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where L is the characteristic size of the system, r

and F

are, respectively, the radius and the energy of

the disclination core, a region in which the distor-

tions are too strong to be described by a pheno-

menological theory.

The restriction of planar director distortions does

not allow the model to grasp the crucial difference

between half-integer and integer k’s. The lines of

integer k, as already discussed, are fundamentally

unstable, as the director can be reoriented along the

axis. This ‘‘escape in the third dimension,’’ is usually

energetically favorable, since the singular core is

eliminated. When opposite directions of the

‘‘escape’’ meet, a point defect hedgehog is formed,

as illustrated in Figure 5c.

Unlike point defects s uch as vacancies in

solids, topological point defects in nematics

cause disturbances over the whole volume.

The curvature energy of the point defect is

proportional to the size R of the system. For

example, for the radial hedgehog with

n = (x, y, z)=

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

þ y

þ z

, and the hyperbolic

hedgehog with n = (x, y, z)=

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

þ y

þ z

one finds, respectively,

¼8 Rð K

 K

ÞþF

and

¼8R



þ F

½14

Defects in Smectics

Layered structure of smec tics leads to line ar

defects of positional order, dislocations, in addi-

tion to disclinations. There is also a special class

of distortions known as focal conic domains

(FCDs) that are associate d with large-scale cur-

vatures of layers. Imagine that because of the

boundary conditions, flow, or the external fields,

the smectic layers are curved over the scale much

larger than the thickne ss of the layers. It is easy

to see from e qn [9] that the curved laye rs will

prefer to maintain their equidistance, as the

curvature en ergy is much smaller than the layers

dilation energy at t he large scal es of deforma-

tions. Generally, the family of equidistant curved

surfaces is ass ociated with the focal surfaces at

which the principal curvatures diverge. These

focal surfaces are thus e nergetically very costly.

A r adical way to reduce the elastic energy would

be to decrease the dimensionality of t he focal

surfaces,say,bytransformingthemintolinesand

points. The latter case corresponds simply to a

system of concentric spherical layer s. The former

is more complicated and corresponds to FCDs in

which the focal surfaces are represented by pairs

of confocal l ines: ellipse and hyperbola (limiting

case: circle and straight line), and the pair of

confocal parabolae. Experiments c onfirm that the

FCDs are the most frequent type of structural

deformations in smectic materials see Figure 8.

Conclusion

To summarize, over the last few decades, liquid

crystals transformed from a mysterious and

curious form of condensed matter into a key

technological material, thanks to the progress in

the understanding of their elastic, optical, and

viscous properties. However, the intrinsic com-

plexity of these materials s till leaves plenty of

room for further studies, not only of an applied

nature, but also fundamental. In the f ield of

thermotropic liquid crystals, re searchers continue

to discover new types of structural organization,

such as the phases formed by ‘‘banana-shaped’’

molecules that are dramatically different from the

phases formed by ‘‘regular’’ rod-like and disk-like

molecules. There is a continuous work to sharpen

our understanding of even the ‘‘old’’ problems, such

as mechanisms of surface alignment, nature and

quantitative values of the elastic constants K

, K

and



K. Even in the case of the electric Frederiks

effect that is at the heart of modern applications, the

search continues as the corresponding process of

director reorientation is generally very complex. In

addition to the dielectric torque, it is controlled by

various factors, for example, a nonlocal character of

the electric field in the anisotropic medium, finite

electric conductivity, flexoelectric effect (i.e., electric

polarization brought about by the director deforma-

tions), surface electric polarization at the bounding

plates, dependence of the dielectric and other

material properties on the frequency of the applied

field which might be comparable with the

60 µm

Figure 8 SmA phase with FCDs based on the confocal pairs

of ellipses and hyperbolas; the scheme on the right shows

the arrangement of the elliptic bases and smectic layers

wrapped around the confocal pairs of defects. Reproduced

from Lavrentovich OD (2003) In: Arodz et al. (eds.) Patterns of

Symmetry Breaking. Dordrecht: Kluwer Academic Publishers,

with kind permission of Springer Science and Business Media.

Liquid Crystals 327

characteristic frequency of dielectric relaxation, cou-

pling of the director reorientation and the material’s

flows, appearance of topological defects, etc. Many

research efforts nowadays are focused on composite

systems, such as liquid crystal colloids and polymer–

liquid crystal composites. Over the next decade or so,

one would expect that the emphasis in fundamental

studies will gradually shift from the thermotropic

liquid crystals to their lyotropic counterparts, as the

lyotropic type of orientational order is featured by

many systems of biological significance, such as

solutions of DNA, f-actin, etc.

See also: Non-Newtonian Fluids; Topological Defects

and Their Homotopy Classification.

Further Reading

Barbero G and Evangelista LR (2001) In: Ong HC (ed.) An

Elementary Course on the Continuum Theory for Nematic

Liquid Crystals, Series on Liquid Crystals. Singapore: World

Scientific.

Blinov LM and Chigrinov VG (1996) Electrooptic Effects in

Liquid Crystal Materials. New York: Springer.

Chaikin PM and Lubensky TC (1995) Principles of Condensed

Matter Physics. Cambridge University Press.

Chandrasekhar S (1992) Liquid Crystals, 460 pp. Cambridge:

Cambridge University Press.

Frank FC (1958) On the theory of liquid crystals. Transactions of

the Faraday Society 25: 19–28.

de Gennes PG and Prost J (1993) The Physics of Liquid Crystals.

Oxford: Oxford Science Publications.

Hartshorne NH and Stuart A (1970) Crystals and the Polarizing

Microscope. New York: American Elsevier.

