φ
ν
instead. Since the asymptotically
N
1
= −
Φ
z
z
ν=0
2
ενϕ
+ O(|z|−2)=−z−1+ O(|z|−2)
2ν
z
§8. Hill’s Equation 125
we have
N
(z −uj)
j=1
= −
Φ
where u
z
are the zeroes of Φz. Thus Φzis characterized by the zeroes and poles.
j
N
(z −λ2ν)
ν=0
Theorem. The finite gap potentials q(x) defined by (7) are characterized
by u
j
= λ
(j =1, ..., N) i. e.
2j−1
N
(z −λ
z
= −
j=1
ν=0
N
(z −λ2ν)
Φ
2j−1
)
.
This follows from a scrutiny of Proposition 1, p. 175 of McKean and
Trubowitz.
Corollary 1. For a given simple spectrum λ
, λ1, ..., λ2Nthe function Φ
0
is well defined, i. e. Fkhave specified values, defining an N-dimensional torus
on the tangent bundle of
ν=0
2
ενϕ
=1. All solutions on this torus are periodic
2ν
N
of period 1 defining the potentials q(x + t).
z
Corollary 2. Since the λ
all the F
have positive values.
k
All solutions ϕ
are hyperelliptic functions of x and so are the correspond-
2ν
interlace the λ2jit follows that on this torus
2j−1
ing potentials according to (13), both periodic of period 2 or 1.
One can determine a particular potential with given simple spectrum λ
λ
, ..., λ2Nby setting μj= λ
1
the «origin» on the torus T
N
.) Then q(x) is an even function and the λ
are roots of y2(1,λ) while the λ2jare the roots of y
. (In McKean – Trubowitz this is chosen as
2j−1
(1,λ). Therefore for this
1
choice of the potential q(x) one has
y
(1,z)
Φ
2
=
z
y
1
(1,z)
.
2j−1
= μ
,
0
j
126 Various Aspects of Integrable Hamiltonian Systems
Since the roots of y2(1,z), y
(1,z) in λ
1
λ λ2jcoalesce if j>Nthe
2j−1
above expression is in fact a rational function.
It is suggestive to investigate the potentials belonging to other tori, i.e. to
different values of the constants c
= Fj. All these solutions are hyperelliptic, but
j
not in general quasi-periodic. Since very little is known about the spectral theory
of quasi-periodic potentials it would be worthwhile investigating even these
special examples which present themselves through this surprising connection
between Hill’s equation and the mechanical problem. (This connection was
found by E. Trubowitz and the author.) However, this approach has not yet been
carried out.
References
[1] A. 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. Surveys 31 (1976) 59–146.
[2] V.A.Marˇcenko, and I. V. Ostrovskii, A characterization of the spectrum of
Hill’s operator, Mat. Sbornik 97 (139) 1975, 493–554.
[3] H. Hochstadt, On the determination of Hill’s equation from its spectrum,
Arch. Rat. Mech. Anal., Vol. 19 (1965) 353–362.
[4] H. P. McKean, and P. van Moerbeke, The spectrum of Hill’s equation, Inven-
tiones Math. 30 (1975) 217–274.
[5] 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. 29 (1976) 14–226.
[6] E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl.
Math. 30 (1977) 321–337.
[7] N. Levinson, The inverse Sturm – Liouville problem, Mat. Tidsskr. B. (1949)
25–30.
Geometry of Quadrics and Spectral Theory
1
§ 1. Introduction
a. Background
In this paper we are concerned with integrable Hamiltonian systems. This
concept goes back to classical analytical dynamics of the last century. Briefly
these are nonlinear systems of ordinary differential equations described by a
Hamiltonian function and possessing sufficiently many integrals (or conserved
quantities) so that they are more or less explicitly solvable by quadrature. Therefore these systems played a crucial role in the last century before more qualitative
methods for differential equations were developed at the turn of the century. Subsequently interest In these systems decreased, partly due to the realization that the
existence of global integrals can be established only for exceptional Hamiltonian
systems.
