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§1. Introduction 131
Salmon and Fiedler [17]). Under the geodesic flow the orthonormal frame ϕ
will undergo a motion described by an antisymmetric matrix B,sothat
= Bϕj,˙L =[B, L].
˙ϕ
j
This is the Lax representation of the geodesic flow, where B turns out to be the
matrix
−1
B = −(α
The fact that the eigenvalues λ
−1
α
(xiyj− xjyi)).
i
j
are preserved under the geodesic flow
j
means obviously that the tangents of one geodesic o f an ellipsoid will touch the
same n − 2 quadrics confocal to the ellipsoid — also a well-known result of
geometry.
These results will be derived in Section 3. There are further properties of the
isospectral manifold M(λ
, ..., λ
1
which will be established in Section 4. If we identify the lines x = x
on M(λ) to points and also take the quotient under the reflections x
then we arrive at an (n − 1)-dimensional manifold M
) of matrices L(x, y) with fixed spectrum,
n−1
(λ) which is isomorphic
j
+ sy
0
→±xj,
to the Jacobi variety of the hyperelliptic curve
2
w
= P
(z)=z−1det(zI − L)det(zI − A),
2n−1
which is of genus n− 1. Thus the Jacobi variety has complex dimension n−1 and
is a torus with 2n −2 periods. The geodesic flow is linear in the variables of the
Jacobi map, and thus the geodesic flow is closely related to Abel’s theorem for
hyperelliptic integrals. This fact was used by Staude [20] to give a geometrical
interpretation of the addition theorem for hyperelliptic integrals.
j
c. Perturbations of Rank 2
In the above approach the choice of the matrices (1.3) was unmotivated and
it is difficult to make the right guess. At present there seems to be no systematic
way for finding such isospectral matrices. In this case I owe the essential hint
to M. Adler, who suggested looking at matrices of the form A + x ⊗y − y ⊗ x.
The matrices (1.3) can be obtained as limit case of similar matrices, which we
will now discuss.
If A is again a fixed symmetric matrix and x, y, ξ, η four n-vectors, we call
A + x ⊗ ξ + y ⊗ η
132 Geometry of Quadrics and Spectral Theory
a rank-2 perturbation of A. We will study the special case where
ξ = ax + by, η = cx + dy,
so that
L(x, y)=A + ax ⊗ x + bx ⊗ y + cy ⊗x + dy ⊗ y (1.6)
is a matrix which depends on two n-vectors x, y while a, b, c, d are fixed with
Δ=ad − bc =0.
We will take this 2n-parameter family of matrices as starting point, study the
algebraic manifold M(λ
the fixed spectrum λ
, λ2, ..., λn, and investigate the isospectral deformations
1
of these matrices. The basic observation is the following: If we consider the
symplectic manifold (R
, ..., λn) of those x, y ∈ Rnfor which L(x, y) has
1,λ2
2n
,ω) with the symplectic two-form
n
ω =
dyj∧dxj,
j=1
then the eigenvalues of L(x, y) given by (1.6) are in involution,
{λ
j,λk
} =0,
where
{F, G} =(F
G
− F
G
x
y
j
j
)
y
x
j
j
denotes the standard Poisson bracket. Again it is better to use the symmetric
functions of the eigenvalues of the λ
(x, y)=1−
Φ
z
or the functions
j
det(zI − L)
det(zI − A)
,
which are quartic polynomials in x, y. With the partial-fraction expansion
n
j=1
Gj(x, y)
z − α
H
,
j
for any Hamiltonian H =
Φ
(x, y)=
z
we have n quartic polynomials G
The Hamiltonian vector fields X
= ϕ(G
, ..., Gn) — or any Hamiltonian depending on the spectrum of L
1,G2
which are in involution.
j
only — is tangential to M(λ),and
n
X
H
∂H
=
j=1
∂G
· X
,
G
j
j
§1. Introduction 133
where because of
,X
[X
G
]=−X
G
j
k
{Gj,Gk}
=0
all these vector fields commute. In particular, the isospectral manifolds are
Lagrange manifolds. All these Hamiltonian systems are integrable, by which
we mean a vector field having n integrals G
in involution, for which dGjare
j
linearly independent in an open dense set. In Section 2 we will show how these
Hamiltonian vector fields can be written in the form
d
L =[B, L].
dt
In Section 3 it will be shown that the geodesic flow on an ellipsoid can be derived
as the limit case of matrices (1.6) with a =0, b = −c = ν, d = ν
2
for ν →∞.
d. Hyperelliptic Curve
In Section 4 we show how M(λ) is related to the Jacobi variety of a
hyperelliptic curve. The manifold M(λ) is n-dimensional, but after factoring
out a 1-dimensional group one is led to an (n −1)-dimensional manifold M
which is isomorphic to the Jacobi variety of a hyperelliptic curve of genus n − 1.