Kitzerow HS and Bahr C (eds.) (2001) Chirality in Liquid

Crystals, 502 pp. New York: Springer.

Kleman M and Lavrentovich OD (2003) Soft Matter Physics: An

Introduction. New York, NY: Springer.

Larsen RG (1999) The Structure and Rheology of Complex

Fluids, 664 pp. New York: Oxford University Press.

Lavrentovich OD, Pasini P, Zannoni C, and Zumer S (eds.)

(2001) Defects in Liquid Crystals: Computer Simulations,

Theory and Experiments, 344 pp. Dordrecht: Kluwer Academic

Publishers.

Sonin AA (1995) The Surface Physics of Liquid Crystals, 180

pp. Australia: Gordon and Breach.

Wu ST and Yang DK (2001) Reflective Liquid Crystal Displays,

336 pp. Chichester: Wiley.

Ljusternik–Schnirelman Theory

J Mawhin, Universite

de Louvain, Louvain-la-Neuve,

Belgium

Introduction

Using Lagrange multipliers, the smallest and

the largest eigenvalue of a symmetric quadratic form

QðuÞ¼

j;k¼1

ða

¼ a

can be obtained by minimizing and maximizing Q

on the unit sphere S

n1

= {u 2 R

: kuk= 1}. If the

corresponding extremum is reached at u



, then u



an associated eigenvector.

In the setting of integral or partial differential

equations, a ‘‘recursive variational method’’ has

been proposed to determine all the eigenvalues 







and corresponding eigenvectors

, u

, ..., u

of Q or, in modern terms, of the

associated symmetric matrix A = (a



¼ min

kuk¼1

QðuÞð¼Qðu

ÞÞ



¼ min

kuk¼1;uu

¼0;...;uu

j1

¼0

QðuÞ

ð¼ Qðu

ÞÞ ðj ¼ 2; ...; nÞ

Further considerations have led to a nonrecursive

minimum–maximum principle:



¼ min

R

: dim X

¼jg

max

fu2X

: kuk¼1g

QðuÞð1  j  nÞ

and to a dual maximum–minimum principle

(Weyl):



¼ max

;...;p

j1

min

fkuk¼1;up

¼0;1ij1g

QðuÞ

ð1  j  nÞ

These principles have been widely used in various

existence and approximation questions of math ema-

tical physics, and extensions have been made to the

abstract setting of symmetric bilinear forms in

Hilbert spaces.

Around 1930, Ljusternik and Schnirelman have

extended this theory beyond the frame of quadratic

forms, replacing Q by a differentiable real-valued

function f and the unit sphere by a finite-

dimensional compact differentiable manifold M.

Their aim was the obtention of the ‘‘critical points’’

of f on M, that is, the points u 2 M where the

differential f

(u)off at u (as a linear functional on

the tangent space T

M to M) is equal to zero, and of

the corresponding critical values, that is, the values

of f at critical points. When M is a sphere, the

328 Ljusternik–Schnirelman Theory

critical points are nontrivial solutions of the

equation

ðuÞ¼u ½1

for some  2 R (nonlinear eigenvalue problem).

Ljusternik and Schnirelman have replaced the

dimension of the vector spaces occurring in

the minimum–maximum principle for eigenvalues

by the concept of ‘‘category’’ of a closed set A in a

topological space X. An early success of their

approach was the existence of three geometrically

distinct closed geodesics without self-intersections

on any compact surface of genus zero. In 1960,

their theory has been extended to infinite-

dimensional manifolds and to other measures of

the ‘‘size’’ of a set than the category, allowing many

theoretical developments as well as various

applications to nonlinear differential equations.

Ljusternik–Schnirelman Category

Let X be a topological space (e.g., a normed vector

space, or a differentiable manifold, or a metric

space), and A a closed subset of X. The category of

A in X, cat

(A), is the least integer k such that A

can be written as

j = 1

, with A

closed and

contractible in X, that is, continuously deformable

in X into a single point. If no such k exists, one sets

cat

(A) = þ1. We write cat(X) for cat

(X). For

example, if X is contractible (in itself), cat(X) = 1.

This is the case for any normed space X. For the

hypersphere, cat

n1

) = 1, but cat(S

n1

) = 2.

The Ljusternik–Schnirelman category satisfies the

following properties, which are not too difficult to

prove. If A, B  X are closed,

1. cat

(A) = 0 if and only if A = ;;

2. if A  B, cat

(A)  cat

(B);

3. cat

(A [ B)  cat

(A) þ cat

(B);

4. if  : [0, 1]  X ! X is a continuous deformation

of X((0, A) = A), cat

(A)  cat

((1, A)); and

5. if X is a finite-dimensional manifold and A  X

is compact, there is a neighborhood B of A such

that cat

(B) = cat

(A).

Computing or even estimating the category of a

given set is in general difficult, requiring techniques

of algebraic topology. In particular, one can show

that, for the n-torus T

= S

 S

S

(n times),

cat(T

) = n þ 1, and for the n-dimensional projective

space P

= S

, obtained by identifying the anti-

podal points of S

,cat(P

) = n þ 1. It is clear that a

setofcategoryp must contain at least p points. If X

is connected, any compact subset of category p þ 1

has (topological) dimension larger or equal to p.