In the last 15 years the subject of integrable Hamiltonian system has regained
considerable interest with the discovery of some partial differential equations
which can be viewed as such systems with infinite degrees of freedom. In this
case the integrals form an infinite sequence of conserved functionals. The most
celebrated example is the Korteweg – de Vries equation: u
+ uux+ u
t
=0. Extensive investigations of this equation have led to surprising links with
scattering theory, spectral theory, complex analysis of hyperelliptic curves and
their θ-functions, and differential geometry.
The purpose of this paper is to establish a connection of some classical
integrable Hamiltonian systems with the elementary geometry of quadrics. The
motivation starts with the following observation: The classical approach to
finding the relevant integrals was based on solving the Hamilton– Jacobi equation
by separation of variables (St¨ackel [19], Jacobi [6]). This required the appropriate
choice of variables and computational skill. A case in point is Jacobi’s integration
of the geodesics on an ellipsoid or C. Neumann’s study [14] of a mass point
moving on a sphere under the influence of a linear force.
On the other hand, in the recent studies of partial differential equation
the integrals were found as eigenvalues of some linear operators which depend
1
Geometry of quadrics and spectral theory. The CHERN Symposium 1979, Berkely, Springer
Verlag New York, 1980, 147–187.
xxx
=
128 Geometry of Quadrics and Spectral Theory
on the solution of the partial differential equation but have the feature that
their spectrum is conserved for each solution of the partial differential equation
considered. Thus under the time evoluation of this equation the linear operator
changes in such a way that its spectrum remains fixed, i. e., it undergoes an
isospectral deformation. The eigenvalues, viewed as functionals, represent the
integrals. This approach of using isospectral deformation of a linear operator has
been developed by P. D. Lax in connection with the Korteweg– de Vries equation
and has been applied by other investigators to many other examples.
The question arises naturally whether all integrable Hamiltonian systems can
be described by isospectral deformation. The question is shifted from finding
the integrals of the systems, provided they exist, to finding the linear operator
whose spectrum is preserved. We will not attempt to answer this question in any
generality, but consider some classical examples, such as Jacobi’s geodesic flow
on the ellipsoid, and construct an isospectral deformation for them. The relevant
matrix turns out to be symmetric, and we will give a geometrical interpretation
for the eigenvalues and eigenvectors. This does not lead to new results for this
old problem, but to an interesting geometrical interpretation of the eigenvalues
and eigenvectors of these operators. In the course of this investigation we will
see that our approach also is applicable to the Korteweg –de Vries equation, thus
establishing a link between this partial differential equation and the theory of
confocal quadrics.
b. Geodesics on an Ellipsoid
We begin directly to illustrate our approach with the geodesic flow on an
ellipsoid, which had been first integrated by Jacobi. In December 1838 he wrote
to his friend and colleague Bessel: «Yesterday I solved the equations for the
geodesic lines on an ellipsoid with three different axes by quadrature. These are
the simplest formulae of the world, Abelian integrals, which turn into elliptic
integrals if two of the axes become equal.» This quotation shows how much the
Abelian integrals were m vogue at the time; below we will see how this theory
ties in with isospectral manifolds.
If A is a positive definite symmetric n by n matrix with distinct eigenvalues
n
and x ∈ R
ellipsoid as
an n-vector, then we write the equation for the (n − 1)-dimensional
−1
x, x =1 (1.1)
A
and the differential equations of the geodesics as
2
d
x
= −νA−1x, ν =
2
dt
−1
A
|A−1x|
y, y
2
,y=
dx
, (1.2)
dt
§1. Introduction 129
where we restrict ourselves to solutions which lie on the ellipsoid. Here ·, ·
denotes the inner product in R
n
.
