lbYFILI:
-_
3·16y:-i
TOPOLOGY OF THE NODAL AND CRITICAL POINT SETS
FOR EIGENFUNCTIONS OF ELLIPTIC OPERATORS
by
Jeffrey H. Albert
A.B. Columbia College
(1966)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 1971
Signature of Author......
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Depa
Certified by..... ........
;me•
0
hematics,
August 1....
Mathematics,
August 16,
.......
1971
This Su....risor
Thesis
Sunervisor
Accepted by................
Chairman, Departmental Committee
on Graduate Students
Archives
SEP 30 1971
kIBRARIE2
2.
TOPOLOGY OF THE NODAL AND CRITICAL POINT SETS
FOR EIGENFUNCTIONS OF ELLIPTIC OPERATORS
by
Jeffrey IH. Albert
Submitted to the Department of Mathematics on
August 16, 1971 in partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
ABSTRACT
The theorems announced here are the initial step of
a program aimed at describing solutions of elliptic
partial differential equations on a manifold by means
of geometric properties, such as the layout of the zero
set and the nature of the critical points.
Let M be a compact, connected C manifold
without boundary. Let C0 (M) denote the real
Cw functions on M. Let L be a second-order,
self-adjoint C0 elliptic operator on M and let
We say an eigenfunction
p e C0 (M).
where
P = L + p,
0
(1) all the
u E C (M) of L + p is generic if:
critical points of u are non-degenerate, i.e. u is
(2) the zeros are all regular
a Morse function; and
points (implying that the zero set is a submanifold
of codimension 1).
There are examples of operators, e.g., the
Laplacian on the 2-sphere, having eigenfunctions
which are not generic. However, if dimension M = 2,
then for most p s C'(M), P = L + p has only simple
eigenvalues and generic eigenfunctions. The statement
"for most p" means that the set of p with these
properties is a countable intersection of sets which
are open and dense in C"(M) in the Cs topology.
(It is proved without any restriction on dimension M
that most operators P = L + p have simple
eigenvalues.)
3.
In addition to the above results, we have the
if M and P are real-analytic, an
following:
eigenfunction of P has only a finite number of
critical values.
Thesis Supervisor: Victor W. Guillemin
Title: Associate Professor of Mathematics
iI
M
Acknowledgement
The author would like to thank Professors
Victor Guillemin and I.
M. Singer for serving as thesis
advisors and for their generous contributions of time
and effort towards furthering his mathematics
education.
In addition, special thanks are due to
Professor Guillemin for his continual encouragement
during the long, fallow period where the counterexamples
outweighed the positive results.
I
MM
I
Table of Contents
Page
1.
Introduction and Examples: Eigenfunctions of
6
the Laplacian on the Sphere
2.
The Critical Values of Eigenfunctions
17.
A.
18.
Finite Critical Value Theorem:
Analytic Case
3.
B.
The Cm Case
21.
C.
Nodal Critical Points in Dimension 2
27.
Generic Elliptic Operators:
Preliminaries and
36.
Outline of the Main Theorem
4.
Results from Elliptic Operator Theory and
42.
Perturbation Theory
5.
Operators with Simple Eigenvalues
50.
6.
Generic Eigenfunctions I: Openness
57.
7.
Generic Eigenfunctions II: Density
67.
Appendix I: Pseudo-Differential Inverses
84.
Appendix II: Perturbation Theory
92.
References
L---
106.
1.
Introduction and Examples
(1.1)
Outline
(For definitions see (1.2).)
We are interested in investigating the question
of what kinds of functions occur as solutions, and in
particular, as eigenfunctions, of second-order elliptic
partial differential operators.
The aim is to
characterize these functions by means of geometric
properties, for example, by describing the nodal and
critical point sets.
In the case of ordinary differential equations,
results of this nature are well-known (the Sturm-Liouville
theory).
(2)
For examples:
(1)
the zeros are isolated;
the critical points are non-degenerate, so they
are isolated; (3)
the nodal set;
n zeros,
[Ref.
(4)
the critical points occur away from
the n-th eigenfunction has exactly
Incel].
We have aimed at deriving theorems analogous to
these for partial differential equations.
is much more complicated, as expected,
The situation
In (1,3), we
discuss the properties exhibited by several eigenfunctions
of the ordinary Laplacian on the 2-sphere,
r,
(1) to (4) are
- W1,40W.. -rr
7.
all false.
Except at critical points, the zero set is
a 1-dimensional submanifold.
but can be replaced by:
(4) no longer makes sense,
(4')
there are exactly
n connected components of the complement of the zero
set of the n-th eigenfunction.
It is known that there
are always < n components if the nodal set is
sufficiently regular, but there are also examples of
eigenfunctions yielding only two components, with
arbitrarily high eigenvalue.
[Ref. Co urant-Hilbertl
The first of our theorems discusses the nature of
the set of critical values of an eigenfunction.
In the
case of a real-analytic operator on a compact real-analytic
manifold, the critical value set is finite.
This is
proved in section 2A and is followed in 2B by a discussion
of the situation in the
CC
case.
show that if the dimension of
M
In section 2C, we
is 2, the nodal
critical points can be described very precisely.
The
diagram of a nodal set of an eigenfunction will always
resemble the examples in (1.3).
The remaining sections (3 to 7) are devoted to the
proof of one theorem describing a generic class of
operators.
iI
The examples of (1.3) are peculiar because
I
For "almost all" operators on a
of their symmetry.
two-dimensional manifold, the eigenvalues are simple and
the critical points of non-zero eigenfunctions are
non-degenerate and occur away from the nodal lines.
I.e., (2)
[See (3.2) for the precise statements.]
and (3) above are generically true and the nodal set
consists of some number of 1-dimensional submanifolds,
i.e. circles.
A very natural next question (related
Is the number of circles related
to (4')) to ask is:
to the eigenvalue and if so, how?
Terminology and Notation.
(1.2)
Manifolds will be connected and without boundary,
and at least Co .
the real-valued
If
Cm functions on
with compact support.
nodal set of
u
critical point of
a regular point.
on
M
about
p,
u E CO(M),
For
u
If
if
(x 1 ,...,x )
du(p) = 0
means
is a critical value of
point
p
u
such that
denotes those
the zero or
u
is
p eM
du(p) = 0; otherwise
a E R
of
CO(M)
M.
{p e M : u(p) = 0}.
is
C (M) denotes
is a Co manifold,
M
p
a
is
are local coordinates
(au/axi)(p) = 0, i=l,...,v.
if there exists a critical
u(p) = a,
A critical point
p
of
u
is non-degenerate or Morse if
This notion is independent
is non-singular.
((axiaxu
ax (p)))
the matrix
of the choice of coordinates about
p.
•
u E C (M) is
non-degenerate or Morse if all the critical points of
are non-degenerate.
u
Non-degenerate critical points are
isolated, hence there are a finite number of them on
any compact manifold.
Let
a = (al,...,av)
be a multi-index, i.e. a v-tuple
V
tal =
of non-negative integers,
DC
=
(a/ax 1 )
... (a/xV)
zero of order k of u if
a : Jal < k
but
.
E aj
j=l
A point
Dau(a) = 0
Dau(a) y 0
and
a E M
is a
for all
for some
a
with
lal = k. A nodal critical point is a zero of order > 2.
A C_ differential operator on
P : C (M)
U CM
CW(M)
C
P
a multi-index and
I L,
a linear map
such that on .any open coordinate nbd
with coordinates
represented
M is
Z=
lal<m
a
(x1,...,x ), PICO(U)
aaDa
where
EaC(M).
can be
a = (al,...,a )
m is theý order of P.
is
_,M
--
- A lvwlbý .. 7r
10.
If
M
is a real-analytic manifold,
operator if the
Let
of
P
f
as
M.
is an analytic
are real-analytic.
ff ,
f f(x)dx, or Just
M
over
P
denote the integral
<f,g> = f fg
Let
for
is self-addjoint.if <Pf,g> = <f,Pg>
The smmbol of
f,g e Cm(M).
for all
f,g.
P
is a real-valued function
on the cotangent bundle defined as follows.
Let
ap
V
(X1,...,x V )
be coordinates on
is a cotangent vector at
p,
P
ap(p,F) 7 0
is elliptic at
p
if
non-zero cotangent vectors
Let
M
=
UCM. If
a.(PE) =
E
i=l
a=(p)
idxi(P)
".
for all
p.
at
be a compact C manifold,
We will be
considering second-order, self-adjoint Co elliptic
differential operators
is a number
such that
X
P
(real, since
ker (P+X) # {0}.
eigenspace corresponding to
an eigenfunction of
I
P
M. An eigefnValue of
on
P
is self-adjoint)
ker (P+X)
X
P
and
is the
u e ker (P+X)
with eigenvalue
X.
[Note
is
- Aw-fýý
:--
.- -.
11.
that
is really the negative of an eigenvalue
X
P
according to the usual definition.]
+0
set of isolated eigenvalues having
limit point.
of ker (P+X); it is finite.
X
as its only
is the dimension
X
The multiplicity of
has a countable
is a simple
eigenvalue if its multiplicity is 1; otherwise it is
multiple.
We can arrange the eigenvalues of
<
increasing order
X
<
...
<
n < "
where an eigenvalue of multiplicity
h times in the listing.
h
P
in
+00
is repeated
Thus we speak of the
"n-th eigenvalue" or the "first n eigenvalues" of P.
The "n-th eigenfunction" is defined inductively to
mean any eigenfunction which is in the n-- eigenspace
and is orthogonal to the first n-l eigenfunctions.
It
is not uniquely defined.
(1.3)
Examples
On the sphere
and
y 2,
yl
rectangular coordinates
(xl, x 2 , x3 )
x
L-
of radius r, we use local
(spherical) coordinates
have:
i
S2 C R
In terms of
on
R 3 , we
xl = r sin yl sin y 2 ; x 2 = r sin yl cos y2;
= r cos yl.
(Y1, Y 2 )
are valid coordinates
~aiL-.·· ·
12.
The
everywhere except at the north and south poles.
S2
Laplacian on
in these coordinates (in the metric
induced from R3) is
S
2
r
{sin yl
The eigenvalues of
2
(sin yl
,
y2
1
yl
sin
X = n(n-l)/r
~
A
have the form
n = 1,2,...
and a basis for the
eigenfunctions in the n-th eigenspace is given by
Ym
y
'
Ynm(Y1 ,'
where
Y2)
2)
Pm(x) = K
n
2
0 < m < n-I
(cos Yl)cos my 2
0 < m < n-i
= P1(Cos
= P
nm
1
(1 - x 2 )m/2 (d/dx)n+m(l - x 2 )n
the Legendre polynomials
pm(0)
n
Yl)sin my
(Knm normalizes
Pm
n
are
so that
= 1).
At the end of this section, there are diagrams
of the nodal and critical point sets of some eigenfuncttons
in the second and fourth eigenspaces.
I
These are
i
__
13.
Y0
=
cos Yl
Y2
Y40
=
(5 cos 3 Y1 - 3 cos yl)/2
Y43 = sin
Y41= sin yl(5 cos 2
For
Y~0, n = 2, 4,
=
sin 2 ylcos ylcos 2y2
Yl c os 3y2
l - l)cos y 2 /4
the zero set consists of
n-1 distinct circles separating the sphere into
n regions.
For
Y20, there are exactly two non-degenerate
critical points, a maximum and a minimum.
For
Y40o
the absolute maximum and minimum are isolated, but
there are degenerate local maxima and minima occurring
in latitudinal circles lying between the components
of the zero set,
For
Y41' Y4 2
and Y3, the nodal sets consist of
intersecting circles, with a different pattern in each
case.
Y41
At each intersection is a critical point.
and
Y42 , they are non-degenerate and two circles
of the nodal set intersect at each place.
is-
For
Y43
has
i
14.
two nodal critical points, at the north and south poles.
They are degenerate; in fact, they are zeros of order 3.
There are thre longitudinal circles of the nodal lines
intersecting at each nodal critical point.
To summarize, we have exhibited critical points
on the nodal lines, degenerate critical points both on
and off the nodal lines, and (non-nodal) critical points
which are not isolated.
And each eigenspace exhibits
a huge range of phenomena,
15.