Ljusternik–Schnirelman Minimax Method

The Ljusternik–Schnirelman category of M provides a

lower bound for the number of critical points of a

smooth function f on suitable finite-dimensional

manifolds M. Namely, if M is a compact Riemannian

-manifold without boundary, any f 2 C

(M, R)

has at least cat(M) distinct critical points, with

critical values

¼ inf

A2A

sup

u2A

f ðuÞð1  k  catðMÞÞ ½2

where

¼fA  M: A closed; cat

ðAÞkg

ð1  k  catðMÞÞ ½3

A fundamental technique in the proof is a deformation

lemma along the trajectories of the gradient system

associated to f (method of steepest descent). If r f

denotes the gradient of f in the Riemannian structure

of M, the Cauchy problem for the gradient system

d

¼rf ðÞ;ð0Þ¼u ½4

has a unique globally defined continuous solution

 (t, u), which is such that

f ð ð1; uÞÞ  f ðuÞ¼

f ððt; uÞÞdt

¼

krf ððt; uÞÞk

dt ½5

Notice that, by property (4) of the category, each

deformation by  of a set in A

remains in A

. For

c 2 R, define

:¼fu 2 M : f ðuÞcg

:¼fu 2 M : rf ðuÞ¼0; f ðuÞ¼cg

½6

From [5] it follows that given c 2 R andanopen

neighborhood U

of K

, one has (1, f

cþ"

n U

)  f

c"

for all sufficiently small ">0. This implies that if

c := c

= c

jþ1

=  = c

jþq

for some q  0, then

cat

)  q þ 1. Assume, by contradiction, that

cat

)  q,letU

be an open neighborhood of K

such that cat

) = cat

)(U

= ; if q = 0), ">0

such that (1, f

cþ"

n U

)  f

c"

,andA 2A

jþq

such

that sup

f  c þ " ,thatis,A  f

cþ"

.Then

cat

ðð1; A n U

ÞÞ  cat

ðA n U

 cat

ðAÞcat

ðUÞj

giving the contradiction c  sup

(1, A)

f  c  " .

Notice that, for each j, c

= inf {c 2 R :cat

)  j},

which shows that the c

are precisely those levels of f

where cat

) changes. The presence of critical

values is detected by changes in the topology of the

Ljusternik–Schnirelman Theory 329

sublevel sets f

when c varies, a common feature of

many techniques for finding critical points of

functions.

A direct consequence is that for each even

f 2 C

, R), system [1] has at least n pairs of

solutions (u, u)withkuk= 1. Indeed, the solu-

tions of [1] are the critical points of f on S

n1

.Asf

takes the same values at antipodal points, it is well

defined on the projective space P

n1

,and

cat(P

n1

) = n.

The Ljusternik–Schnirelman theorem can be extended

to the C

-situation. The category of M gives a lower

bound for the number of critical points of f on the closed

manifold M.IfCrit(M) denotes the minimum of

the number of critical points of all C

-functions on M,

so that Crit(M)  cat(M), an interesting question is

to estimate the gap Crit(M)  cat(M). For M closed

connected, Crit(M)  dim(M) þ 1 (Takens). If

Crit(M) = 2, M is homeomorphic to a sphere, so that

the equality Crit(S) = cat(S) for homotopy spheres is

equivalent to Poincare´’s conjecture! Manifolds with

Crit(M) = cat(M) þ 1 are known, but not with

Crit(M) > cat(M) þ 1.

Ljusternik–Schnirelman Theory

in Infinite-Dimensional Manifolds

The main difficulty in extending the results of the

previous section to functions defined on infinite-

dimensional manifolds lies in the lack of compact-

ness. J T Schwartz and Palais have shown that such

an extension is possible for functions f satisfying on

M a compactness property (allowing an infinite-

dimensional deformation lemma), now referred to as

the Palais–Smale condition: each sequence (u

)with

(f (u

)) bounded and lim

k !1

rf (u

) = 0hasacon-

vergent subsequence. Such a condition can be

localized at level c by replacing the boundedness of

(f (u

)) by lim

k !1

f (u

) = c. The infinite-dimensional

extension of Ljusternik–Schnirelman’s theorem goes

as follows: Let M be an infinite-dimensional Rieman-

nian (or even Finsler) connected complete manifold

of class C

without boundary. Any f 2 C

(M, R)

bounded from below and satisfying Palais–Smale

condition has at least cat(M) distinct critical points.

A simple application can be given to the periodic

solutions of period T (T-periodic solutions) of

Lagrangian systems

þrVðuÞ¼hðtÞ½7

where V 2 C

, R), 2-periodic in each compo-

nent u

(1  j  n), h is continuous, T-periodic and

has mean value



h equal to zero. By the least action

principle, the T-periodic solutions of [7] are the

critical points of the action functional

f ðuÞ¼

ðtÞk

 VðuðtÞÞ þ hðtÞuðtÞ

on the Hilbert space H

obtained by completion of

the space of T-periodic C

functions for the norm

associated with the inner product

hu; vi :¼

uðtÞvðtÞdt þ

ðtÞv

ðtÞdt

It follows easily from condition



h = 0thatf is bounded

from below and that f (u þ 2e

) = f (u)forallu 2 H

with e

the jth unit vector in R

(1  j  n). Conse-

quently, we can see f as defined on the Riemannian

manifold T



,where

= {u 2 H



u = 0}. It is

easy to show that cat(T



) = cat(T

) = n þ 1and

that f satisfies Palais–Smale condition on T



Consequently, system [7] has at least n þ 1 geometri-

cally distinct T-periodic solutions. The same result

holds for the more general systems

þ Au þrFðuÞ¼hðtÞ

occurring in the theory of multipoint Josephson

junctions or in space discretizations of the

sine-Gordon equation. In particular, the classical

forced pendulum equation

þ a sin u ¼ hðtÞ

has at least two geometrically distinct T-periodic

solutions when h is T-periodic and



h = 0, a result

first proved, in a different way, by Mawhin and

Willem.

