Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Semiclassical Approximation see Stationary Phase Approximation; Normal Forms and Semiclassical Approximation

Semiclassical Spectra and Closed Orbits

Y Colin de Verdie`re, Universite

de Grenoble 1,

Saint-Martin d’He

res, France

Introduction

The purpose of this article is to describe the so-

called ‘‘semiclassical trace formula’’ (SCTF) relating

the ‘‘spectrum’’ of a semiclassical Hamiltonian to

the ‘‘periods of closed orbits’’ of its classical limit.

SCTF formula expresses the asymptotic behavior as

h !0(h = h=2) of the regularized density of states

as a sum of oscillatory contributions associated to

the closed orbits of the classical limi t.

We will mainly prese nt the case of the Schro¨ din-

ger operator on a Riemannian manifold which

contains the purely Riemannian case.

We start with a section about the history of the

subject. We then give a statement of the results and

a heuristic proof using Feynman integrals. This

proof can be transform ed into a mathematical

proof which we will not give here. After that we

describe some applications of the SCTF.

About the History

SCTF has several origins: on one side, Selberg

trace formula (1956) is an exact summation formula

concerning the case of locally symmetric spaces; this

formula was interpreted by H Huber as a formula

relating eigenvalues of the Laplace operator and

lengths of closed geodesics (also called the ‘‘lengths

spectrum’’) on a closed surface of curvature 1.

On the other side, around 1970, two grou ps of

physicists developed independently asymptotic trace

formulas:



M Gutzwiller for the Schro¨dinger operator,

using the quasiclassical approximation of the

Green function (the ‘‘van Vleck’s formula’’); it

is interesting to note that the word ‘‘trace

formula’’ is not written, b ut Gutzwiller instead

speaks of a new ‘‘quantization method’’ (the old

one being ‘‘Einstein–Brillouin–Keller (EBK)’’ or

‘‘Bohr–Sommerfeld rules’’).



R Balian and C Bloch, for the eigenfrequencies of

a cavity, use what they call a ‘‘multiple reflection

expansion.’’ They asked about a possible applica-

tion to Kac’s problem.

At the same time, under the influence of Mark

Kac’s famous paper ‘‘Can one hear the shape of a

drum?,’’ mathematicians became quite interested in

inverse spectral problems, mainly using heat kernel

expansions (for the state of the art around 1970, see

Berger et al. (1971)).

The SCTF was put into its final mathematical

form for the Laplace operator on closed manifolds

by three groups of people around 1973–75:



Y Colin de Verdie` re in his thesis was using the

short-time expansion of the S chro¨ dinger kernel

and an approximate Feynman path integral. He

proved that the spectrum of the Laplace operator

determines generically the lengths of closed

geodesics.



J Chazarain derived the qualitative form of the

trace for the wave kernel using Fourier integral

operators.

512 Semiclassical Spectra and Closed Orbits



Using the full power of the symboli c calculus of

Fourier integral operators, H Duistermaat and

V Guillemin were able to comput e the main term

of the singularity from the Poincare´ map of the

closed orbit. Their paper became a canonical

reference on the subject.

After that, people were able to extend SCTF to:



general semiclassical Hamiltonians (Helffer–

Robert, Guillemin–Uribe, Meinrenken),



manifolds with boundary (Guillemin–Melrose),



surfaces with conical singularities and polygonal

billiards (Hillairet), and



several commuting operators (Charbonnel–

Popov).

Recently, some researchers have remarked about the

nonprincipal terms in the singularities expansion

which come from the semiclassical Birkhoff normal

form (Zelditch, Guillemin).

Selberg Trace Formula

We consider a compact hyperbolic surface X.

‘‘Hyperbolic’’ means that the Riem annian metric is

locally (dx

þ dy

)=y

or is of constant curvature

1. Such a surface is the quotient X = H= where 

is a discret e co-compact subgroup of the group of

isometries of the Poincare´ ha lf-plane H. Closed

geodesics of X are in bijective corre spondence with

nontrivial conjugacy classes of . More precisely,

the set of loops C(S

, X) splits into connected

components associated to conjugacy classes and

each component of nontrivial loops contains exactly

one periodic geodesic.

Theorem 1 (Selberg trace formula). If  is a real-

valued function on R whose Fourier transform

 is

compactly supported and 

= 1=4 þ 

is the spec-

trum of the Laplace operator on X, we have:

j¼1

ð  

Þ¼

2

ð þ sÞs tanh s ds

2P

n¼1



2 sinhðnl



=2Þ

 Reð

ðnl



Þe

inl



where A is the area of X, P the set of primitive

conjugacy classes of  and, for  2P, l



is the length

of the unique closed geodesic associated to .