Diagramas of the Nodal and Critical Sets
for Some Eigenfunctions of the Laplacian on the
2-sphere
"
-
ImULm
odal
et
Y20
=
In all diagrams
except the one at the
left, only a planar
projection of the
front hemisphere is
drawn.
cos yl
maxim,
minimi
40
Effi-
1i
=
5 cos 3 yl - 3 coys y
--
16.
critical
points
tical points
Y42
=
sin 2 y1 cos Yl eos 2y2
nodal
critical
point
sin y1 (5cos2y
Lodal critical point
Yj3 = ain 2 yl cos 3y
2
- 1 ) cos Y2
17.
2.
The Critical Value Set of an Eigenfunction
This section is concerned with the problem of
whether or not an eigenfucntion of an elliptic operator
on a compact manifold always has a finite ctitical
First we prove that if the manifold and
value set.
operator are analytic, the critical value set is finite.
Notice that the example of the Laplacian on the sphere
is analytic.
The result follows from the fact that
any analytic function on a compact manifold has a
finite critical value set.
situation is different.
In the
case, the
C
There are plenty of
having infinite critical value sets,
C"
functions
In fact, we show
that locally there are examples in which eigenfunctions
of elliptic operators have infinite critical value sets.
However, these local examples cannot be extended to
global counter-examples.
So the general question of
whether or not there is a finite critical value theorem
case is still open.
C
for the
The section concludes with a discussion of what
the nodal critical points look like if the dimension
of
U
L
M
is
2.
They are isolated (hence
0
is an
U
18.
isolated critical value) and near any nodal critical
point the nodal set consists of a finite number of
rays emanating from the critical point.
The Analytic Case
A.
In this section, "analytic" will always mean
"real-analytic",
M be an analytic manifold.
Let
cw(M)
will denote the analytio functions on
(2.1)
Definitions.
(1) Q CM
there is
is an analytic variety if for all
a nbd
Pfl'"'fm
on
V
V
of
p
such that
...
= fm(x)
By an analytic arc connecting
we mean a continuous function
that
y(O) = p,
and
y
(3)
A set
y(l) = q,
is analytic on
Q C- M
6-
in
M such that
(0,
= 01.
p
to
y : [0,
1]
y(t) E Q
q
-
M
for all
through
[FO,
t
1).
p c Q, there is a nbd V of
q E Qf
V
Q,
such
is locally analytically arcwise
connected if for all
p
p e P,
and analytic functions
Q CV = {x E V : fl(x) =
(2)
M.
implies there is
an
1]
i
19.
p
analytic arc connecting
(2.2)
Theorem.
to
q
Q.
through
An analytic variety is locally
analytically arcwise connected.
Pf:
M
For
an open subset of
RV
there is a proof of
this theorem in the paper bry 'Whitney and 'Bruhat.
Since
this is a local theorem, it translates to a manifold
V
p E ,Q and choose a nbd
of
easily.
Let
that
is the domain for a coordinate chart
V
4 : V
{x
C
RV
Q
and
nfV
V : fl(x) =...
(2 ( V)
such
p
=
= fm(x)
= 0},
a CW(V).
fi
is an analytic variety in
Then
4(V) C:V .
There
W of
0(p)
such that any point in
can be connected to
4(p)
through
exists a nbd
analytic arc.
Pulling back to
can be connected to
p
through
M,
p(2
(r V)
via an
-1 (w)
any point in
0 ~ V
W
via an
analytic arc.
(2,3)
Proposition.
If
9 = {p e M : df(p) = 0}
Pf:
f e C
(M ) ,
is an analytic variety in
M.
On any coordinate nbd U, with coordinates
Uf:
= f(p) =
(Xl,..,Xv)
,
V
Qnu = {P C U : TxI p) --..
(-)
=0
0}.
Id
20.
(2.4)
THEOREM.
f e CO( M )
M is compact,
If
has a
finite number of critical values.
Pf:
a c R
Let
be a critical value of
f(p) = a.
be a critical point such that
set
f.
p s M
Let
Since the
of critical points is an analytic variety, it
Q
is locally analytically arewise connected, so there is
a nbd
of
Np
such that, for all q s Pn
p
y : [0, 1] - Np
is an analytic are
y(O) = p, y(l) = q, y(t) E 9
Np, there
with
t.
for all
This
implies
V
d
=
dt (fo)(t)
for
t0 E (0,
constant on
and thus
1),
Eax
i=1
since
(0, 1)
d(xioy)
;f
y(t
0
) s P.
Mi-
Hence
f(q) = f(y(l)) = f(y(O)) = f(p)
N
o
) = 0
0
foy
and, by continuity, also on
Therefore for each critical point
is a nbd
dt
(y(t))
is
[0, 11
= a.
p E f- (a), there
on which all critical points have critical
p s f-1 (a), there is
value
a.
For each regular point
a nbd
NP
containing no critical points of
f
at all,
I
21.
N = U{Np : p e f
Let
(a)}.
N
is a nbd of
f-l(a)
containing no critical points with critical value 3 a.
Also since
f(N)
f
M
is compact,
is a nbd of
different from
f(M)
a
f
is an open map, so
containing no critical values of
a. Therefore
a
is isolated,
Since
is also compact, the set of critical values is
finite.
(2.5)
Theorem.
Let
Pu = f,
If
f E Cw(M)
and
P
is an analytic elliptic differential operator, then
u E C
(M).
For a proof in
is' entirely
R
,
see Bers and Schechter,
But this
a local theorem, since analyticity
(of functions and operators) is a local notion,
In particular, an eigenfunction of
P
is analytic
and has a finite number of critical values,
B. The
(2.6)
C.
Lemma.
case
Let
F(y)
=
e-1/y
p(l/y)sin (1/y)+q(l/y)cos (l/y)]
for y # 0
0
60I
for y = 0
L--·I~-~3IYi--=
~.-
22.
where
p
q
and
Then
are polynomials.
F : R -) R
is
continuous and differentiable and its derivative has
the same form as
Corollary.
F
is
F.
y = 0.
its derivatives at
F
Pf. of lemma:
y $ 0.
for
0 < lim
and vanishes together with all
Co
Hence
is not analytic.
F
is clearly continuous and differentiable
Also we have:
p(l/y)e-1/y2 sin (1/y)
y4O+
2
sin
=lim
Ip(t)et
< lim
t*+m
Ip(t)e-t I
since Jsin tJ < 1 and
t > 1 => e
= 0
et
since
·I
Hence
lim
p(l/y)e -1
/y)e-0os
lim
y*O+
L.,
q(1/y)e - 1 / y
dominates all polynomials"
I
/Y
sin 1/y = 0.
Similarly
/yO+
cos 1/y = 0
and
lim
y*O +
F(y) = 0.
-
ý
- W".Pqm-..ý
-
--
23.
F(y) = 0.
lim
Also
F'(0) = lim [F(y)
Therefore,
F
is continuous
- F(O)]/(y -
0)
= lim F(y)/y = 0
y+O
y0O
+
F(y)/y
continuous
0.
since
y
F'(O) = 0,
and
has the same form as
F
Thus
is
hence is
differentiable at
y y 0,
For
I
[Pl (l/y) sin (1/y) + ql (l/y) cos (l/y)
F'(y) = e-/
where
pl (t)
ql(t)
so
F'
(2.7)
= -t2p' (t)
= -t2p(t)
+ 2t
-
3
p(t) + t2 q(t)
t 2 q,(t) + 2t3 q(t)
has the same form as
Lemma.
F(y)
=
Let
e
-1/y
0
F : R + R
be given by
2
sin (1/y)
if
y 3 0
if
y = 0
24.
Then
and there is an infinite sequence of
F e C (R)
critical points of
which converge
Pf:
That
F
with distinct critical values
to the critical point zero,
F e C (R)
and zero is a critical point with
critical value zero follow from the corollary to
lemma (2.6).
Sin6e ý(y) = l/y is a diffeomorphism of R+
onto itself,
y
is a critical point of
is a critical point of
F
critical values of
suffices to show that
1/y
iff
F
G = Foe; the corresponding
and
Thus it
are the same,
G
G(t) = e
-t2
sin t
has an
infinite sequence of critical points converging to +o
We have
and having distinct critical values.
G'(t) = e
Therefore
cot t -
G
t
[-2t sin t + cos t]
has a critical point
2t = 0.
Since
t
(cot t -
lim
whenever
2t)
= +o
and
t-+n+
lim
(cot t - 2t) = -
o
for all integers
n, there is
t-+nr
at least one zero of
in every interval
M
0
G',
i.e.
(nw, (n+l)f).
0
critical point of
Let
t
0
be such a
G,
25.
Then
critical point.
sin t = + 1//1 + 4t 2 .
If
(n+l)w)
for
R , so these critical values
n
even, having distinct critical
Lemma:
+oo.
Hence the sequence
F.
Let
f : R2
-
R
be defined by
-l/x
f(x1,x 2 ) =
1 + Xl2 + e
1 + X1 2
Then
1/et2J1 + A4t
satisfies the conclusions of the
yn = 1/tn
theorem for
(2.8)
+
of critical point of G, one in each interval
values and converging to
{Yn )
t
Therefore there is an infinite sequence
are distinct,
(n7r,
and
But the function
C.
is strictly decreasing on
{t }
2
is the critical value for the
J1 + 4t
critical point
even, then sine is
n .is
sin t = 1//1 + 4t
positive, so
G(t) = l/e
cot t = 2t, implying
f
C (R 2 )
and
f
2
sin (1/x2 )
if x2 # 0
if x 2 =0
has an infinite sequence of
critical points which converge to the critical point zero
and have distinct critical values,
~CI
W,
26.
Pf:
Clearly
;f/3x
2
f E C (R2 ).
= F'(x
2
),
where
The critical points of
F
f
3f/x
x
f(0, x 2 ) = F(x 2 )
= 2x!,
is the function of lemma 2.7.
must have the form
is a critical point of
x2
where
We have
F.
Also
f
so the critical values of
exactly the same as those of
(0, x2 ),
are
By lemma 2.7, we
F.
can construct an infinite sequence of critical points
of
f
lying on the x2 -axis, converging to zero, with
distinct critical values converging to 1.
Remark.
If
U
is any nbd of 0, flU e CC(u)
and the
conclusion of the above lemma is valid by extracting
from the previous sequence a subsequence lying in
Thus
has an infinite sequence of distinct critical
flU
values converging to
Proposition.
(2.9)
0.
There is a nbd
and an elliptic operator
Lf = f,
Pf.
on
Let
21
R2
U.
f
L
on
U
U
of
0
in
R2
such that
as in lemma 2.8.
A =
2
2
2
2
2/3x2 +2 /3x2
Af(x 1 , x 2 ) = 2 + F"(x
be the ordinary Laplacian
2
2
), so
Af(0, 0) = 2.
27.
4 = Af
Let
4 ý 0.
and choose a mbd
L = (f/4)A.
Define
operator on
U
by a non-zero
U
of
L
Then
0
on which
is a
C
elliptic
since it is a multiple of the Laplacian
C"
function.
Lf = (f/4)Af = f
Also
4 = Af.
since
Remarks.
We have just constructed a local example of
a function which is
an eigenfunction of a CO elliptic
differential operator and has a non-isolated critical
value.
To see that this cannot be extended to a global
L
example, notice that
fails to be elliptic at any
point where the eigenfunction
f/4
then vanishes,
annihilates
f
vanishes, for
But an elliptic operator which
constants,
such as
L
above,
has zero
as an eigenvalue and an eigenfunction in a different
eigenspace must be orthogonal to constants and hence
must vanish somewhere,
C.
The Nodal Critical Points
(2.10)
(1)
chart
M
Notation.
U CM
Suppose
: U
-
RH
is the domain for a coordinate
with coordinates
6(x) = (X1,...,XV).
~
28.
a E U
Let
series of
and
f
f e C (M).
about
a
We will write the Taylor
as
f(x) = E Daf(a)[x - a]a
a = (al,...,,
where
)
is a multi-index; it is understood
that we are writing the Taylor beries of
( , )
(2) Let
of
M.
about
be a metric on the cotangent bundle
This defines a norm in every cotangent space.
f E C"(M), ldf(p) I
Given
- 1
4(x),
and evaluating it at
4(a)
foe
will mean
(df(p), df(p))1/2
_df.
p(
Define
f(1 )(P)
= If(p)j + Idf(p)!
Then:
(i)
f(
only if
(iii)
rC
) ()
p
>
for all
p
f(
1
f(1) (p ) = 0
if
and
) ( ) + g( 1 ) ( p)
IfI(1) = sup {f( ) ( p )
1
C (M)
and
is a nodal critical point of
(f + g)(1)(p) (
(ii)
on
1
e M}
defines a norm
29.,
Idri
UCM,
i=l
fa/ax~j
I:
U.
on
on an open set
(xl,...,x~)
U,
2
2
V
~
i=l
let
A
to
isequivalent
Given coordinates
(3)
on an open set
(x1,,.,x~)
given coordinates
(iv)
be the ordinary Lanlacian
/~x
in these
coordinates.