Another way to study nonlinear eigenvalue pro-

blems of the form

ðuÞ¼g

ðuÞ

in a Hilbert or a suitable reflexive Banach space X

is based upon a Rayleigh–Ritz approximation

through a sequence of finite-dimensional problems,

where the classical theory is applied. Conditions

upon f , g 2 C

(X, R) are given, generalizing

Ljusternik–Schnirelman’s ones, which ensure the

existence of infinitely many solutions. Again, some

compactness is needed to justify the limit process,

and expressed by some assumptions upon f and g

too lengthy to be reproduced here. The following

application is exemplary. Let   R

be a bounded

domain and X = W

1, p

(), p > 1, be the Sobolev

space of functions u :  ! R obtained as the comple-

tion of the smooth functions with compact support

330 Ljusternik–Schnirelman Theory

in  for the norm kuk

1, p

= (



kru (x)k

dx)

1=p

Define the functionals f and g on W

1, p

()by

f ðuÞ¼



kruðxÞk

dx; gðuÞ¼



juðxÞj

The critical points of f on {u 2 X : g(u) = 1} corre-

spond to the nontrivial solutions of the Dirichlet

eigenvalue problem



u ¼ juj

p2

u in ; u ¼ 0on@ ½8

for the p-Laplacian operator 

defined by



uðxÞ:¼r kruðxÞk

p2

ruðxÞ



which occurs in the modelization of various

problems in a porous medium. An eigenvalue is

any  2 R such that problem [8] has a nontrivial

solution. The Ljusternik–Schnirelman technique

implies the existence of a sequence of eigenvalues

going to infinity, with the usual minimax character-

ization. When N = 1, direct computations show that

this sequence gives all eigenvalues, but the problem

remains open for N  2. The corresponding forced

problem



u  juj

p2

u ¼ hðxÞ in ; u ¼ 0on@

is always solvable (although not uniquely) when  is

not an eigenvalue, but solvability conditions at the

higher eigenvalues (Fredholm alternative) remain

almost terra incognita.

Index Theories and Critical Points

of Symmetric Functionals on

a Banach Space

Closely related to the Ljusternik–Schnirelman category

is the concept of index associated to the action of a

compact topological group G on a normed space X,

that is, to a continuous map G  X ! X,[g, u] 7!gu

such that 1  u = u,(gh)u = g(hu), u 7!gu is linear.

The action is isometric if kguk= kuk, A  X

is invariant if gA = A for all g 2 G, f : X ! R is

invariant if f  g = f for all g 2 G,andh : X ! X

is equivariant if g  h = h  g for each g 2 G.Let

Fix G = {u 2 X :

gu = u for all g 2 G}. The aim of an

index is to measure the size of invariant sets.

Explicitly, an index theory associates to each closed

invariant subset A of X a non-negative (possibly

infinite) integer G-ind(A), its G-index, such that

1. G-ind(A) = 0 if and only if A = ;;

2. if R : A ! B is equivariant and continuous,

G-ind(A)  G-ind(B);

3. G-ind(A [ B)  G-ind(A) þ G-ind(B); and

4. if A is compact, there is a closed invariant

neighborhood U of A such that G-ind(U) =

G-ind(A).

A first example of index is Krasnosel’skii’s genus

or Z

-index which corresponds to the action

0  u = u,1 u = u of G = Z

. The invariant sets

are the ones symmetric with respect to the origin

and Z

-ind(A) is defined by Z

-ind(;) = 0 and, for

A 6¼;, as the smallest integer k such that there

exists an odd h 2 C(A, R

n {0}). A consequence of

the Borsuk–Ulam theorem in algebraic topology is

that any symmetric bounded neighborhood of the

origin in R

has Z

-index equal to n. Furthermore,

for a compact A  R

n {0} symmetric with respect

to the origin, and

A = A=Z

(A with antipodal

points identified), one has Z

-ind(A) = cat

n{0}

(

A).

A second example, the S

-index, is important in

the study of periodic solutions of autonomous

Hamiltonian systems. S

-ind(;) = 0 and for a non-

empty closed invariant A  X, S

-ind(A) is defined

as the smallest integer k such that there exists a

positive integer n and h 2 C(A, C

n {0})

with h  g = g

 h for all g 2 S

. A Borsuk–

Ulam-type theorem for S

-equivariant mappings

implies that if Z is a finite-dimensional invariant

subspace of X such that Fix S

\ Z = {0} and D is

an open bounded invariant neighborhood of 0 in Z,

then S

-ind(@D) = (1=2)dim Z.

As the category of a Banach space X = 1, the

classical Ljusternik–Schnirelman approach does not

provide any information about the multiplicity of

the unconstrained critical points of f 2 C

(X, R). If f

is invariant under the action on X of a compact

group G and satisfies Palais–Smale condition, a

Ljusternik–Schnirelman minimax method associated

to a G-index provides multiplicity results for

unconstrained critical points. Letting

¼fA  X : A is compact, invariant,

and G-indðAÞjg

¼ inf

A2A

sup

f ðj ¼ 1; 2; ...Þ

one shows as in classical Ljusternik–Schnirelman

theory that if c := c

= c

jþ1

=  = c

jþq

for some j

and some q  0, then G-ind(K

)  q þ 1. The proof

uses an equivariant deformation lemma.