For this problem it will turn out that the relevant isospectral matrices L,
which we give here without motivation, are of the form
L(x, y)=P
where the tenser product x ⊗y denotes the matrix (x
projection into the orthogonal complement of the vector y. Thus the symmetric
matrix L(x, y) depends on two vectors x, y ∈ R
(A − x ⊗ x)Py, (1.3)
y
),andPythe orthogonal
iyj
n
, where, however, the length
of y =0is irrelevant.
If we identify x with the position on the ellipsoid and set y = dx/dt,then
the eigenvalues λ
Actually one eigenvalue, say λ
, λ2, ..., λnof L are preserved under the geodesic flow (1.2).
1
is equal to zero and belongs to the eigenvector y
n
of L. But the other n−1 eigenvalues are nontrivial algebraic integrals of (1.2). It
is better to form the symmetric functions of the λ
and look at the characteristic
j
polynomial l(z)=det(zI − L) of L, which is a polynomial of x, y. In fact, the
ratio
is a rational function of z with poles at the eigenvalues α
and zeros at λ
, ..., λ
1
of x, y the function Φ
expansion of Φ
(x, y) is
z
2
det(zI − L)
|y|
z
det(zI − A)
the nontrivial eigenvalues of L(x, y). As a function
n−1
(x, y) is a quartic polynomial. The partial-fraction
z
Φ
=
z
j=1
n
Gj(x, y)
=Φ
z − α
(x, y)(1.4)
z
, α2, ..., αnof A
1
,
j
where the G
(x, y) are also quartic polynomials of x, y which are integrals for
j
the flow (1.2). Actually only n −1 of them are independent on the ellipsoid,
since there the relation
n
j=1
−1
α
j
Gj(x, y)
0=Φ
0
= −
holds.
We wish to indicate the connection with confocal quadrics to the ellipsoid (1.1), which are given by the equation
−1
(z − A)
x, x +1=0.
130 Geometry of Quadrics and Spectral Theory
We will set
(x, y)=(z −A)−1x, y,Qz(x)=Qz(x, x), (1.5)
Q
z
and denote the quadric
Q
(z)+1=0
z
.
by U
z
To interpret geometrically the eigenvalue equation Φ
we first establish the identity
(x, y)=0of (1.4)
z
(x, y)=Qz(y)(1 + Qz(x)) − Q
Φ
z
2
(x, y),
z
so that for fixed z, x this represents a quadratic form. The equation
(x, y)=0
Φ
z
represents the quadratic cone of tangents to U
, going through the point x,after
z
the point x is translated to the origin. Secondly one has
(x + sy, y)=Φz(x, y),
Φ
z
so that Φ
sees that for a given line x = x
is constant along any line x = x0+ sy, y =0. From these facts one
z
+ sy, the roots z = λ1, λ2, ..., λ
0
n−1
of the
equation
,y)=0
Φ
z(x0
are such that the above line is tangent to the confocal quadrics U
2, ..., n− 1). Generically a line in R
and the set of lines tangent to U
, ..., U
λ
1
n
touches just n −1 confocal quadrics —
forms a normal congruence; see
λ
n−1
λ
j
(j =1,
Bianchi [2].
Thus the «isospectral» manifold of matrices L(x, y) with a fixed distinct
spectrum λ
tangents to n − 1 confocal quadrics U
, λ2, ..., λ
1
is identified with the normal congruence of common
n−1
(j =1, 2, ..., n− 1), which can be
λ
j
considered as a geometrical interpretation of the spectrum of L(x, y).
Also the eigenvectors ϕ
The eigenvalue λ
=0corresponds to ϕn= y as was mentioned above, while
n
the other eigenvectors ϕ
line x = x
+ sy.SinceL = L(x, y) is symmetric, the n vectors are pairwise
0
of L have a simple geometrical interpretation:
j
are the normals to U
j
at the point of contact of the
λ
j
orthogonal — which is the content of an old theorem of Chasles (Bianchi [2],
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