(14)
u
(see
defn,;
u or
a
such that
c > 0
x E U.
clx-alk, all
<
a e
at
k(
9 ) iff there is a nbd
p.
and a const~ant
Iu(x)I
zero of order
hats a
C03(r'I)
E
F~acts
(i)
If
P
i~s a
differential operator of order m and u. has a zero of
a,
order k at
f s C~(N)
If
(ii)
k
a zero of order
order
in
a,
at
k~j
has a zero of order
a,
at
(2.11)
k
at
u
has
a non-zero polynomial
(Iii)
k,
has a zero of
a.
Let
~Re'mark.
P
differential operator on
coordinates
,j and
has a zero of
fu
then
x-a, homogeneous of degree
order
a,
has a zero of order k-n at
Pu
then
(
1
be a second order C
I/I For
.,x)on
a nbd
a e ~,
UJ
of
elliptic
there exist
a
such that
30.
o(a,S)
2
=
Let
Theorem.
(2.12)
in these coordinates,
P
differential operator on
be a second-order
and suppose
M
(x1,...,x,)
a e M, there exist coordinates
at
u
Pu = 0. Given
on a
has a zero of order
k
a, the k-th Taylor polynomial
Dau(a)[x
uk(x)
=
Auk
0.
Pf.
such that if
a
nbd U of
CWO elliptic
=
-
ala
u
of
about
a
satisfies
k =0O or 1, this is true for any coordinates,
If
for a second order operator annihilates linear functions.
k
Assume
>
2. Using (2.11), choose coordinates
(X1".'V)
on a nbd
i=C
a (a,)t;
at
is
a
P=
i,
U
of
a
so that
i.e. the principal part of
P
A.Then
C
i,J=l
"i
a/ax~axj + T
where T is a first order
operator
V2
V2
=
i~=l1
=
A+ iC
J a~
/axiaxj +
Cpj-
C
ax
lal'/x1 x
gjRjal
+ T
/axiaxj + T
31.
v = u - uk
Now let
v
polynomial.
be the remainder of the k-th Taylor
has a zero of order > k+l
at
a.
We have
0 = Pu = Auk + Av +
V
2
E_ [g j-gi (a)a]
/axiaxj + Tu
i,j=1
= Au k + f
Claim f has a zero of order > k-l at
Auk = 0
imply
of order
k-2
at
must have a zero
Auk
for otherwise
This will
a.
a, hence so must
f.
To show the
claim,
(1) v
has a zero of order > k+l
so
Av
has a zero of order > k-l.
(2)
u
has a zero of order
k
so
Tu
has a zero of order
k-l.
(3) a2u/axiaxj
gij-gij(a)
i,j=l
order > k-l.
4
and
has a zero of order
vanishes at
a
A
T
is
of order 2,
has order 1,
k-2, but
(zero of order > 1),
[gij-gij(a)] 2u/ýxix
so
and
has a zero of
32.
dim M = 2.
Assume
(2.13)
of
a e M.
at
a
(x1 ,x2 )
Let
Proposition.
u
Suppose that
Idu(x)I > Klx-a
Pf.
R2
as
C; this allows
us to use complex variables techniques
Since
function
f.
implies that
Hence
uk =
Auk = 0,
Requiring
f
Re f,
f(a) = 0, which we may do,
is homogeneous of degree k in z - a.
for some
uk(z) = 1[(z-a)
+ (••)k]
= 71
Iduk(x)I = 21
;
1
(z = x l + x 2 VCT).
for some complex analytic
f(z) = c(z-a)k
Now note that
and
k- 1
dim M = 2, view
Since
V CU
such that
K > 0
a constant
U
has a zero of order k
Then there is a nbd
Auk = 0.
and
be coordinates on a nbd
2
But
.uk
c s C, so
k(z)j
where
u k//z = k (z-a) k
C _~_~
33.
so
= kl
jduk(x)
Idu(x) I
IIx-a
k-l
> lduk(x)l - ldv(x)l
Ix-al k-1
> kl
clxalk
-
since v has a zero of
order k+l
> provided
x-ax-al
provided
Remark.
Pu = 0
If
Ix-al
< kI!/2c
(or equivalently,
(2.12) and (2.13) imply that if
order k of u,
then there is
lu(x)l < Klx-al k
constants
K, K'
Corollary.
Let
(i)
IL
0),
a e M is a zero of
a nbd V of
> 0.
Pu = 0
with
P
as above.
Nodal critical points of
u
are isolated
(iii)
a
(P+X)u =
a on which
Idu(x)l > K' x-al k-I
for
and
(ii) The nodal set
Pf:
(this defines V)
u- 1(0)
has measure zero
Zero is an isolated critical value
(i)
is immediate for the proposition implies that
is the only critical point in
V.
For
(ii),
the
3 14.
set
N
M
of regular nodal points has measure zero in
since there exists a countable covering of
coordinate nbds
Ui
in which
hence has measure zero.
N n Ui
N
by
is one-dimensional,
Since the nodal critical points
M is compact, there are only a
-1
finite number of them. Hence u- (0) has measure
-l
p is a
u - (0), if
(iii): For each p
zero.
are isolated and
N
critical point, there is a nbd
critical points other than
a regular point, there is a nbd
critical points at all.
Then
u(N)
is a nbd of
Let
0
(i)
by
p
containing no
N
N = U{N
and if
p
is
containing no
: pE U-e(0)},
containing no critical
values other than zero.
Remarks.
(1) Notice that in the example in section B above, the
critical value is 1.
(2)
These results also enter into the proof of the
theorem on generic eigenfunctions (see section 7).
(3)
We actually know more about the structure of the
nodal set of u, that in
a nbd of a nodal critical
35.
point
a, the nodal set consists of 2k rays (curves)
emanating from
a
(where
For this is true of
nbd W of
a
open set in
f.
is the order of the zero).
f(z) = Re zk
and there exists a
and a homeomorphism from
R2
such that
and the nodal set of
set of
k
u
a
W
onto an
gets mapped onto
0
gets mapped onto the nodal
This follows from theorem I in Kuo.
36.
3.
Generic Elliptic Operators
(3.1)
Let
The CS norms.
M
be a compact C
of dimension v.
Fix open sets
with coordinates
,
(x(i)
1 ' ,.,x
{i) }
(i) ) on Ui
covering
and let
be a partition of unity subordinate to
each multi-index
Da(i)
For
... (a/ax(i))V . In addition to
for pointwise differentiations, we will
C0 (Ui )
consider it as a map
Definition.
{Ui).
a = (al,,,,,a ), set
Di) = (/x(i)) al
using
U1,.,, Ur
manifold
Let
Ifl= sup
C (1).
Cs(M).
f
{jf(p)I : p
M}
r
Ifl
Si=l1 Jal<s
Note that
equivalent.
Ifl < I
Also
I
1
ID4)
0
I 1 (1)
rfl
•l
•
so
these norms are
of (2.10) is equivalent to Ifll.
37.
By the Cs topology on C(OM), we mean the topology
induced by the norm
1... 1.
This is equivalent to
the usual definition,
Facts:
(1)
If
P
is a differential operator of order m, then
jPfls < (const) Ifs+m
(2)
(3.2)
jfg s < (const) flIsl~
s
Main Theorem .
Let
L
be a second order, self-adjoint, Cw elliptic
differential operator on
M.
L
will be fixed from now
on and we will study operators of the form
p e Co (M)
L + p, where
is called a forcing function.
Definition.
Let
u
be an eigenfunction of
L + p.
u
is generic if (1) all the critical points of u are Morse
and non-nodal or (2)
Let
Let
t
A = {p
u
is identically zero.
E C00(M) : all eigenvalues of L + p are simple).
B = {pE A : all eigenfunctions of
L + p
are generic).
T~12U-0
d
38.
MAIN THEOREM:
dim M = 2,
If
B
is
a countable
intersection of sets which are open and dense in the
C s -topology on
Remark,
It follows that
B
particular
For
A
B
is dense in
C0(M); in
is non-empty.
n = I, 2,..., let
= {p e Cm(M)
and let
s > 2,
C (M), provided
: the first n eigenvalues of L +p are simplel
A0 = C00(M),
Then
00
A C ...
CA
n
C ...
C A2 C
A 1 C A0
= C (M)
and
A= n
n=l
THEOREM I:
for all
is
open in the Cs topology on
An
n
is dense in
n,
An- 1
in the
Cs topology,
toolgy
n.
Theorem II implies
all
C (M),
n.
THEOREM II:
for all
An
An
is dense in
Thus theorems I and II
imply that
Cw(M)
A
countable intersection of open, dense sets in
Theorems I and II are proved in section 5,
for
is
a
Cm(M).
An.
39.
Bn ={p
Let
for
n = 1,2,,,,
THEOREM
Ann
and let
B0 = C"Oi),
is open in the CS topology for all
+
TV:
n,
1,
means the greatest
[xl
(Lp+h+j) implies u is generic}
k~er
£
s > Fy12]
provided
Note:
Bn
I~II:
THEOREI'
V.
s A,: u
Assume dim M = 2,
integer
B,
< Xe
is dense in
Bnl, in the CS topology for all
'n*
From theorems II and TV, we have
Bn CA n (~ B n-l CA n-l (NB n-2
c ~~c 1
dense dense
dense
dense
implying
Bn
is dense in
have the main theorem,
Cm(M).
Since
0\R A1~ m
dense
B = (Th Bn, we
n~l
Theorem ITT is proved. in section 6
and theorem TV is proved in section 7,
theorem IV is the only place
see also the introduction
dim M = 2
to section 7,
Note that
is mentioned;
~g---------
40.
Intuitive Discussion:
Theorem I just says that the
property of having the first n eigenvalues of
L+p
simple is
p,
stable under small perturbations of
Theorem III says that the property of having generic
eigenfunctions is stable under small enough perturbations
(which must be at least small enough to keep the first
n eigenvalues simple),
Both of the density theorems
use these openness results to show that you can change
the n-th eigenvalue and/or eigenspace appropriately by
a small perturbation while not really changing the
previous
n-l.
has multiplicity
In theorem II, if the n-th eigenvalue
h > 1
but the previous
n-l
eigenvalues
are simple, we may break up the n-th eigenvalue into
h simple eigenvalues by a small perturbation, thus
n+h-l simple eigenvalues.
producing
Similarly, in
theorem IV, we show that starting with the first
n eigenvalues simple and the eigenfunctions in the
first n-l eigenspaces generic, it is possible to preserve
these properties while making the n-th eigenfunction
generic,
So we perform a perturbation in
to get into
WC
Bn.
An (
Bn-1
d
41.
Having simple eigenvalues is extremely important
for these perturbation arguments, for if the n-th
eigenvalue is simple, it is only necessary to produce
one non-zero generic eigenfunction in the n-th
eigenspace,
All others will then be generic since they
will be constant multiples of the first one,
I
_
__
_
_·_
~___·_
42.
4.
Results from Elliptic Operator Theory and Perturbation
Theory
The Hs norms:
(4.1)
For
f2
llfll
fE C (M),
Let
r
11 1S
=
z
if)
I
C (M[) in the norm
The completion of
Sobolev space
IIDfi)(
ajI<s
i=r
Note that
HS(M).
II... I
I Ifl
=
lfI
is the
and
HO(M) = L 2 (M).
Lemma 1.
(Sobolev),
Ifl
provided
Lemma 2.
_<
For
(const)
f E C (M),
lf llt
t > [v/2] + s + 1
l fgjjl s
(const)
If
IglI s
The use of the partition of unity enables you
Remark.
to globalize the standard local results,
Bers and Schechter,
0
1
for f,g E C (M),
0
-0
0
-0
Reference:
-~-
.9r±--
.
43.
The Milnimax
(4.2)
Let
Princip~~le,
order, self-adjoint
P
be a second
elliptic differential
operator on
u E CQ)(N), we define a auadratic form
For
vn
)
be a linearly
independent set of C~ functions on
N
and define
A(u)
L~et
u(P7u).
=
An(Y1.'vnl=
{v1 *,
Ih(u):u
Let the eigenvalues of
X1 < X <
-2
P
*<X n
-'
E Cm(M),IIUII"1~(UIVI>=0,l~in-1~
and let
U.1 , U 2 ,
-
eigenfunctions
be correspondingf
in increasing order
be arranged
Ilunll
with
...