- and S

-Invariant Functionals

InthecaseoftheZ

-action, the following multiplicity

result holds for possibly unbounded even f 2 C

(X, R)

satisfying the Palais–Smale condition and having the

mountain pass geometry: if Y \ {u 2 X : f (u)  0} is

bounded for each finite-dimensional subspace Y of X,

Ljusternik–Schnirelman Theory 331

f (0) = 0, and f (u)  a > 0on@B(r), then f has

infinitely many couples of critical points. As an

application, the semilinear Dirichlet problem

u þ u þjuj

p1

u ¼ 0in

u ¼ 0on@

½9

has infinitely many solutions when   R

bounded, 1 < p < (N þ 2)=(N  2), and <

, the

smallest eigenvalue of  with Dirichlet boundary

conditions. The corresponding energy functional,

defined on W

1, 2

()by

f ðuÞ¼



kruðxÞk

 

juðxÞj



juðxÞj

pþ1

p þ 1

satisfies the Palais–Smale condition. This condition

fails in the critical case where p = (N þ 2)=(N  2), at

least at some levels c, and this lack of compactness

creates both difficulties and interesting phenomena.

This situation, which occurs in many important

problems of geometry and physics (harmonic maps,

Yang–Mills connections, Yamabe problem, equations

of constant mean curvature, closed geodesics pro-

blems, etc.), reveals indeed, in physical terms, ‘‘phase

transitions’’ or ‘‘particle creations’’ at the levels where

the Palais–Smale condition fails. In the special case of

eqn [9] with p = (N þ 2)=(N  2), if N  4, a positive

solution exists when  2 [0, 

], and, if N = 3, the

same is true for  2 [



, 

]andsome



2 [0, 

with the optimal value 



= 

=4when is a ball. For

N  4, [9] has at least cat() nontrivial solutions

when  2 [0, 



]forsome



<

.Suchalackof

compactness, which can also occur for eqn [9] in R

(nonlinear Schro¨ dinger equation), is associated to the

invariance of f with respect to the action of some

noncompact group, coming, for example, from scale or

gauge invariance. P L Lions’ concentration–compact-

ness method is useful to analyze those problems.

The following multiplicity theorem holds for an

-invariant f 2 C

(X, R) satisfying Palais–Smale

condition. Let Fix(S

) = {0} and Z be a closed

invariant vector subspace of X of positive finite

dimension. If f is bounded from below, f (u)  c < 0

whenever u 2 Z and kuk= r, and f (0)  0 for u 2

Fix(S

) \ (f

)

1

(0), then f has at least dim Z=2

distinct S

-orbits of critical points of f with critical

values less or equal to c. This abstract theorem

provides multiplicity results for the periodic solu-

tions (closed orbits) of autonomous Hamiltonian

systems in R

þrHðuÞ¼0 ½10

where J is the symplectic matrix, H 2 C

, R),

and c 2 R is such rH(u) 6¼ 0 for u 2 H

1

(c). If

1

that B[r]  C  B[R] for some 0 < r < R <

ﬃﬃﬃ

then [10] has at least n closed orbits on H

1

(c). The

problem is reduced to finding the critical points of a

suitable dual action functional acting on some space

X of 2-periodic functions having mean value zero.

The S

-action on X is defined by time translations

[, u] 7!u



= u( þ) for all  = e

i

2 S

. One takes,

in the abstract result above, Z = {(cos t)e þ

(sin t) Je : e 2 R

}, so that dim Z = 2n. The complete

proof is quite involved, and, although some

improvements of Ekeland–Lasry conditions have

been obtained, the problem remains open to know

if some pinching condition of the energy surface

between spheres or ellipsoids is necessary.

Some Extensions

When dealing with unbounded functionals, it may

be convenient to replace the Ljusternik–Schnirelman

category cat

(A) by a relative category cat

X, Y

(A)

with respect to a closed subset Y where, in the

covering of A occurring in the classical definition, a

set A

 Y is added, which is continuously deform-

able in X into a subset of Y in such a way that

points of Y remain in Y during the deformation.

Clearly cat

X, ;

(A) = cat

(A). This allows us to prove,

under some restrictions on the coefficients and the

period, the existence of at least four periodic

solutions for the double pendulum with periodic

forcing of mean value zero. The classical Ljusternik–

Schnirelman category gives at least three periodic

solutions without restrictions, and the question of

their necessity to obtain four solutions is open.

The relative category also gives a simpler proof of

Conley–Zehnder’s version of the Arnol’d conjecture

(the existence of at least 2n þ 1 geometrically distinct

1-periodic solutions for the Hamiltonian system

þrHðt; uÞ¼0

with H 1-periodic in each variable), under minimal

regularity assumptions upon H. The general con-

jecture, namely that the minimum number of fixed

points of all Hamiltonian symplectomorphisms of a

closed symplectic manifold M is larger than the

minimum number of critical points of smooth

functions f on M, remains open.

In another direction, a Ljusternik–Schnirelman

theory for functionals defined on closed convex sets

of a Banach space has been developed, which is

specially well suited for the study of the Plateau

problem for minimal surfaces, for surfaces of

constant mean curvature, as well as for variational

inequalities.

332 Ljusternik–Schnirelman Theory

See also: Bifurcations of Periodic Orbits; Compact

Groups and Their Representations; Floer Homology;

Ginzburg–Landau Equation; Inequalities in Sobolev Spaces;

Minimal Submanifolds; Minimax Principle in the Calculus

of Variations; Saddle Point Problems; Sine-Gordon

Equation; Spectral Theory for Linear Operators.

Further Reading

Ambrosetti A (1992) Critical Point and Nonlinear Variational

Problems, Cours de la Chaire Lagrange. Paris: Socie´te´

Mathe´matique de France.

Cornea O, Lupton G, Oprea J, and Tanre´ D (2003) Ljusternik–

Schnirelman Category. Providence: American Mathematical

Society.

Courant R and Hilbert D (1953–62) Methods of Mathematical

Physics, vol. 2. New York: Wiley-Interscience.

Ekeland I (1990) Convexity Methods in Hamiltonian Mechanics.

Berlin: Springer.