This generalizes the statement for the geodesic flow on the ellipsoid in which
case M
factoring out the reflections x
second matrix M = M(y) whose spectrum μ
is obtained by identifying the straight lines x → x + sy to points and
→±xj. The proof is based on introducing a
j
together with that of L = L(x, y)
j
determines x, y.Inotherwords,x, y are described by two spectra, each of which
is given by a set of functions in involution. The spectrum of M is viewed as a
divisor in the Jacobi map. The computation of the symplectic form ω in these
variables takes the form
ω =S
λjμ
k
dλj∧ dμk,
where S = S(λ, μ) is the function to which one is also led by the separation of
variables of the Hamilton – Jacobi equations. In this way one sees the connection
1
of the two approaches. For details we refer to Section 4
.
e. Applications
In Section 5 we describe various classical integrable examples for which
the integrals can be obtained in terms of the eigenvalues of matrices of the
1
ln this connection we refer to the forthcoming «Lectures on θ-functions and Their Applications»
by David Mumford, held at Bombay and Harvard 1978–1979.
134 Geometry of Quadrics and Spectral Theory
form (1.6). Here we have to distinguish three cases according as the rank of the
symmetric part
of the matrix
illustrated by a subclass of geodesics of the orthogonal group with a left-in-
ab
is 2, 1, or 0. We give three examples, the last one being
cd
⎛
⎝
a
b + c
2
b + c
2
d
⎞
⎠
variant metric, as it was studied first by Arnold. (For literature see Dikii [4],
Manakov [8], Mischenko [11].)
Finally we show how these systems relate to the finite band potential in
the Hill’s equation, in the periodic and the quasiperiodic case. This is based on
a connection between the translation flow for the above finite-band potentials
and a mechanical problem of a particle moving on the sphere |x| =1under the
influence of a linear force. This connection was found by E. Trubowitz; it was
described previously (Moser [12]). Finally, in the Appendix we describe a class
of matrices L involving x
−1
whose eigenvalues are also in involution. They are
j
also integrals for a classical mechanical system discussed in the dissertation of
Rosochatius [16].
f. Connection with M. Reid’s Result [15]
We want to mention a related result which we learned from a letter of Horst
Kn¨orrer. In his unpublished dissertation of 1972 Miles Reid established that the
set of (m −1)-dimensional linear subspaces of a nonsingular intersection of two
quadrics in P
(C) is — as algebraic manifold — isomorphic to the Jacobi
2m+1
variety of a hyperelliptic curve. It is tempting to guess a connection to the above
result about the common tangents of n −1 confocal quadrics U
, ..., U
λ
1
λ
n−1
in Cn. Such a connection really exists, and Kn¨orrer communicated to me a
n−1
beautiful construction of a 1-to-2
to M. Reid’s
1
Jacobi variety for some appropriate quadrics.
mapping of the variety of common tangents
g. Final Remarks
The above approach is obviously very unsystematic and relies on lengthy
calculation. Why are the eigenvalues of matrices of the form (1.6) in involution? The deeper reasons have still to be revealed. M. Adler, who gave the
1
Note added in proof :H.Kn¨orrer on the ellipsoid, Inv. Math. 59, 119–143 (1980).
§2. Perturbation of Rank 2 135
initial hint for the form of the isospectral matrices required, found a general
framework based on the coadjoint representation of certain Kac – Moody algebras in extension of his previous work [1], which allow him to encompass the
above examples as special cases of a general theory. Originally it was the plan
to publish a joint paper from this point of view; however, since the theory is
formidable and lengthy, it was necessary to separate off the general approach.
Adler’s Lie-algebraic approach will appear elsewhere. Here we merely want to
show the baffling connection between the spectral theory of the matrices (1.6)
and the geometry of quadrics.
I want to express my thanks to Horst Kn¨orrer for informing me about his
geometrical construction in connection with M. Reid’s work. After presenting
this paper in Berkeley, I visited in Warwick and lectured on this topic. I want
to thank D. Epstein for his hospitality and Adrian Douaday for interesting discussions. He supplied an elegant alternative proof for the involuntary character
of the eigenvalues of (1.6). Because of length restrictions we could not present
his argument here. I am grateful to P. Deift for suggestions and for reading the
manuscript. Finally, I am particularly indebted to M. Adler, who contributed
essential ideas to this work.