,
Un,
0
0
= I.
The ore m.
(1)
Xn
=
max {An
l"
~n- ·
V
):{V 1 .,
1
linearly
independent in C"(r4)}
(2)
u = Un,
1
-U
extremals
for the problem.
and
=-f
and~ unPun,
X
n
1~
(I.e. Xn
are a set of
=
An(Ul,9oeunl)
the latter being obvious.
I
i
44.
(4.3)
let
Pseudo-Differential Inverse.
N = ker P
Let
CC (M).
TN
With
P
as in (4.2),
: L2 (M) - C (M) be
the projection onto
THEOREM.
There is a pseudo-differential operator
Q : C (M)
-
C (M) of order
-2
such that
QPu = u - WNu
Furthermore
y
+ n(x,y)
u c Cs ( M)
for
Qw Nu = 0
and
Qf(x) = f K(x,y)f(y)dy,
M
is
in
for fixed
L P (M)
where the kernel
x E M, where
2
if v
= 2
(v-l)/(v-2)
if v >
2
A proof is given in appendix I.
Regularity.
If
of order m, then
Y
is a pseudo-differential operator
lIyulI
s
_<(const)JluJ s+m for u e CC(M).
·ia-;·r--
----1
_
~
r
i
.
.
145*
(4*)4)
Perturbation Theory
De fn.
Let
(I..I)
be a function defined (at least) on the
f
and let
interval
(-T,T),
analytic
in E for
itt
T
<
<
T.
f
IIf(~)
lim
such that
I-ri
(1)
(-T,T)
-~
C
fri~ll
Iit < T
IIf 1 II I~ji
0
if
converges
l~1
<
is a continuous function, hence
I-cl
{T
(2)
f(r)
analytic for
for
I'd<
T'
f('r)
is analytic for
analytic in
<
T'
for all
T'}
j~1
<
T
<
If(-r)Ii
T.
analytic
f(w)
means
T,
T < T'.
where
(3) If we write
(1)f
i=0
{fi}
-+i=01
is analytic for
f(~r)
If
E
is bounded on
F Cr)
is
< T.
Remarks.
f
m
X
-
is absolutely analytic for
in addition, the real series
if
f
if there is a sequence
m
itt
We say
E.
with values in
E
of elements of
for
T > 0
Let
be a normed space.
R
=
m-
Lul
f
< T
F(j
if
f
te
is.
means with reepect to the usual norm.
In this case, analytic and absolutely analytic are equivalent.
__
T=--
~
46.
(4.5)
Let
Perturbation Theorem.
of multiplicity
h
of
L+p.
absolutely analytic in
nbd of
with
T = 0,
a number
h
> 0,
TO
R
analytic in
for
Suppose
Then:
= p.
IJn
(a)
x(i)(0)
(b) u
is
Tr
I < TO
there exists
X(1)(T), ...
functions
a nd
U(1) (T),...,U ( h) (T) analytic in
for
p(T)
I...Is) for all s, on some
(Co(M),
p(O)
be an eigenvalue
X
h
,
(h) (T)
functions
(CO(M), I.. .Is)
< T0 , such that:
(1)
eigenvalue
i = l,...,h;
= X,
is
an eigenfunction of
X(i)(T)
for
I
I
T0 ,
L+p(T), with
i
= 1,...,h;
(c) {u(1)(T),... ,u(h) (T )} is an orthonormal set in
L2 (M) for each
(d)
S:
i < TO;
for every open interval
contains only the eigenvalue
Id
MhL-
IC R
such that
X
L+p, there is a
of
T
CC~-U-rr~LL=z~h
r
47.
T > 0
such that for
IT]' < T
there are exactly
h eigenvalues (counting multiplicity)
x (h)(T)
(1)( )
of
in
L+p(T)
I,
This theorem follows from a minor variation of a
theorem of Rellich (Theorem I,
p.
57).
However since
neither the theorem norRellich's book seem to be
well-known, we give a proof in Appendix II.
(4.6)
One can write down explicit expressions
Remarks,
for the coefficients of these power series by eouating
coefficients in the equations
(L + p(T)
+ X
)(T))u(J)(T)
(1)
=
However, rather than using the entire power series, we
will generally only need the first two terms.
write
X
(
(T)
eigenvalues,
=
X
where
power series in
T
+ a T + RIj()T)2
aj c R
for
and
We will
for the perturbed
% (T)
ITI < TO; and
is an ordinary
r
4 8.
+lj vjT + w (T)T
eigenfunctions,
wj (T) EC (M)
CS-norm on
where
2
u., v.
for the peturbe
perturbed
E C
C(M)
and
is analytic with respect to the
C*(M) for
ITI < To
need linear perturbations
Also we will only
p(T) = p+To
where
a E C (").
Differentiating the equation (1) with resnect to T
and setting
T = 0
yields
(L + p + X)vj + (a + ai)uj
= 0
j
= l,...,h
(2)
From this it follows that
(a + aj)uj i. ker
where
L
p + X)
f(a + aj)ujuk =
implying
and
(L +
-fauj uk = a 6jk
6jk
is the Kronecker delta,
j
= 1,...,h
j,k = 1,...,h
(3)
49.
h
Also
vj
Q=[(o + aj)u ] +
Cj
k
(4)
k=l
h
= Qx Euj] + k=l
E cjkuk
for some
cjk E R, where
inverse of the operator
Q0
is the pseudo-differential
L + p + X
(see 4,3).
I
50.
5.
Operators with Simple Eigenvalues
I.
Openne ss
(5.1)
Let
p,
xn
and
L+p
and
ln'-Xn
Pf:
(Continuity of the Eigenvalues),
Theorem.
Let
p'
E C0 (M)
n'
be two forcing functions,
the n-th eigenvalues of the operators
L+p'
respectively.
Then:
Ip'-pIl
A(f) =-ff(Lf) - fpf 2
and
A'(f) = -ff(Lf) -
JP'T
be the quadratic forms of the minimax principle (4,2),
L+p
corresponding to the operators
define
to
A(f)
A (fl "'f
and
n-1)
A'(f)
and
and
L+p'
An '(fl'"' fn-1)
respectively.
Then
IA'(f) - A(f)j = I/(p'-p)f 2 1
<Ip'-PI ff 2
= tp'-pt
Let
r = Ip'-pl,
Hence
if Ifll = 1
and
relative
51.
A(Mf)
-
r
{fl***
Let
for llfl
< A(f)
with
An (fl'***
=
1
(*)
be a linearly independent set of
fn-l }
By taking the minimum of (*)
functions,
f C C °(M)
< A'(f)
Ifl I
= 1
and
over all
f - fl'"
1,
n-l )-r < An '(flJ'''
fn-1
)
n-l , we get
)+r
• fn-
< An(f
Similarly by taking the maximum over all linearly
}
{fl'''
Ifn-i
independent sets
of
Cw functions on
and applying (1) of the minimax principle, we get
Xnn -r
< Xn
-n
- XnI
< --Xn
n
hence
Ixn
(5.2)
Proof of Theorem I:
on
C (M)) for all
Let
n
p E An
be the eigenvalues
< r
as required.
= Ip' - Pl
topology
is open in the
n.
be given and let
of
X1 < X2 <
.
<
L+p; the first n are simple,
n
< Xn+1< .
Let
52.
j : j
6 = min {Xj+l-
=
l,,...,n; 6 > 0.
U = {p' E C (M) : Ip'-pIs < 6/2}.
of
in the Cs topology.
p
eigenvalues of
Claim
Xj'
J
x j- <
-
IX+1
< 6/2 + IXj - X!+1
j j+
- + (Xj+l•
implying
-
+1 -
j+l
be the
{X1'}
p' s U,
Then, by (5,1),
j = l,...,n.
for
Xj+l
J+l
6 <X j+l
U is an open nbd
I<
p'- pl < Ip'- p's < 6/2
- ji
IX1'
for
L+p'
Let
Let
-
We have
+ Ijx
-
X411 +
xIt
il -
+ 6/2
J)
J
S0 ,
n eigenvalues of
L+p'
U C An,
An
implying
- SjXj
all j.
j = 1,.,.,n,
are simple if
is
open.
Hence the first
p' e U, so
xLI
id
-·14F-4~-
-.
.~C~i~
53.
II.
Density
We want to show that by perturbing the forcing
function slightly, we can break up a multiple eigenvalue
into eigenvalues with lower multiplicities.
Then we
can start with the smallest eigenvalue and break up
the multiple eigenvalues one at a time until they are
simple.
We use the perturbation theorem (see 4.5 and 4.6
for notation).
Assume
of
L+p
X
is an eigenvalue of multiplicity
p(T)
and that
= p+Ta
is
a perturbation of
We have expressions
Q•i (T)
=
+ T
+ T2(T)
j
u
(t) = u.
'v.
= 1,...,h
+ t 2wj(t)
for the perturbed eigenvalues and eigenfunctions,
valid for
ITI < To.
h
p.
Id
54.
(5.3)
There exists
Proposition
of the
Pf.
aj
=
p(T)
for the perturbation
f=fg.
<f,g>
(f,g)
given by
p+Ta, at least two
G
linear operator
<G f,g> = -
-
-
f afg,
is a vector space of
N
C (M),
contained in
h
dimension
product
: N
-
N
f,g E N.
for all
Furthermore,
with respect to this basis has
as its jk-th entry; so:
- f au uk = a 6jk
(by 4.6,
diagonal in
eqn (3)).
Fix a basis
a = f.fk"
3 k
{fl.'.*f
<Gauj ,uk
So the matrix
this basis; and its
and hence its eigenvalues are
(2)
ful,...,uh}
a, and the orthonormal basis
of unperturbed eigenfunctions, we have
is
", the
of
flh}
<G fj ,fk>
Ga
N
such that
G
of
has an inner
N
f afg, so there exists a self-adjoint
matrix of
Given
so
There is a quadratic form on
given any orthonormal basis
(1)
such that
are distinct.
N = ker (L+p+X).
Let
a E C(MT)
h}.
Then the matrix of
diagonal entries
al,...,a.'
Let
Ga
j Z k
has
-
and take
f1
I fkI
Jk
2
F-
Id
55.
in its jk th place.
is not a multiple
Ga
Therefore
of the identity so it must have at least two distinct
Hence for the perturbation
eigenvalues.
a,
with this
aj
at least two of the
p(T). = p+Ta
must be
distinct by (1).
(5.4)
Proof of Theorem II.
p e
Let
so that
(i)
s0 > 0
and let
Ip'-pls < c0
[Xn - E0 , Xn +
interval
of
1
is dense in
p' E An-l
implies
]0.
An-
1'
be small enough
L+p
is the only eigenvalue of
Xn
(ii)
n_
A
h
Let
and
in the
be the multiplicity
X
if
exists
p' s C (M)
multiplicity < h.
such that
IPj+l-Pj
n-th eigenvalue of
Ip'-p
< e
s
and
there
nv has
By applying the claim repeatedly, we
can obtain a sequence
so that
e : 0 < e < E0,
h > 1i, then for all
Claim:
p = pl,...P*
< s/h
L+pj
h
(J = l,...,h-l)
of forcing functions
and the
has multiplicity < h -
j + 1.
id
56.
Thus
'ph-PI
L+ph
are simple, proving the theorem.
< E
and the first n eigenvalues of
Now we prove
the claim:
(1)
A
By (5.3),
we can choose
a e Ce(M)
+ ac.T + 8(T)T2
(
(T) =
such that if
are the perturbed
eigenvalues corresponding to the unperturbed
eigenvalue
X = An
and the perturbation
then at least two of the
T1 > 0
exists
A~J)(T)
(2)
A
[X-E,
of
is the only eigenvalue of
T2 > 0
(1) (T),...,(h)(T)
L+p(T)
0 < IT I < Tl,
in
(A-E,
see thm. 4.5 (d).
= p+TU,
Then there
at least
are distinct.
A+E], there exists
implies
are distinct.
such that for
two of the
Since
a.
p(T)
X+s)
L+p
such that
in
0 < ITI < T 2
are the only eigenvalues
(counting multiplicity) -
If in addition
JT I < Tl, at least
two of these are distinct, hence they must all have
multiplicity < h.
(3)
Then
Let
T =
min {Tm, T2, E/als)
IP'-PIs = ITIJaos
< E.
and let
By (5.1), I3I - XnI <
so by (2), A' must be one of the A~J)(T)
multiplicity < h.