Ekeland I and Ghoussoub N (2002) Selected new aspects of the

calculus of variations in the large. Bulletin of the American

Mathematical Society (NS) 39: 207–265.

Fuc

ik S, Nec

as J, Souc

ek J, and Souc

ek V (1973) Spectral Analysis

of Nonlinear Operators. Berlin: Springer.

Ghoussoub N (1993) Duality and Pertubation Methods in Critical

Point Theory. Cambridge: Cambridge University Press.

Gould SH (1966) Variational Methods for Eigenvalue Problems,

2nd edn. Toronto: Toronto University Press.

Hofer H and Zehnder E (1994) Symplectic Invariants and

Hamiltonian Dynamics. Basel: Birkha¨user.

Krasnosel’skii MA (1963) Topological Methods in the Theory of

Nonlinear Integral Equations. Oxford: Pergamon.

Ljusternik LA (1966) The Topology of the Calculus of Variations

in the Large. Providence: American Mathematical Society.

Ljusternik LA and Schnirelman LG (1930) Me´thodes topologi-

ques dans les proble` mes variationnels (Russian). Moscow:

Trudy Scient. Invest. Inst. Math. Mech. (French translation

(1934) Actualite´s scientifiques et industrielles no. 188. Paris:

Hermann).

Mawhin J and Willem M (1989) Critical Point Theory and

Hamiltonian Systems. New York: Springer.

Palais RS (1970) Critical point theory and the minimax principle.

In: Proceedings of Symposia in Pure Mathematics, vol. 15,

pp. 115–132. Providence: American Mathematical Society.

Rabinowitz P (1986) Minimax Methods in Critical Point Theory

with Applications to Differential Equations. Providence:

American Mathematical Society.

Schwartz JT (1969) Nonlinear Functional Analysis. New York:

Gordon and Breach.

Struwe M (1996) Variational Methods, 2nd edn. Berlin: Springer.

Willem M (1996) Minimax Theorems. Basel: Birkha¨user.

Zeidler E (1985) Nonlinear Functional Analysis III. New York:

Springer.

Localization for Quasiperiodic Potentials

S Jitomirskaya, University of California at Irvine,

Irvine, CA, USA

Introduction

Discrete Schro¨dinger operators with quasiperiodic

potentials are operators acting on ‘

)anddefined



¼  þ V ½1

where  is the lattice tight-binding Laplacian

ðn; mÞ¼

1; dist ðn; mÞ¼1

0; otherwise



and V(n, m) = V

(n, m) is a potential given by

= f (T

T

),  2 T

, where T

 =  þ !

, and

! is an incommensurate vector. In certain cases 

may also be replaced by a long-range Laplacian

L(n, m) = L(n  m) with L(n) !0 sufficiently fast.

The questions of interest in the study of quasiper-

iodic and other ergodic operators are the nature and

structure of the spectrum, behavior of the eigenfunc-

tions, and the quantum dynamics: properties of the

time evolution 

= e

itH



of an initially localized

wave packet 

Of particular importance is the phenomenon of

Anderson localization which is usually referred to

the property of having pure point spectrum with

exponentially decaying eigenfunctions. A stronger

property of dynamical localization (see the section

‘‘Dynamical localization’’) indicates the insulator

behavior, while ballistic transport, which for d = 1

follows from the absolutely continuous spectrum,

indicates the metallic behavior.

Operators with ergodic potentials always have

spectra (and pure point spectra, understood as closures

of the set of eigenvalues) constant for a.c. realization of

the potential. The individual eigenvalues however

depend very sensitively on the phase. Moreover, the

pure point spectrum of operators with ergodic

potentials never contains isolated eigenvalues, so pure

point spectrum in such models is dense in a certain

closed set. An easy example of an operator with dense

pure point spectrum is H

which is operator [1] with



1

= 0, or pure diagonal. It has a complete set of

eigenfunctions, characteristic functions of lattice

points, with eigenvalues V

. H



may be viewed as a

perturbation of H

for small 

1

.However,sinceV

are dense, small denominators (V

 V

)

1

make any

Localization for Quasiperiodic Potentials 333

perturbation theory difficult, for example, requiring

intricate KAM-type schemes.

Various methods developed for the Anderson

model (where V

are i.i.d.r.v.’s) such as Fro¨ hlich–

Spencer multiscale analysis and its enhancements, or

Aizenman–Molchanov method, do not work for

quasiperiodic potentials as, among other reasons,

quasiperiodicity does not allow for nice perturba-

tions. The situation here is more difficult and the

theory is far less developed than for the random

case. With a few exceptions, the results are confined

to the one-dimensional setting, and also the case of

one frequency (b = 1) has been much better under-

stood than that of higher frequencies.

One might expect that H



with  small can be

treated as a perturbation of H

=, and therefore

have absolutely continuous spectrum. It is not the

case though for random potentials in d = 1, where

Anderson localization holds for all . The same is

expected for random potentials in d = 2 (but not

higher). Moreover, in one-dimensional case, there

is strong evidence (numerical, analytical, as well as

rigorous) that even models with very mild stochas-

ticity in the underlying dynamics (and sufficiently

nice sampling functions) have point spectrum for

all values of  , like in the random case (e.g.,

= f (n



 þ ), for any >1). At the same time,

for quasiperiodic potentials, one can in many cases

show absolutely continuous spectrum for  small

as well as pure point spectrum for  large (see

below), and therefore there is a metal–insulator

transition in the coupling constant. It is an

interesting question whether quasiperiodic poten-

tials are the only ones with metal–insulator

transition in 1D.