A nice recent presentation of the Selberg trace

formula can be found in Marklof (2003).

Semiclassical Schro¨ dinger Operators

on Riemannian Manifolds

If (X, g) is a (possibly noncompact) Riemannian

manifold and V : X !R a smooth function which

satisfies lim inf

x !1

V(x) = E

> 1, the differential

operator

H = (1=2)h

 þ V is semibounded from

below and admits self-adjoint extensions. For all

those extensions, the spectrum is discrete in the interval

e1, E

d and eigenfunctions

H’

= E

’

are loca-

lized in the domain V  E

.IfX is compact and V = 0,

we recover the case of the Laplace operator.

We will denote this part of the spectrum by

inf V < E

ðhÞ < E

ðhÞE

ðhÞ< E

For the Laplace operator, we have E

=h



, where



 



 is the spectrum of the

Laplace operator.

The SCTF can also be derived the same way for

Schro¨ dinger operators with magnetic field. One can

even extend it to Hamiltonian systems which are not

obtained by Legendre transform from a regular

Lagrangian. In this case, Morse indices have to be

replaced by the more general Maslov indices.

Classical Dynamics

Newton Flows

Euler–Lagrange equations for the Lagrangi an

L(x, v):= (1=2)kvk

 V(x) admit a Hamiltonian

formulation on T

X whose energy is given by

H = (1=2)kk

þ V(x). We will denote by X

the

Hamiltonian vector field

:¼

@





Preservation of H by the dynamics shows immedi-

ately that the Hamiltonian flow 

restricted to H <

is complete.

The Hamiltonian H is the ‘‘classical limit’’ of

in more technical terms, H is the semiclassical

principal symbol of

If V = 0, H = (1=2)g



and the flow is the geo-

desic flow.

Periodic Orbits

Definition 1 A periodic orbit (, T) (also denoted

p.o.) of the Hamiltonian H consists of an orbit 

of X

which is homeomorphic to a circle and

a nonzero real number T so that 

(z) = z for all

z 2 . We will denote by T

() > 0 (the primitive

period) the smallest T > 0 for which 

(z) = z.

Semiclassical Spectra and Closed Orbits 513

If (T, E) are given, W

T, E

is the set of z’s so that

H(z) = E and 

(z) = z.



The (linear) Poincare´ map 



of a p.o. (, T) with

H() = E: we restrict the flow to S

:= {H = E}

and take a hypersurface  inside S

transversal to

 at the point z

. The associated return map P is a

local diffeomorphism fixing z

. Its linearization



:= P

) is the linear Poincare´ map, an

inversible (symplectic) endomorphism of the

tangent space T

.



The Morse index (): p.o. (, T) is a critical point

of the action integral

L((s),

 (s)) ds on the

manifold C

(R=TZ, X). It always has a

finite Morse index (Milnor 1967) which is denoted

by (). For general Hamiltonian systems, the Morse

index is replaced by the Conley–Zehnder index.



The nullity index () is the dimension of the

space of infinitesimal deformations of the p.o. 

by p.o. of the same energy and period. We always

have ()  1and() = 1 þ dim ker (Id  



Example 1 (Geodesic flows)



Riemannian manifold with sectional curvature < 0:

in this case, we have for all periodic geodesics

() = 0, () = 1.



Generic metrics: for a generic metric on a closed

manifold, we have () = 1 for all periodic

geodesics.



For flat tori of dimension d: we have () = 0 and

() = d.



For sphere of dimension 2 with consta nt curva-

ture: if 

is the nth iterate of the great circle, we

have (

) = 2jnj and ( 

) = 3.

It is a beautiful result of J-P Serre that any pair of

points on a closed Ri emannian manifold are end-

points of infinitely many distinct geodesics. Count-

ing geometrically distinct periodic geodesics is much

harder especially for simple manifolds like the

spheres. It is now known that every closed Riemannian

manifold admits infinitely many geometrically distinct

periodic geodesics (at leas t, in some cases, for

generic metrics, (Berger 2000 chap. V). There exists

significant knowledge concerning more general

Hamiltonian systems as well.

Nondegeneracy

There are several possible nondegeneracy assump-

tions. They can be formulated ‘‘a` la Morse–Bott’’

(critical point of action integrals) or purely

symplectically.

Definition 2 Two submanifolds Y and Z of X

intersect cleanly iff Y \ Z is a manifold whose

tangent space is the intersection of the tangent

spaces of Y and Z.