OIL-
p' = p(T).
and have
~n~-~r~a-
57.
6.
Generic Eigenfunctions I: Openness
The object of this section is to prove theorem III,
that by perturbing the forcing function a small amount,
the property of having generic eigenfunctions,
i.e.
Morse with non-nodal critical points, is stable.
Throughout this entire section and the next we
consider one eigenvalue
of
A
L+p
at a time.
always assume that all the eigenvalues
If
A = X,n
function
'
1
are simple.
then corresponding to a perturbed forcing
p'
we get a sequence of eigenvalues
< ... < A' <
n
(for small
< X
We
T)
'
n+1
< ...
A'
n = An (T)
If
is
p'
= p(T),
then
the perturbed eigenvalue
discussed in the perturbation theorem.
Thus there is
no confusion; the perturbed eigenvalue corresponding to
the n-th eigenvalue of the unperturbed operator is the
n-th eigenvalue of the perturbed operator and it is
simple.
A'
or
(6.1)
So we will talk about the "corresponding eigenvalue"
and the "corresponding eigenfunctions."
A(T)
Let
p e An
eigenvalues of
L
L+p.
and let
X
be one of the first n
Let
60 > 0
be such that
58.
%
/P'P1<
implies
t + [v/2
Theorem:
IP'-P1 5
< 6
s'ker (L + p' + X')
Iu'-ult
<
Assume
s > 0, there exists
X'
and if
corresponding to
L+p'
Let
1.
For all
such that if
of
-
s A.
n
Ilul I = 1.
u s ker (L + p + X),
5 >
p'
X
of
is the eigenvalue
L+Q, there exists
Hu' II
with
6 : 0 < 6 < 8
= 1
and
5.
Proof:
I.
that
Discussion.
IP"-PI,
pt
E ~~
Let
X'
Let
< 60.
eigenvalue and let
with eigenvalue
u'
X'
and
be arbitrary except
be th·e
be an eigenfunction of
Lu
Lu'
u'
is one-dimensional.)
?X')
pu + Xu
+
+
L+p'
ilut II = 1. (Note that there
are only two such eigenfunctions,
ker (L + p' +
corresponding
=
0
p'tu' + X'u'
=
0
and
-u', since
We have
_*Y
_
T-
59.
Subtractracting and letting
v = u' - u
gives
Lv + p'u' - pu + X'u' - Xu = 0
<=>
Lv + (p'-p)u' + pv + (X'-X)u' + Xv = 0
<=>
(L + p + X)v = [(p-p') + (X-X')]u'
Let
r = (p-p') + (X-X')
exists
n
R
QX
L + p +A
Ivit
both
In
are small.
u'
(1)
+ nu
3
and dim ker (L + p + X) = 1).
To
is small, we will use (1) and show that
and the appropriate Sobolev norms of
Since
= QX(w)
Qw
v = u' - u, we have
(2)
+ (1 + q)u
(2) is a decomposition of
MiL
Then there
is the pseudo-differential inverse of
(by 4.
show
w = ru'.
such that
v = Qx(w)
where
and let
u'
into components in and
I.~~~C-,
T--a~L
60.
ker (L + p + X), so by the Pythagorean
orthogonal to
theorem,
1 = I QwI 12 + (1+ n)
since
Iull
2 + 2r
If
n2
+ 82 = 0
0
= -1 + J1-
=
-1
-
/l -
82
-2 < n2 < 1 < n1 < 0.
nln2 = 62
= e2/ In
2 < 6 since 6 < 1
1n1
if
Ilu'll = 1. Letting 6 = lIQ wll, we have
e < 1, there are two distinct real roots
nI
so
=
2
n = n2
since
<
In21.
implies that
Finally,
in (1), we are not about to show
n121 > 1.
Iln
However, in this case we have
small,
61.
ut
=
Q(w)
-u t
=
-QX(rut)
+
(1 +
(1 + l)
-
(1 +
+
QX(r[-u'1)
=
)
there exists
Thus we have shown:
such that
1)
(2) holds with
'i= ri
.1
Ut
S
ker (L + p' + \,)
We will assume this
in part TII
TI.
Estima~tes.
c1
I , c2
etc., will denote various
positive constants which never depend on
'
or
u'.
We have, for all
InI
~.IP'-P1 5 +
IIQxwI Is+2
rC
IX'-XI
<
21p'-p1
llwlI
(3)
by (14.14)
<c2IrISIIu 'Is
5 2c211ulll,
5
by lemma 2, (14.1)
<2c2luuIS
P'-P Ic by (3)
(14)
1ý
- ýW-4=ý .-
A
62.
For
s= 0 ,
Ilu'll
since
IIQwl i
1=
= 1i,this implies
IIQXwlI2
< C21rl
S(5)
so if Irs < 1/c , i.e. Ip'-Pls < 1/2c ,
2
2
and
nll1 < e < 2c2 1P'-PI
Claim that for all
that
For
s,
lu'ls < Ks.
K0 = 1
then
(6)
s
there is a constant
Ks > 0
such
This follows by induction on even s.
and
Iu'l I ss<-- l IQI(w)
+ (1 + n1 )ul
I
<_IlQx(w) I S+ (1 + In
1 )lulls
< c31
<
Iwi s-2 + 211ul I
(by 4.4 and
c3 rl s-2 1 u'IIs-2 + 211 u
< 3(1/c2)Ks-
2
+ 21 ul I
i Ks
(4.1,
In1l
< 1)
lemma 2)
(by (5)
and the
induction hypothesis)
T63.
Finally, using Sobolev's lemma, the claim and (4),
we have
1QXwjt
Choose
The proof:
and let
X.
to
<
s > t + [v/21 - 1
< 6,
Jp'-ps
6 < min {60, 1/2c 2, c/(c5+2c 2 1u tl
with
A'
the eigenvalue corresponding
By the concluding remark in part I, there exists
with
lu' ll=
u' = Q (w) + (1 + n)u, where:
w = ru',
u'
for
c5 1P'-P s
c 5 = 2c 2 c 4 K s.
where
III.
cC411Qxwil s+2
+ X')
E ker (L + p'
(and
6 < 1 since
n = -
1 + J1 -
0
6 < 1/2c
by (5))
2,
and
1
e = IIQxw I
and
.
Then
lu'-ult = (QXw + ault
< QXwl t
+ lal lult
< c5lP'-P s + 2c2 1ultlP'-~l
<
(c5 + 2c2 1ult)6
< C.
s
(by (6),
(7))
(7)
CI-uu~~qeb
`r
64.
(6.2)
If
Proposition.
U Im
is an open set containing
all the nodal critical points of
exists
such that if
6 > 0
f c C (M), then there
< 6,
If - gl1
g
then all the nodal critical points of
..
Since
Pf:
I1
and
I...
suffices to show this for
1
= 0.
)(P)
=
< 6}.
Recall that
There exists
critical point of
f(
1
Claim:
)(p)
lie in
g, then
g(
1
)(p)
= 0.
V6 .
(1) and (2)
U.
Proof of (1): If
p
is a nodal
Then
<- (f - g)(1) ( p ) + g() (p)
< If - gj(1) <
Proof of (2):
V
such that
6 > 0
g
Let
If-gl(l) < 6,
if
(1)
the nodal critical points of
prove the proposition.
L
1(1) .
(See 2.10 and 3.1 for notation.)
{p E M: f( ) (P)
1
then all
(2)
also.
U
is a nodal critical point if and only if
p E M
f(
lie in
are equivalent, it
(1)
I...
c(M),
C
g
Let
6 ,
hence
p
C
I
6 = mi n {fl1)(p)- : p cM -
U}.
For
I
65.
p E M - U, we have
f(
critical points of
f
> 0
)(p)
1
lie in
since all the nodal
U.
But
f(1)
is a
continuous function, hence it attains its minimum on
the non-empty compact set
p 4 U implies
(6.3)
f(
1
)(P)
M - U.
6
Lemma (Morse).
{f
and
p ! V6 .
CO(M) : f
open and dense in the Cs topology on
Proof of Theorem III:
(6.4)
topology on
CW(M), for all
Bn
For
if
n = 0.
p E Bn C Bn-I
Ip'-PI
< 60
ni An,
then
u n c ker (L + p + Xn ) ,
un
Hence
V6 CU.
is Morse}
Cm(M)
n, provided
Assume
Finally,
is
s > 2.
if
is open in the Cs
The proof is by induction on
it is true for
6 > 0.
Thus
n.
B0 = C0(M)
Since
Bn
is
open in
60 > 0
there exists
p' E Bnl
s > [v/2] + 1.
A .
C (M)
such that
Let
Hunl1 = i1. p E Bn
means that
is a Morse function, so by (6.3) there is an
0' > 0
such that if
If - Unl 2 < E',
then
f
is a
Morse function.
0
0
E
0
66.
Also
p E Bn
implies
un
has no nodal critical
E" > 0
points, so by proposition (6.2), there exists
such that
if - Un l < c"
critical points.
implies
The hypothesis
f
has no nodal
s > [v/2] + 1
us-to apply theorem (6.1) with
t = 2.
E = min (E',S").
6 : 0 < 6 < 60
that
(2)
lu'u
n
IP'-Ps < 6
there exists
= 1
and
There exists
implies
(1)
Let
ker (L + p' + XI)
Id
EhL_
is
is one dimensional, every non-zero
sp'-pI
< 6}C B
{p' E CcO()
Ths{' •M
u'
n
Since
function in it has these properties, so
Thus
and
such that
Therefore
Morse and has no nodal critical points.
such
Bn-l
p' A A
u'
n E ker (L + p' + X')
n
Ju'n - u n 22 < c.
enables
:I'Ps
p E Bne
so
n'
Bn
n"
is open.
- -AwýAmp
.. :
67.
7.
Generic Eigenfunctions II:
Density
Finally we show that the set of forcing functions
yielding operators with generic eigedPunctions is dense.
First we show that by a small linear perturbation we may
eliminate the nodal critical points, which are isolated
and whose structure we know from section 2C.
Then we
will show that we can make all the non-nodal critical
points non-degenerate.
We assume throughout this section that
dim M = 2.
This is used in two places: in prop. 7.1 we use the
corollary of (2.13) to assert that the nodal set has
measure zero; in theorem 7.5 we use (2.13) to assert
Idu(x)I > Kx - alk - 1
of an eigenfunction
if
u.
a
is a zero of order
k
Neither of these statements
involves dimension 2 although we have used complex
variables to prove them.
(7.1)
Proposition.
simple.
a s Cm(M)
If
a E M
Let
is a zero of
such that
u(T) = u + Tv + T2 w(T)
1i
u E ker (L + p + X), with
X
u, there exists
v(a) # 0, where
is the perturbed eigenfunction
·- ci--·-_a
-·r~r-.-
;L~L-
___
68.
p(T) = p + To.
corresponding to
By contradiction.
Pf.
a perturbation
Let
a E M, u(a) = 0.
p(T) = p + TO,
3
n E R
Given
such that
v = QX(au) + nu
(*)
by remarks (4.6) and the fact that
QX
= /K
where
K
for all
since
L + p + X,
is the kernel of
of the form (*),
v
By (4.3),
y
-
u(a) =
(a,y)a(y)dy
Qx
(see 4.3).
K (a,y)
v(a) = 0
e.Co (M
for all a s
belongs to
2
p =
If
we have
0 = fK (a,y)a(y)u(y)dy
ML_
is simple;
is the pseudo-differential inverse of
v(a) = [Qx(au)](A)
Id
X
v-l)/(v-2)
L p(M), where
if
v = 2
if
v > 2
)
7
I
r-
69.
Choose. q
Since
so that
C (M)
is
L
q
dual to
is dense in
1 + 1- =
p
Lp,
Lq(M), we have
0 = f K (a,y)o(y)u(y)dy
M
for all
a c L (M). Therefore
zero in
L (M),
so it vanishes almost everywhere.
{x : u(x) = 01
has measure zero by the corollary
to (2.13), we conclude that
everywhere, and henee
f e C ( M)
Thus if
complement of
have : g s N•
onto
\ C (M)
(M
constant multiple of
L
which is
K (a,y)
K (a,y)
=
is
0
g(x) = f/K(x,y)f(y)dy,
)).
in
L 2 (M),
g(a) = 0
impossible.
we
for all
we
(for
But any element of
g(a) = 0
LP (M).
be the orthogonal
(L + p + X)
implies
Since
almost
zero in
u, so vanishes at
Co(M) = N $ NI'/r\ Cm0(M),
g e C (M),
K (a,y)u(y)
-
Letting
N = ker
NI nCo
-
and
g(a) = 0.
must have
y
y
a.