Perturbative and Nonperturbative

Approaches

It is probably fair to say that much of the theory of

qusiperiodic operators has been first developed

around the almost-Mathieu operator, which is

;!;

¼  þ f ð þ n!Þ½2

acting on ‘

(Z), with f : T !T; f () = cos (2).

Several KAM-type approaches, starting with the

pioneering work of Dinaburg–Sinai in 1975, were

developed, in 1980s and 1990s, for this or similar

models in both large and small coupling regimes. Of

those, the most robust and detailed is the reduci-

bility result of Eliasson (1998) that settled the case

of small couplings for sufficiently regular potentials.

The common feature of those perturbative

approaches is that, besides all of them being rather

intricate multistep procedures, they rely extensively

on eigenvalue and eigenfunction parametrization

and perturbation arguments.

The common feature of the perturbative results in

the quasiperiodic setting is that they typically provide

no explicit estimates on how large (or small) the

parameter  should be, and, more importantly, 

clearly depends on ! at least through the constants in

the Diophantine characterization of !.

In contrast, the nonperturbative results allow

effective (in many cases even optimal) and, most

importantly, independent of !,estimateson.The

latter property (uniform in ! estimates on ) has been

often taken as a definition of a nonperturbative result.

Recently developed nonperturbative methods are

also quite different from the perturbative ones in that

they do not employ multiscale schemes: usually only

a few (from one to three) sufficiently large scales are

involved, do not use the eigenvalue parametrization,

and rely instead on direct estimates of the Green’s

function. They are also significantly less involved,

technically. One may think that in these latter

respects they resemble the Aizenman–Molchanov

method for random localization. It is, however, a

superficial similarity, as, on the technical side, they

are still closer to and do borrow certain ideas from

the multiscale analysis proofs of localization.

Lyapunov Exponents

Here for simplicity we consider the quasiperiodic

case, although the definition of the Lyapunov

exponents and some of the mentioned facts apply

more generally to the one-dimensional ergodic case.

Let d = 1. For an energy E 2 R the Lyapunov

exponent (E) is defined as

ðEÞ¼lim

n!1

ln kM

ð; EÞkd 

½3

where

ð; EÞ¼

n¼k1

E  f ð!n þ Þ1



is the k-step transfer matrix for the eigenvalue

equation H=E.

In physics literature, positivity of the Lyapunov

exponent is often taken as an implicit definition of

localization, as Lyapunov exponent is often called

the inverse localization length. Thus, we will be

interested in the regime when Lyapunov exponents

are positive for all energies in a certain interval

intersecting the spectrum. If this condition holds for

all E 2 R, there is no absolutely continuous

334 Localization for Quasiperiodic Potentials

component in the spectrum for all . Positivity of

Lyapunov exponents, however, does not imply

localization or exponential decay of eigenfunctions

(in particular, neither for the Liouville ! nor for the

resonant  2 T

Nonperturbative methods, at least in their original

form, stem to a large extent from estimates invol-

ving the Lyapunov exponents and exploiting their

positivity.

The general theme of the results on positivity of

(E), as suggested by perturbation arguments, is that

the Lyapunov exponents are positive for large .

This subject has had a rich history. The strongest

result in this general context up to date is the

following theorem (Bourgain 2003):

Theorem 1 Let f be a nonconstant real-analytic

function on T

, and H given by [1]. then, for

>(f ), we have (E) > (1=2) ln  for all E and all

incommensurate vectors !.

Corollaries of Positive Lyapunov Exponents

The almost-Mathieu operator On one hand the

almost-Mathieu operator, while simple looking,

seems to represent most of the nontrivial properties

expected to be encountered in the more general case.

On the other hand it has a very special feature: the

duality (essentially a Fourier) transform maps H



4=

; hence  = 2 is the self-dual point. Aubry and

Andre in 1980, conjectured that for this model, for

irrational ! a sharp metal–insulator transition in the

coupling constant  occurs at the critical value of

coupling  = 2: the spectrum is pure point for >2

and purely absolutely continuous for <2. This

conjecture was modified based on later discoveries

of singular-continuous spectrum in this context for

frequencies or phases with certain arithmetic proper-

ties. The modified conjecture stated pure point

spectrum for Diophantine ! and a.e.  for >2

and pure absolutely continuous spectrum for <2

for all !,. The spectrum at  = 2 is singular

continuous for all ! and a.e. (this follows from a

combination of works by Gordon, Jitomirskaya,

Last, Simon Avila, and Krikoryan).

As with the KAM methods, the almost-Mathieu

operator was the first model where the positivity of

Lyapunov exponents was effectively exploited

(Jitomirskaya 1999):

Theorem 2 Suppose ! is Diophantine and (E, !) > 0

for all E 2 [E

, E

]. Then the almost-Mathieu operator

has Anderson localization in [E

, E

] for a.e. .

The condition on  can be made explicit (arithmetic)

and close to optimal. This, combined with the

mentioned results on the Lyapunov exponents,

critical value  = 2, and duality, gives the following

description in the Diophantine case:

Corollary 3 The almost-Mathieu operator H

!, , 

has



for >2, Diophantine ! 2 R and almost every

 2 R, only pure point spectrum with exponen-

tially decaying eigenfunctions.



for  = 2, all ! 62 Q, and a.e.  2 R purely

singular-continuous spectrum.



for <2, Diophantine ! 2 R and a.e.  2 R,

purely absolutely continuous spectrum.

Precise arithmetic descriptions of ! ,  are available.

Thus, the Aubry–Andre conjecture is settled at

least for almost all !, . One should mention,

however, that while 1



can be made optimal by

existing methods, both 2



and 3



are expected to

hold for all  and all ! 62 Q, and such extension

remains a challenging problem (see Simon (2000)).