§ 2. Perturbation of Rank 2
a. Isospectral Manifolds
We take as a starting point the spectral problem for a perturbation of rank 2
of a symmetric bilinear operator. Let V denote a real (or complex) finite-dimensional vector space, , a real inner product, and let A be a matrix symmetric
with respect to the inner product, i. e. Av, w = v, Aw. Moreover, we will
assume the eigenvalues of A to be distinct.
A perturbation of rank r is given by
r
xρξρ,v,
ρ=1
where x
Lv = Av +
, ..., xrand ξ1, ..., ξrare two sets of linearly independent vectors
1
in V . We write the above formula in terms of the tensor product ⊗ as
r
L = A +
xρ⊗ ξρ. (2.1)
ρ=1
136 Geometry of Quadrics and Spectral Theory
It is well known that the spectrum of L is determined by the formula
where W
det(z −L)
det(z −A)
is the r-by-r matrix given by
z
W
= Rzxρ,ξσ,ρ,σ=1, ..., r; Rz=(zI − A)−1. (2.3)
z
=det(I −W
)(2.2)
z
This formula has been extended to infinite-dimensional vector spaces (see
Kato [7]); the right-hand side is called the Weinstein– Aronszajn determinant.
We specialize the above to rank two and
= x, x2= y; ξ1= ax + by, ξ2= cx + dy,
x
1
so that
L = L(x, y)=A + ax ⊗ x + bx ⊗ y + cy ⊗ x + dy ⊗ y, (2.4)
where a, b, c, d are constants with determinant Δ=ad −bc =0and x, y linearly
independent vectors of V . This defines a 2n-dimensional family of matrices in
whose spectra we are interested. In particular the n-dimensional foliation given
by isospectral matrices will be of interest to us.
The main result of this section is the observation that the eigenvalues of these
matrices are «in involution» with respect to the symplectic structure
∗
i. e. the natural symplectic structure of T
any two functions F = F (x, y), G = G(x, y) in C
V = V∗× V ∼ V × V .For
1
(V × V ) we define the
n
dyj∧dxj,
1
corresponding Poisson brackets
where F
{F, G} = F
, Fyare defined by the relation
x
dF = F
x
−Fy,Gx,
x,Gy
,dx + Fy,dy.
One calls a family of functions F «in involution» if for any two of the F, G ∈ F,
one has
{F, G} =0.
§2. Perturbation of Rank 2 137
1(R2
Such a family can be extended by closure under composition: If ϕ,ψ∈C
,R)
then
{ϕ(F, G),ψ(F, G)} =
∂(ϕ, ψ)
∂(F, G)
{F, G},
i. e., if F, G are in involution, so are ϕ(F, G), ψ(F, G). Instead of showing the
involutary character of the eigenvalues, we will establish this for the symmetric
functions of the eigenvalues which are rational in x, y.
If we apply the formulae (2.2), (2.3) to the case (2.4), we obtain a
two-by-two matrix
W
Q
(x) Qz(x, y)
=
z
z
Q
(x, y) Qz(y)
z
ac
, (2.5)
bd
where
Q
(x, y)=Rzx, y,Qz(x)=Qz(x, x),
z
and (2.2) becomes
det(z −L)
det(z −A)
=1−tr W
+detWz=1−Φz, (2.6)
z
where
Φ
(x, y)=aQz(w)+(b + c)Qz(x, y)+dQz(y)−
z
− (ad − bc)(Q
(x)Qz(y) − Q
z
2
(x, y)).
z
Thus the eigenvalues of L are the values of z for which the rational function Φ
manifold of matrices (2.4) with spectrum λ
takes the value 1. If these eigenvalues λjare distinct, the isospectral
z
, λ2, ..., λnis given by
1
{x, y | Φ
(x, y)=1for j =1, ..., n}, (2.8)
λ
j
and hence it is an algebraic manifold.
b. Isospectral Deformations
Theorem 1. For any z, z
i. e. the fun ctions Φ
(x, y) are in involution.
z
in the resolvent set of A one has
, Φ
{Φ
} =0,
z
z
(2.7)
138 Geometry of Quadrics and Spectral Theory
This theorem can be verified by a direct but lengthy calculation. We do not
present this, since the result will be a consequence of Theorem 2. Clearly this
theorem remains valid if multiple eigenvalues of A are permitted.