Q
N
Since
maps
is a
- ------ =1·-
-7
i~--
id
70.
fl** 'fm
such that
function
g
for
Pf.
By induction.
for
m - 1.
fl''o'f
-1
If
0 <
- 0,
le0
Let
i )
Given m funct:ions
there exists a
f 0,
i
- l·o
Trivial for
such that
let
m = 1.
h(a i ) d 0
you're done with
0
Suppoee true
for i
g = h.
If
Then
.,m-1 and fm(ai) r
Ig(ai)I = Ih(ai) + efm(a )I
- lel Ifm(ai)l
> Ih(a i)/2
g(a ) = h(a )
(7.3)
and let
Proposition.
al,...,am
Rik
of
,...,m-1.
> Jh(a i)
and
h
be chosen so that
< min {lh(Ai)I/21fm(ai): i=1,..
g=h + ef m.
fis
. • •
There exists a linear combination
h(a ) Z 0,
h(am)
fi (a
M.
which is a linear combination of the
g(a i ) 0 0
with
al,. S.,aa m
Let
Lemma.
(7.2)
> 0
+ efm(am ) - efm(am ) : 0.
Let
u E ker (L + p + X),
be zeros of
u.
X
There exists
simple
01
71.
C (M)
a
u(T) = u + vT + w(T)T2
such that if
the perturbed eigenfunction of
v(a i ) / 0
Pf:
p(T) = p + Ta, then
i = 1,...,m.
for all
By prop. (7.1), there exist
such that if
al,'...,m
u.(T) = u + v.t + w (T)T
2
perturbed eigenfunction corresponding to
we have
v (aj)
j
0,
j = l,...,m.
E
C (M)
is the
pj(T) = p + ajT,
Furthermore
vj = Q[a ju] + nu
for some
j
function
v'
v'
=
m
E e.v.
j=l J
E R.
is
(*)
By lemma (7.2),
there exists a
which is a linear combination
and satisfies
v'(a)
= 0
for all
m
j = l,...,m.
id
MMOL.
Let
a =
e 6a.
j=Il i
If
u(T) = u + TV + T2w(T)
of
L + p(T),
and let
p(T) = p + Ta.
is the perturbed eigenfunction
~-~c~--_
-L
~IIE~-i~~h
72.
v = Qx[au] + nu
n E R
for some
m
= Q[IE
ejju]
+ nu
J=-1
m
m
ej (v
=
- fju) + nu
by (*)
J=1
m
m
=
n6 v
j=1
+ n'u
where
= nn'
- J
e.n
j=l
'J
= Vt + nfU
Since the
ai
are all zeros of
i = 1,...,m.
v(ai ) # 0, all
(7.4)
Proposition.
IT I < T,
1 norm on
U
all
MMMI
v(T)
C
u, we have
Suppose that
is analytic in
(M ) ;
and all
a E M
x e U,
and for
with respect to the
let. u(T) = u + TV(T).
be a coordinate nbd of
T : ITI < T
T
u E CO(M)
Let
and suppoee that for
73.
(C) lu(x)l < Kix -alk
(ii) jdu(x) I > K' x - alk-1
Iv(T)(x)I > J
(iii)
Then there exists
implies
If
u(T)
T' > 0
K,K'
J > 0
such that
> 0,
k an integer > 2
independent of
0 < IjT
critical points in
is a critical point
0 = d[u(T)](x)
Then
for some
has no nodal
x s U
for
of
u(T),
< T'
U.
then
= du(x) + Td[v(T)](x)
Idu(x)
Ix - al k-l 1<
by (ii)
=-1 IlI Idv() ](x)l
< clIT
lu([)(x)
=
where
RTsup
c = 1
Iu(x) + tyv(T)(X)I
> ITIv(T)(x)l - lu(x)
> ITIJ - Klx - alk
>
by (i)
and (iii)
IT J - K(cTj)I)k/k-1
> lT J/2
> 0
LI
{Idv(T)I:T < T}
if
provided
r
0.
ITi < T' = 1(J/2Kc)k-l
74.
(7.5)
Assume that
Theorem.
be an eigenvalue of
p' E C (M)
s uch that
eigenvalue of
For all
corresponding t o
L+p'
there exists
A'
and if
<
pt-p S
X -< An
and let
e > 0,
s ker (L + p' + A') - {0}
u'
then
L+p.
p e A
n
X
of
is the
L+p,
implies
u
has no
nodal critical points.
Pf:
We will take
p(T)
= p + T~
p' = p(T)
for a linear perturbation
to be chosen and some
u e ker (L + p + X),
Ilull
= 1.
number of nodal critical points
coordinate nbds
vi
of
K , K' > 0
Choose
ki _ 2
so that
an intege r
v(a
i )
? 0
Wi = {X E
: iv(x)l > lv(ai3)112}
Iv(T)(x)
u
and
such that if
> Iv(a i ) 1/4
for all
InT
.
(see section 2C).
i = 1,...,m
for
Look at each critical point
Ti > 0
of
m
Idul(x) > K'lx - a.
- 1
1
(prop 7.3).
exists
61
al,...,a
i = 1,...,m, on which
,
and
and
a E C (M)
There are a finite
k.-1
ju(x)! < Kilx - ail
for
ai
Let
T.
ai
Claim:
< Ti
x E W.
1
.
Let
there
then
(v(T) = v + TW(T)).
id
75.
Let
Si = sup {Iw(T)I : ITI < TO},
the radius of convergence of
S.
= 0, let
1
T i1 = T 0
Sii1*$ 0,
T. = Iv(a i ) l / 4 Si .
1
let
(TO
u(T); see
is essentially
4.5).
If
and the claim is immediate.
Then for
and
ITI
x S Wi,
Iv(T)(x) I > Iv(x)l - ITI l(WT)(x)l
> 1v(a
i)
> l1v(a
i
=
K = Ki,
> 0
-
Iv(ai ) I
v(a )I
proving the claim.
are satisfied
)
- TiS
i
Thus the conditions of prop 7.4
[with
U = Wi, a = ai, T = Ti,
K' = K!, J = Iv(a i)/4],
such that if
so there exists
0 < Ij-I < T' u(t)
il
nodal critical points in
has no
Wi .
m
Now let
T' = min {T ,..., T
and
W=
i=1
For
0 <
TII < T', u(T)
Wi
.
has no nodal critical points
W
76.
in
W.
6 > 0
By prop. 6.2, there exists
such that if
lu(T) - ull < 6, then all the nodal critical points of
u(T)
lie in
W.
Since
u(T)
is analytic with respect
to the C 1 norm on
Cw(M), there is
ITI < T"
ju(T) - ull < 6.
implies
T = min {T', T", E/IcI s
for
0 < It! < T.
ITla ls < e;
(ii)
(iii)
It! < T'
such that
Now let
p' = p(T), u' = u(t)
and let
(i) IP'-p s =
Ip +
TU - p1s
implies that all the
ITI < T"
nodal critical points of
T" > 0
W;
lie in
u(t)
implies that there are no nodal
critical points of
u(t)
in
W.
Hence
u' = u(T)
has no nodal critical points at all, nor does any
non-zero multiple of it, so the theorem is proved.
(7.6)
eigenvalue of
assume
Suppose
Theorem.
u
there is
L+p.
Let
p c An
n
X -< Xn
n
and
u c ker (L + p + X),
has no nodal critical points.
such that
p' s An
eigenvalue of
L+p'
and if
IP'-PI s <
u'
u / 0;
For all
C,
e > 0,
is an
u' e ker (L + p' + X) -
then all the critical points of
non-nodal, i.e.
is an
is generic.
u'
are Morse and
{0},
77.
Remark:
In the proof, we choose a
u'
satisfying
the required conditions and then show that there is a
such that
u'
is an eigenfunction of
with eigenvalue
X.
Also
p' E C (M)
L+p'
close to
Pf:
if
u'
is sufficiently
is close enough to
It suffices to assume
that
I.
p
p'
Ip'-pI s< e
implies
Construction of
u':
E
is small enough so
p' E Ane
u
Since
has no nodal
u
example, let
U
=
{peM
:
a nbd of
has no critical points in
y = inf {ju(p)I
u(p)j
u-1 (0)
y/2}
such that
is
such
VCU
a
nbd.
= 1
M - U.
v E C (M)
6 > 0.
Let
Iv - uls+2 < 6
u' =
V
= 0 on 7,
By the Morse lemma (6.3),
such that
Morse function.
Let
and let
be a bump function: 0 < V
9 < 1,
on
T. For
: du(p) = 0}; then
SE O(M)
Let
NL
<
u- 1 (o)
U of
critical points, there exists a nbd
such that
u.
and
(1 - O)u + *v.
there exists
v
is a
We have
W
78.
(1)
u'
on
V
(hence
(2)
u'
on
N
M
U
has no critical points in V)
u'
has only non-degenerate
u'
(hence
M - U)
critical points in
(3) lU' - uls+2
c1 Ips+ 2Iv = 1I(v - u)ls+ 2 -<
< c21v -
du'(p) I > Idu(p)I
(4)
ls+
2
uls+2
< c2 6
- Id(u'-u)(p)I
Idu(p) I - Iut-uI (1)
> Idu(p)I - c31u'-ull
> Idu(p)
If
t - V, then
p
-
c3 c2 6
Idu' (p) > c 4 /2 > 0
p e 9 - V};
c4 = inf {Idul(p):
(hence
(5)
If
u'
provided
has no critical points in
p ý U,
ju'(p)I
> Iu(p)j
where
6 < c 4 /2c
S- V)
- Icu'-u)(p)I
> y/2 -
Iu'-ul > y/
> y/4
provided
2
-
6 < y/4
2
c3.
jilL··i~~
r
79.
(recall
y
from definition of
no zeros except in
since
u = u'
U).
Therefore:
(u'Y"1 (0) = if1 (0)
U, so
u'
has
and
in a nbd of this common zero set,
u'
has
no nodal critical points either.
II.
Conlstru~ct'ion
eigenl'unction
Pt:
Next we show that
of an operator
The choice of
s$tisfy
of
p'
L+p'
with eigenvalue
is dictated by the fact that
CL + pt + X)ut
the equation
[
p
is an
u'
0.
Ut
Xe
must
Define
on
(6)
I
These are
overlap
u eker
M
-
C
V
-
Lut
reduces to
Also
by definition, and on
CL + p + X)u,
u 1 (O)
M -V.
-
"o
0
<
on
V, the left hand side
which is zero.
such that
be a bump function:
on
14
CL + pt + X)u' = 0
To estimate the size of
nbd of
o
where defined and agree on the open
-1
i (0) since Ut = u on V and
CL + p + X).
1 CO)
f
+ Xu'
IP'-P1 ,
5 let W be a
WCV
4
<
andij let
1, ~ = 0
on
sC(M
¶W, 4,= 1
80.
Then
+ (1-0)(p-p')i
since
-IO(P-P')I
(1-W)(p-p')
$
(for q(x)
=IQ[
1 => x E V =>
p = p')
+ Lu' + Au'
U
S
= I(4/u')(L + p
+ A)u' I
c51/u'I
<-5
1s/u I(L + p + X)u'j
< c5/u'IsI(L +
p+ X)(u' - u)l
since
< o61e /u' l lu'
Claim:
with
There exists
K
L
(L + p + X)
- ul s+2
K > 0
depending only on
and using (3),
(7)
u e ker
Ip-p'I_ < c2 c6 K6
such that
ý, u, s.
I/u' Is < K,
Assuming the claim,
81.
[Note:
a nbd of
¢/u'
is
well-defined and
Co
since
0 = 0
in
(uI)-l(0).]
To prove the claim, write
I /u'
Is
+ (1
=
-
p)Is
(l-
< Ip2/u'I +
)/u'I5
= I¢2/u'i s + I(1 = I2/u
'
)/uls
s + K1
0(l -
(where we used the fact that
x
this follows from the fact that if
and both sides are zero and if
Now
Nows
2 /u'1
< 12/uI
4)/u' = 0(1 -
%
V,
x E V, u'
then
4)/u:
W(x)
= 1
= u)
+ I2(u - u')/uu'Is
(since
1/u' = 1/u + (u - u')/uu')
< K2 + K3'l/ulsl /u'/lu - u'ls
< K2 + K461/u'ls
82.