The method in the above work, while so far the

only nonperturbative method available allowing

precise arithmetic conditions, uses some specific

properties of the cosine. It extends to certain other

situations, for example, quasiperiodic operators

arising from Bloch electrons in a perpendicular

magnetic field, where the lattice is triangular or

has next-nearest-neighbor interactions. However, it

does not extend easily to the multifrequency or even

general analytic potentials. A much more robust

method was developed by Bourgain–Goldstein

(2000), which allowed them to extend (a measure-

theoretic version of) the above localization result to

the general real analytic as well as the multi-

frequency case. Note that essentially no results

were previously available for the multifrequency

case, even perturbative.

Theorem 4 Let f be nonconstant real analytic on

and H given by [2]. Suppose (E, !) > 0 for

all E 2 [E

, E

] and a.e. ! 2 T

. Then for any ,

H has Anderson localization in [E

, E

] for a.e. !.

Combining this with Theorem 1, Bourgain (2003)

obtained that for >(f ), H as above satisfies

Anderson localization for a.e. !. Those results were

recently extended by S Klein to potentials belonging

to certain Gevrey classes. One very important

ingredient of this method is the theory of semialge-

braic sets that allows one to obtain polynomial

algebraic complexity bounds for certain ‘‘excep-

tional’’ sets. Combined with measure estimates

coming from the large deviation analysis of

(1=n)lnkM

()k (using subharmonic function theory

and involving approximate Lyapunov exponents),

Localization for Quasiperiodic Potentials 335

this theory provides necessary information on the

geometric structure of those exceptional sets. Such

algebraic complexity bounds also exist for the

almost-Mathieu operator and are actually sharp

albeit trivial in this case due to the specific nature

of the cosine.

Further corollaries of positive Lyapunov expo-

nents for analytic sampling functions f and b = 1

include Ho¨ lder regularity of the integrated density

of states, zero-dimensionality of spectral measures

for all !, , almost Lipshitz continuity of spectral

gaps, continuity of measure of the spectrum (in

frequency), and vanishing of lower transport

exponents for all !, . Some weaker statements are

available for b > 1orf belonging to certain Gevrey

classes.

Without Lyapunov Exponents

While having led to significant advances, Lyapunov

exponents have obvious limitations, as any method

based on them is restricted to one-dimensional

nearest-neighbor Laplacians. It turns out that the

above methods can be extended to obtain nonper-

turbative results in certain quasi-one-dimensional

situations where Lyapunov exponents do not exist.

For example, nonperturbative localization results

extend to the strip (of arbitrary dimension).

The following nonperturbative theorem deals with

the case of small coupling:

Theorem 5 Let H be an operator [2], where f is

real analytic on T and ! is Diophantine. then, for

<(f ), H has purely absolut ely continuous spec-

trum for a.e. .

We note that an analog of this theorem does not

hold in the multifrequency case (see next section).

The results of this type are obtained by a method

(developed by Bourgain and Jitomirskaya in

2000–02) that studies large deviations for the

quantities of the form (1=n)lnjdet (H  E)



j and

path-determinant expansion for the matrix elements

of the resolvent. Those techniques apply also to

certain other situations with long-range Laplacians,

for example, the kicked-rotor model. Theorem 5 is a

result on nonperturbative localization in disguise as

it was obtained using duality from a localization

theorem for a dual model which has in general a

long-range Laplacian and a cosine potential, and

was in turn obtained by an extension of the method

of Jitomirskaya (1999). A certain measure-theoretic

version of it allowing nonlocal Laplacians but

leading only to continuous spectrum is also available

(see Bourgain (2004)).

Multidimensional Case: d > 1

As mentioned above, there are very few results in

the multidimensional lattice case (d > 1). Essentially,

the only result that existed before the recent

developments was a perturbative theorem – an

extension by Chulaevsky–Dinaburg of Sinai’s

method to the case of operator [1] on ‘

) with

= f (n  !), ! 2 R

, where f is a cos-type function

on T. This also holds nonperturbatively for any real-

analytic f (see Bourgain (2004)). Note that since

b = 1, this avoids most serious difficulties and is

therefore significantly simpler than the general

multidimensional case. We therefore have:

Theorem 6 For any >0 there is (f , ), and, for

>(f ,), (, f )  T

with mes() <, so that for

!=2, operator [1] with V

as above has Anderson

localization.

This should be confronted with the following

theorem of Bourgain:

Theorem 7 Let d = 2 and f () = cos 2 in H = H

defined as above. Then for any  measure of ! s.t.

has some continuous spectrum is positive.

Therefore, for large  there will be both ! with

complete localization as well as those with at least

some continuous spectrum. This shows that non-

perturbative results do not hold in general in the

multidimensional case! Perturbative results, how-

ever, had been obtained, see next section.

A similar (in fact, dual) situation is observed for

one-dimensional multifrequency (d = 1; b > 1) case

at small disorder. One has, by duality:

Theorem 8 Let H be given by [2] with , ! 2 T

and f real analytic on T

. Then for any >0 there is

(f , ) s.t. for <(f , ) there is (, f )  T

with

mes() < so that for !=2, H has purely abso-

lutely continuous spectrum.

And also

Theorem 9 Let d = 1, b = 2 and f be a trigonometric

polynomial on T

with a nondegenerate maximum.

Then for any , measure of ! s.t. H

has some point

spectrum, dense in a set of positive measure, is positive.

Therefore, unlike the b = 1 case (see Theorem 5),

nonperturbative results do not hold for absolutely

continuous spectrum at small disorder.

Perturbative Localization by

Nonperturbative Methods

While the above demonstrates the limitations of

the nonperturbative results, the nonperturbative

336 Localization for Quasiperiodic Potentials