We extend the class of functions (Φ
) by forming for any polynomial f(z)
z
H(x, y)=
1
4πi
f(z)Φz(x, y) dz,
|z|=R
where the circle |z| = R contains the spectrum of A. We express this function
explicitly by introducing a basis in which
A = diag(α
1,α2
, ..., αn)
and set
)=βj,β= diag(β1,β2, ..., βn).
f(α
j
Then one finds
2H = aβx, x+(b + c)βx, y + dβy, y−
For example, for β
ad − bc
−
2
= δikthis becomes
i
G
(x, y)=ax
k
− (ad − bc)
βi− β
j
αi− α
i=j
2
+(b + c)xkyk+ dy
k
(xiyj− xjyi)2.
j
(xiyk− xkyi)
i
2
k
αk− α
−
2
,
i
(2.9)
(2.10)
where the prime indicates that i = k. These functions are all in involution, as a
consequence of Theorem 1, and the Φ
Φ
(x, y)=
z
and
H(x, y)=
are recovered by
z
n
Gj(x, y)
z − α
j=1
n
1
f(αj)Gj(x, y).
2
j=1
,
j
§2. Perturbation of Rank 2 139
If (a, b + c, d) =(0, 0, 0),thendG
are linearly independent on an open
j
dense set, while for a = b + c = d =0we have the relation
n
Gj=0.
j=1
2
In this case we have n − 1 independent commuting functions in |x|
G
, ..., Gn, for example.
3
For any choice of the constants f(α
˙x =
is integrable, since G
, G2, ..., Gnare integrals in involution. Hence the
1
∂
∂y
)=βjthe vector field
j
H, ˙y = −
∂
∂x
H (2.11)
: G2,
spectrum of the matrix (2.4) is fixed under any of these flows, so that in the
case of distinct eigenvalues there exists a nonsingular matrix U = U (t) such
−1
that U
LU is a constant matrix. The infinitesimal version of this statement is
that the differential equation (2.11) can be written in the Lax form
d
L =[B, L](2.12)
dt
with some matrix B. This is the content of
Theorem 2. The vector field (2.11) with H given by (2.9) defines an
isospectral deformation of the matrix (2.4) given by (2.12) where
β
− β
i
αi− α
j
(xiyj− xjyi). (2.13)
j
1
(b − c)β +(ad − bc)
B =
2
The diagonal elements of the last matrix are zero.
Corollary. If H = H(G
, ..., Gn), then the vector field XHcorre-
1,G2
sponds to the isospectral d eformation˙L =[B, L],whereB is of the form (2.13)
with β
where for fixed G
=2∂H/∂Gj. Indeed
j
X
H
= cjthe βjcan be considered as constants. For this vector
j
n
∂H
=
j=1
∂G
X
G
j
1
=
j
β
,
jXG
2
j
field Theorem 2 gives the statement of the corollary.
140 Geometry of Quadrics and Spectral Theory
We show that Theorem 1 is a consequence of Theorem 2. From the
form (2.12) of the differential equation (2.11) it is plain that the eigenvalues
of L, a hence any function of the eigenvalues are constant along orbits. Therefore
1 − Φ
z
(z −α
(z −λ
=
)
j
)
j
is a constant of the motion, for any z,or
d
Φ
= {Φz,H} =0.
z
dt
Taking β
=2δjk,wegetH = Gk; hence
j
{Φ
z,Gk
} =0,
and hence
{Φ
, Φ
} =(z− αk)−1{Φz,Gk} =0.
z
z
The proof of Theorem 2 consists of a calculation, which we break into
several steps. Setting
s =
1
2
(b + c),r=
1
(b − c),
2
we break L into symmetric and antisymmetric parts:
L = A + S + R,
S = ax ⊗x + s(x ⊗y + y ⊗ x)+dy ⊗y,
R = r(x ⊗y − y ⊗x).
β
i
αi− α
2
+ r2,weset
− β
j
(xiyj− xjyi),
j
With the determinant Δ=ad − bc = ad −s
B = rβ +ΔΓ, Γ=
the diagonal terms of Γ being zero. The Hamiltonian H is broken up into its
quadratic and its quartic part:
H = F − ΔG,
1
aβx, x + sβx, y+
F =
2
βi− β
1
G =
αi− α
2
i<j
j
(xiyj− xjyi)2.
j
1
dβy, y,
2
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