Assuming
6 < 1/K
and combining these two estimates
4
yields
I¢/u'
s < K1 + K2 + K461 /u'Is
I0/u'lI
=>
III.
< K = (KI + K 2 )/(l - K 6)
The proof:
e > 0
Let
Jp'-pJs < E implies
6 = min {c 4 /2c
2 c3
and
be given, so that
p' E An.
Choose
, y/4,
4
1/K
, s/c
2 c6
K}
Apply the procedures in part I to construct
u' X 0
to (5) imply
u'
Then
eigenvalue
imply
u'
E,
in ker (L + p' + X)
all generic.
L_
Construct
p'
is an eigenfunction of
X. Finally, our choice of
Ip-p'1S <
u'.
(1)
and all the critical points of
are Morse and non-nodal.
part II.
K > 0.
6
as in
L+p'
with
and (7)
and since any other eigenfunction
is a multiple of
u', they are
83.
(7.7)
Proof of Theoremi IV:
,1A
is dense in
Bn
in the C~stopology on C03(M).
IP"-PIs
with
IP'-PI,
< e·'
~.
<
p' E C CM~) with
S
Ip'-pI 5
A'
-
< 6'/2
{0}
and such that
has no nodal critical
By theorem (7.6), there is a
IP"-P'1 5
with
that
(lnAsince this is
p' s B
ker (L + p' + A')
points.
is
such that
Then by theorem (7.5), there exists a
open.
uA
: 0 < s' < E
E1
Choose
implies
p" e Bn
Claim there is a
p s BnlC'~LAn*
Let
< s'/2
such
n'n that
u"
Utt
is generic.
Ip"pl -IP"-P'I5
+
p" e C (M),
and whose n-th eigfenvalue
s ker CL + p" + A')
n
Therefore
IP'-PI 5
<
s'/2
p" s Bn
+ s'/2
implies
and
=
' < E.,
84.
Appendix I:
Pseudo-Differential Inverses
Theorem (4.3) contains somewhat more information
than the existence of a parametrix of an elliptic
operator modulo operators of order
-
o, so we will
give a derivation based on the standard theorems.
For
reference, see Seeley and Bers & Schechter.
An operator
Defn.
IPull
all
P : C (M)
< csl lulls+m;
order
-
C*(M)
-
*
has order m if
means order m, for
m.
If
THEOREM I.
P
is an elliptic pseudo-differential
operator of order m, then there exists a pseudo-differential
operator
E
-
of order
PE = 1 - S
where
THEOREM II.
Let
R, S
P
m
u e L 2 (M)
in the
L2
and
-
c.
be a self-adjoint elliptic
M.
such that
For all
Pu = f
inner product.
We will assume Theorems I and II.
a-
EP = 1 - R
are operators of order
differential operator on
exists
such that
f e L2 (M), there
iff
f -ker P
I
85.
Claim there exists a linear operator
(1)
Q : L 2 (M) -+ L 2 (M)
u E Cw(M);
all
By theorem II,
f EN
N_
Let
on
N
Q0 : N
.
with
Q0 f = v;
Q0 P = PQ 0 = I
Pv = f.
If
+ N
v'
Pv' = f, then
v -
v = •
Then
v'
= N IN
i
N
on
.
.
Q = Q0N
Let
u =
N-L
-
u = u -
NL
Now
N u)
TN u
N
QfN = QOTNJ N = 0.
Claim Q is a pseudo-differential operator of
- m.
order
M
Let
E
M
E N4
= {0}.
this is well-defined and satisfies
= 0 + Q 0 Pr
(2)
(u)
is any other element
QPu = Q0 fN Pu = Q0 Pu = QOP(7rNu +
Also
such
For theorem II says
=>3u E L 2 ( M ) 3 Pu = f.
and satisfies
of
we can define
rN(u) and Qrru = 0, for
N = ker P.
is projection on
fN
Q0 P = PQ 0 = I
that
QPu = u -
such that
be the pseudo-differential
0
0
86.
parametrix described in
Q = E + T
where
T
Theorem I.
We will show
is an operator of order
-
Write
EP = 1 - R
QP = 1 - "N
implying
(Q - E)P = R - wN
and
0 = EPwN =
so
(Q - E)P = R(1 -
7N
- RwN
N)
Multiplying on the right by
and using
E
PE = 1 - S
gives
R(1 - nN)E = (Q - E)PE
= (Q T = (Q - E)S + R(1 -
E)
wN)E
-
(Q - E)S
is an operator of order -
since the composition of an operator of order
an operator of finite order has order
-
w
with
- c, and R, S have
87.
r
Id
- c.
order
(3)
Q = E + T.
Also
It remains to show that we can write
Qf(x) = / K(x,y)f(y)dy
M
y
where
+
v = 2
if
2
for
L (M)
is in
K(x,y)
v > 2
if
(v-l)/(v-2)
By a partition of unity argument, it suffices to
show that if
is a pseudo-differential operator on
Q
UJC RV,
a bounded open set
where
y
Delt. Let
K(x,y)
-
w
is in
then
L p(u)
Qf(x) = f K(x,y)f(y)dy
for the appropriate
pseudo-homogeneous function of degree
for
and
g(x) is
and such that
w
is
a
if:
f(x) = c log Ixi + g(x), where c e R
w = 0,
(i)
f
be a non-positive integer.
p.
Co
f
on
jIx=l
RV -
{0},
g(x) = 0.
homogeneous of degree 0
88.
(ii)
for
is
f
w < 0,
homogeneous of degree
Lemma.
Let
degree
w.
provided
Pf.
that
I
r
on
V
V
wr<
(01 with
0 =
and
x v-
UC [0,c]
if
1 > 0
-
H-
=lxi
If(x)lDdr
and is
We
For any bounded open set
wp +
(0o
-
be a pseudo-homogeneous function of
f
Identify
where
C
U,
f E L (U)
0.
xS
by
x/lxl.
Let
Assume
w < 0.
c
>
0
x <I>
(r,e),
be such
I
Then
<
[0,c]
=
fc
0
'%
f
If(r,0)IPy~-ldrdO
Sv-I
wp~pv-1 1 f(1,0) I~drd0
=C
0
S
sv-1
since
f
degree
=
fc
0
(iwp+v-1d
w
homogeneous of
<=>
f(r,E3) = rWf(1,0)
If(1,0)
SV-
j~d0
89.
The first integral in the product is finite provided
up + v -
1 > 0
and the second is an integral of a
continuous function over a compact set.
W = 0, observe that the function
If
r(log r) p
for
r > 0
for
r
=
(r,e)
+
rlf(r,e)l P
0
Therefore
and so long as
is continuous at
v > 2,
f If(x) Pdx = /
U
0, for all
p.
0
is continuous on
t,
we have
If(r,e6)rV-ldrd6 < x.
U
THEOREM III.
Let
U C RV
be open.
pseudo-differential operator of order
If
-
Q
is a
m < 0,
then
Qf(x) = / K(x,y)f(y)dy
U
where
if
(i)
v < m
K(x,y)
and
is everywhere defined and continuous
(ii)
if
v-m
K(x,y) =
E K.(x,y) + K'(x,y)
j=0
MI.-
Id
v > m, we have
J
9--
Id
-
-~--
90.
K.(x,y)
where
and for fixed
y X x
for
C
is
x+z)
z + K.(x,
j + m - v;
is a pseudo-homogeneous function of degree
K'
x,
is continuous everywhere.
We refer to Seeley, pp. 208-210 for details.
Pro•.
U, Q be as above.
Let
then for fixed
If
the map
p < (v-l)/(v-m)
provided
Pf:
x c U,
v < m,
If
there is
y
U
-
is bounded,
K(x,y)
is in
L (U),
v > m.
when
no problem since y -+ K(x,y)
is
continuous.
If v > m,
v-m
K(x,y) = E K (x,y) + K'(x,y)
j=0
K'(x,y)
is continuous, and
z
pseudo-homogeneous of degree
y
-
K(x,y)
is in
L
P
if
+
K.(x, x+z)
j + m - v.
each
f IK.(x,y)IPdy = /
IK.(x,
i
U'
i
U
(see above)
Now
y -+ K.(x,y)
J
x+z)lPdz
is
is
and
"49
i"-~
~-~q-=Pi
-
W
-~ibsL
~JsLLi:
~-
Ih ~1III1-
91.
where
is
U'
finite
= {z : y = x + z E U}.
The second integral
by the lemma,
p(j + m -
j = 0,...,v-m.
provided
It suffices to know
which is satisfied for all
the condition
Remark.
if
p < (v-l)/(v-m)
if
and we have taken
+ v -
p(m-v) + v -
m = v
1> 0,
1> 0,
and yields
v > m.
Theorem (4.3) is now proved, since
that case,
v > 2.
p
v)
p = (v-l)/(v-2)
m = 2
when
in
-e
I----
--
I
-3Rmt-
7'--
92.
Appendix II.
Perturbation Theory
Theorem 4.5 states that for perturbations of the
forcing function depending linearly on a parameter
T,
the eigenvalues and eigenfunctions are convergent power
series for small
T.
The purpose of this appendix is
to prove theorem 4.5.
The argument is almost identical
to Theorem I, p. 57 in Rellich's book.
If
E, F
are Banach spaces, let
B(E, F)
the space of bounded linear operators from
B(E, F)
to
F;
is a Banach space with the operator norm.
Lemma 1.
Suppose that in a nbd of
absolutely analytic function in
analytic in
small
E
denote
T.
E;
If
G(T)v(T)
then
v(.)
T
= 0,
B(E, F)
G(T)
and
is an
v(T)
is
F
is analytic in
for
is absolutely analytic, then so
is G(T)v(T).
Lemma 2.
for small
If
T
a small nbd of
G(T)
and
0,
G(0)
G(T)
absolutely analytic in
is invertible, then for
is invertible and
B(E, F)
Abstract Theorem (Statement)
hh-
d
B(E, F)
is absolutely analytic in
for small
[G(T)]
T.
in
T
-
is
93.
Hypotheses:
<
product
,
>.
be a real Hilbert space with inner
E
Let
E
defined operator on
of
P
P
Suppose
and assume
with multiplicity
linear operator
Q(P + X) = 1 -
h.
Q : E - E
X)
.
A
such that
let
absolutely analytic function in
B (E,
self-adjoint for each
Let
T
QrN = 0
and
orthogonal projection
Finally,
N = ker (P +
onto
is an eigenvalue
Assume there is a bounded
wrN is
where
N'
is a symmetric, densely
R(T)
be an
for small
and such that
R(0)
= 0.
P(T) = P + R(T).
There exist
Conclusions:
S(1) T),
,A(h)(T)
small
T,
such that:
(a)
X(j)(0)
(b)
for each
=
functions
analytic in
u (1 ) ([),.
h function
h
) (T)
..u (h
(for
j
R
for small
analytic in
T
E
and
for
= 1,...,h)
X;
u,(j)
u
is
an eigenvector of
P(T)
with eigenvalue
(c)
for each
set in
Y;
E.
T, {u
(1 )
()...
h ) (( T )}
h)
is an orthonormal
94.
The major step in the proof of this theorem is the
following proposition, showing the existence of one pair
of analytic eigenvalue and eigenvector.
Proposition.
Hypotheses as in abstract theorem.
exists an analytic function
and an analytic function
X(T)
in
in
u(T)
E
R
There
for small
for small
T
T
such
that
(a) X(o) = X
(b)
u(T)
is an eigenvector of
P(T)
with eigenvalue
Pf:
(1)
Derivation.
we will assume
arise.
Before launching into the actual proof
X(T)
and
u(T)
exist and show how they
Conclusion (b) says
[P(T) + X(u)]u(T) = 0
=>
[P + X]u(T) = - [R(T) + X(T) - X]u(T)
=>
u(T) = - Q([R() + X(T) - X]u(T)) + v(T)
where
-C
(1)
v(T) e N = ker (P + X).
=>
{1 +
=>
u(T) = (1 + Q[R(T) + X(T) - X1]-lv(T)
Q[R(T) + X(T) - X1 u(T) = v(T)
(2)
95.
provided, of course, that
invertible;
it
will be if
u(T)
can solve for
1 + Q[R(T) + X(T) -
T
provided
)
and
U,...
v(T)
,uh
is
So we
is small enough.
Now fix an orthonormal basis
v(T)
X]
are known.
of
N.
is given by specifying real analytic functions
c (T),... ,ch(T)
with
h
E ci.(')u
(3)
i=l
[R(T) + X(T) - X]u(T)
By (1),
since it is in the range of
and using (2) and (3)
(for
to get
is orthogonal to
P + X.
u(T),
Writing this out
we have
j = l,...,h)
0 = <[R(t) + X(T) - X]u(r), uj>
= <R(t) + X(T) - X]{l +
h
I
Q[R(T)
(r)<[R(T)
+ X(T) - X]{l
+ Q[R(T) + X(T)
\ /LI-L\L/L
i=l
We have n equations in n unknowns,
M
-
-
-
- X]}-V v(T), uj.>
+ X(-r)
so a non-tfivial
-
-
-
X]- 1 ui,
96.
solution
exists if and only if
c 1 (T),...,ch(T)
0 = det
((<[R(T)
-
+ X(T)
X]{1 + Q[R(T)
this last equation can be solved for
If
X]}-lui,
X(T), we can
work these steps backwards and come up with
This is
+ X(t) -
u(T).
what will be done below.
Tihe proof.
(2)
Let
be the complexification of
F
and a typical element of
flf2
E E.
F
is
F
is
E, i.e.
fl + f2 V-
F = E ®R C
where
a complex Hilbert space with inner
product given by
<fl + f2 vrg
<flgl>
1
+ g2
+ <f 22 ,g >
An analytic function on
function on
T>=
+ {<f
E
2
,g 1 > - <f 1 ,g2 > }V-
extends to a complex analytic
F; similarly for operators, and symmetry
and boundedness are preserved.
u j))
__i| __ U
~i·__
_____?:
97.
(3)
Now, using these remarks, define
fij(aT)
for
a,T
= <[R(T)
u >
+ a]{1 + Q[R(T) + a]}-lui,
in a small (complex) lbd of 0.
1 + Q[R(T) + a] = 1, so
{1 + Q[R(T)
+ a]}-1
a,T.
and is analytic for small enough
For
a = ¶ = 0,
exists
[The extensions
of our definitions and lemmas 1, 2 to more than one
parameter and to complex parameters is easy.]
F(a,T)
(4)
= det ((fij(a,T))
Suppose a complex-valued function
F(a(T),T) = 0
show that
T
for small
X(T) = X + a(T)
This implies
is symmetric.
a(T)
not all zero,
h
E
J=1
Fix
T.
satisfies
We will
is an eigenvalue of
T
X(T)
is real, since
(T) = 0
P(T).
P(T)
is an eigenvalue, we
F(a(T),T) = 0 =>
such that
fij(a(T),T)c
a(T)
a(O) = 0.
and
is real if
To see that
argue as follows.
Let
cl(T),...,ch(T) E C,
98.
h
Let
v(T) =
E
cj()u
j=l
and
u(T) = {1 + Q[R(T) + a()]}-i-1v(T)
u(T)
a non-zero eigenvector of
is
eigenvalue
(5)
3
by reversing the steps in
A(T),
a(T),
a real-analytic function
= a6ij
fij(a,0)
= 0
a(O)
such that
with
P(T)
,
and
for small
F(a(T),T) = 0.
F(a,0) = ah .
(1).
T,
Since
By the Weierstrass
preparation theorem,
F(a,T)
where
fcn of
= (ah
E(0,0)
T,
= 1
1
()h-+
and
pj(T)
J = 1,...,h.
must be roots of
...
+ ph(T))E(a,T)
is complex-analytic
So solutions of
ah + p 1 (T)ah-
+ *
F(a,T) = 0
+ Ph(
)
= 0.
These roots are not a priori analytic functions of T,
but they can be written as power series in
[Puiseux series].
Let us fix the branch of
the one which gives a real root if
positive and write
T
T /h
T1/h
is real and
as
9~PF~---
b-
99.
hh
a(T) =
i/h
.
for one root
a T
By (4), we know that
a(T)
is real if
T
This implies that only integral powers of
non-zero coefficients.
For if
am
is real.
T
have
is the first
non-zero coefficient
a(T) =
so
o@
aiti/h
ai/h
i=m
am = lim
T -
0
The right hand side is real because
because of the branch we chose for
a(T)
T1 /h
is real and.
Also
aIn-1
m ( 1 )m/h = lim
T
is real, implying
multiple of
h.
-+ 0-
(-1)m/h
m
is an integral
This argument may be repeated for the
higher coefficients.
Thus the roots may be written as
real-analytic functions of
-L
real and
T.
C
-~---
100.
In (4) and (5), we have constructed an eigenvalue
(6)
X(T) = X + a(T)
of
T.
of
P(T)
as a real-analytic function
To get the eigenvector
steps in (4).
u(T), one repeats the
3 cl(T),...,ch(T)
such that
h
E
f ij((T),)c (T) = 0
j=1
Observe that for real
T, the matrix
is real and symmetric.
By the finite dimensional
analytic perturbation theorem the
T
to be analytic in
c.(T)
is
not real,
also a solution.
((fij(a(T),T)))
c.(T)
can be chosen
[see Rellich, p. 32]
replace it
by
and if
Re c.(T)
which is
Then, proceeding as before,
h
v(T) = E c (T)u.
j=1
is analytic in
E
J
for real
J
T
and so is
u(T) = {1 + Q[R(T) + a(T)]}
by lemmas 1i, 2.
proposition.
V(T)
This completes the proof of the
_
__
_Ii
;
_ _L~J
101.
By the preceding proposition,
Proof of Abstract Theorem:
we know that there is an eigenvector
analytic in
E
for small
T.
u(1)(T)
of
P(T),
We prove the existence
of the other eigenvalues and eigenvectors satisfying
(a),
(b), (c) by induction on the multiplicity
Define
w(T)
as the projection of
subspace spanned by
u
(T)v = <v,u (1 ) (T )>u(1
(1 ) (
T),
) (T ).
h.
E
onto the
...
, uh
i.e.
Let
P'(T) = P(T) - 7(T)
u(1)(0)
Let
= u1
and let
orthonormal basis for
ul,
u2 ,
ker (P + X).
Now for
be an
j = 2,...,n
P'(0)u. = P(O)u. - w(0)u.
3
J
3
SXu.
J
so
X
least
is an eigenvalue of
h - 1.
P'(O)
with multiplicity at
Claim that the multiplicity is exactly h - i.
If not, there would be an element
v e E
and having norm 1i,such that
orthogonal to
2--·I
~jlC:
102.
P' (O)v =
X<v,ul> = <P'(O)v,ul>
Then
= <v,P'(O)ul>
= (x
=> <v,Ul> = 0
-
l)<v,ul>
P(0)v = Xv.
and hence
X
that the multiplicity of
is
But this implies
a contradiction.
> h,
Applying the abstract theorem to the eigenvalue
X
[by the induction hypotheses], there are
P'(O)
analytic functions
in
(J) (T)
R
and
u
j
(T)
(i )
in
E
for
u ( j ) (T)
j = 2,...,h
form an orthonormal set for each
[P'(T) + x
) (T)]u
reasoning.
) ( T) = 0.
9
<u(1)(T),
j
=
2
w
...
0
h
by
We have
= [P'(T) -
+
)(T)
T,
and
In addition
0 = [P(T) + x (1)]u
(
)(1)
(1)
u (1 ) (T )
= [P'(T) + (X(1)(T) - l)]u(1)(0
~----
= x, the
X''(0)
such that
the
following
of
103.
so
u(1) (T)
is an eigenvector of
X(1)(T) - 1. For small
eigenvalue
X(1)(T)T - 1 ~Y
that
<u(1
)
(T),
(T) for all
u
)(T)>
different eigenspaces.
7w(T)u
[P(T)
) (T)
+ X
P' (T)
= 0
= 0
with
T,
so
j = 2,...,h,
since they are in
In particular this implies
and finally
( T) ] u
= [P'(T) + X
(J)T)]u()(T)
Proof o_ Theorem 4.5.
To get conclusions (1) to (3),
we apply the abstract theorem just proved.
E = L2(M) , P = L + p,
inverse of
P + X
analytic function
satisfying
p(0)
and let
Q
p(T)
in
let
be the pseudo-differential
(C (M),
R(T)
=
of theorem 4.5 except that the
L2 (Mr).
are analytic in
I....
p(T) -
abstract theorem gives conclusions
analytic in
Let
Given an absolutely
(see 4.3).
= p,
Q ) (T) = 0
is )
p.
for any
Then the
(2) and (3)
(1).
,V
u (j
) (
)
are only
We will show first that they
(C(Mi), 1...Is)
for any
s.
Recall
from the derivation (1) of the proposition that an
Z•
•
M
104.
u(T)
analytic eigenvector
A(T)
of
P(T)
with eigenvalue
must satisfy
u(T) = - Q[R(T) + X(T) - X]u(T) + v(t)
h
where
v(T) =
E
j=1
{ul
orthonormal basis
Since
, . . . ,
uh}
(some s)
in
CS(M),
+
R(T)
and
is analytic in
is
Hs+2(M)
Q[R(T) + a(T)]u(T)
R,
by lemma 1.
HIs(H)
is analytic
is absolutely analytic
analytic in
X(T)
is analytic in
Q : Hs(M)
ker (P + X).
s ) , any s. Now if u(T)
(Cm(M),1...
Hs
of
v(T)
ker (P + X) C Cw(M),
in
terms of a fixed
in
c (T)u.
[R(T) + X(T) - X]u(T)
Since
a bounded operator
is analytic in
Hs+ 2 (M).
H 0 (M) = L2 (M),
We
so in
know
u(T)
is analytic in
fact
u(T)
is analytic in every Hs-space by induction.
By the Sobolev lemma,
Cs-space.
u(T)
is analytic in every
~~-~----
105.
Secondly we must prove (d).
ininimax principle (see
A1- < X<
2
L + p
<
(X<b
.
p1.2).
Let
h
i~nauso
teegnauso
n<..b
arranged in increasing order.
Xn-i <
n =
multiplicity
n+l
-0O*
h.
Let
I
there exists
Choose
T > 0
such that
Then by (5.1),
Ihn+h(t)
IT < T
n+hI
-
h eigenvalues in
I
jT1
A
Hience
of
L + p.
IC(An~
<
irpljies
<,
~n-i (T), An+h(T) & I.
i~e.
A
has
be an interval such that
such that
cS >0
Alssume
AXh
+h-l
contains only the eigenvalue
and
We do this using· the
T
I
Then
+ 5, A
inplies
I~n~l(T)
Ip(T)
-
-
n- 1
implying
L + p(T)
has at most
counting multiplicity.
But we
have already shown that it has at least
h eigenvalues
the proof.
A I
T,..Ah()
5)
-
This completes
Pj
'
<
5.
106.
References
1.
Bers, L., F. John and M. Schechter.
Differential Equations.
Partial
New York: John Wiley and
Sons, Inc., 1964.
2.
Courant, R.,
and D. Hilbert.
Physics, Vol. I.
Methods of Mathematical
New York: Interscience Publishers,
1953.
3.
Ince, E. L.
Ordinary Differential Equations.
New York: Dover, 1956.
4.
Kuo,
T.
On C -Sufficiency of Jets of Potential
Functions Topology, Volume 8 (1969),
5.
Rellich, F.
Problems.
6.
Seeley, R.
pp.
167-171.
Perturbation Theory of Eigenvalue
New York: Gordon and Breach,
1969.
Topics in Pseudo-Differential Operators.
Notes issued for: Centro Internazionale Mathematico
Estivo.Rome: Edizioni Cremonese,
7.
Whitney, H., and F. Bruhat.
1969.
Quelques Proprietes
Fondamentales des Ensembles Analytiques-Reels
Comm. Math. Helvetici, Volume 33 (1959),
pp.
I,
132-160.
U
~-
-~
107.
Biographical Sketch
The author was born in Brooklyn, New York on
March 2, 1946 and lived in New York City until 1966.
Interest in mathematics started in high school, where
he was active on the school's math team and won several
prizes in competitions.
He earned an A.B. degree
(magna cum laude) from Columbia College in 1966, and
was elected to Phi Beta Kappa and won the mathematics
prize in that year.
While in college, he also worked
part-time as an assistant mathematics editor for
Schaum Publishing Company in New York.
He started
graduate work at M.I.T. in September 1966 on a National
Science Foundation Graduate Fellowship and in September
1970 was appointed to a full-time instructorship in
mathematics at Tufts University.
Outside interests
include recorder playing and music in general, and
tennis and squash.
His favorite activities are those
relating to the mountains: hiking, cross-country
skiing, snowshoeing and canoeing.
He is married, and
his wife Maril works at Somerville Hospital as a nurse.
ON&