Fixed points of a smooth map are clean if the

graph of the map intersects the diagonal cleanly.

Definition 3 We will denote by (ND) the following

property of the p.o. (

, T

): the fixed points of the

associated (nonlinear) Poincare´ map P are clean.

The set W

T, E

is ND if all p.o.’s inside are ND.

T, E

is then a manifold of dimension ().

Example 2



Generic case:  = 1; (ND) is equivalent to ‘‘1 is

not an eigenvalue of the linear Poincare´ map.’’

In this case, we can deform the p.o. smoothly by

moving the energy. This family of p.o.’s is called

a cylinder of p.o.’s. The period T(E) is then a

smooth function of E.



Completely integrable systems:  = d;(ND)isthena

consequence of the so-called ‘‘isoenergetic KAM

condition’’: assuming the Hamiltonian is expressed

as H(I

, ..., I

) using action-angle coordinates, this

condition is that the mapping I ![rH(I)] from the

energy surface H = E into the projective space is a

local diffeomorphism. This condition implies that

Diophantine invariant tori are not destructed by a

small perturbation of the Hamiltonian.



Maximally degenerated systems: it is the case

where all orbits are periodic ( = 2d 1). For

example, the two-body problem with Newtonian

potential and the geodesic flows on compact

rank-1 symmetric spaces.

Canonical Measures and Symplectic Reduction

Under the hypothesis (ND), the manifold W

T, E

admits

a canonical measure 

,invariantby

. In the case

 = 1, this measure is given by jdtj=

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det(Id  )

By using a Poincare´ section, it is enough to

understand the following fact: if A is a symplectic

linear map, the space ker (Id  A) admits a canonical

Lebesgue measure.

We start with the following construction: let L

and L

be two Lagrangian subspaces of a symplectic

space E and !

, j = 1, 2, be half-densities on L

denoted by !

2 

1=2

). If W = L

\ L

, we have

the following canonical isomorphisms: 

1=2

) =



1=2

(W)  

1=2

=W). So 

1=2

)  

1=2

) =



1=2

=W)  

1=2

=W)  

(W). M

= L

=W are

two Lagrangian subspaces of the reduced space

=W whose intersection is 0. Hence, by using

the Liouville measure on it, we get 

1=2

) 



1=2

) = C. Hence, we get a density !

on W. It turns out that the previous calculation is one

of the main algebraic pieces of the symbolic calculus of

514 Semiclassical Spectra and Closed Orbits

Fourier integral operators and the density !

arises in stationary-phase computations.

The graph of a symplectic map is equipped with a

half-density by pullback of the Liouville half-

density. So we can apply the previous construction

to the intersection of the graph of A and the graph

of the identity map.

Actions

Definition 4 If (, T) is a p.o., we define the

following quantity which is called action of :

AðÞ¼



 dx

In the (ND) case, A() is constant on each connected

component of W

T, E

In the generic case and if T

(E) 6¼ 0 (cylinder of

p.o.), p.o.’s of the cylinder are also parametrized

by T (i.e., we note by 

the p.o. of the cylinder of

energy E and 

the p.o. of period T). If

a(E) = A(

) and b(T) = 

L(

(s),



(s))ds, a(E)

and b(T) are Legendre transforms of each other.

Playing with Spectral Densities

We will define the ‘‘regularized spectral densities.’’

The general idea is as follows: we want to study an

h-dependent sequence of numbers E

(h) (a spectrum)

in some interval [a, b]. We introduce a non negative

function  2S(R) which satisfies

(t)dt = 1, and

also D

, ", h

(E) =



(E  E

), where 

(E) =

1

(E="). It gives the analysis of the spectrum at

the scale ". Of course, we will adapt the scaling "

to the small parameter h. If the scaling is of the size

of the mean spacing of the spectrum, we will get a

very precise resolution of the spectrum.

The general philosophy is:



If h is the semiclassical parameter of a semiclassi-

cal Hamiltonian, the mean spacing of the eigen-

values is of order h

(Weyl’s law). The trace

formula gives the asymptotic behavior of

, ", h

(E) for "  h (and hence ">>E except

if d = 1). This behavior is not ‘‘universal’’ and

thus contains a significant amount information of

(in our case, on periodic trajectories).



Better resolution of the spectrum needs the use of

the long-time behavior of the classical dynamics and

is conjecturally universal. It means that eigenvalues

seen at very small scale behave like eigenvalues of an

ensemble of random matrices, the most common one

being the Wigner Gaussian orthogonal ensemble

(GOE) and Gaussian unitary ensemble (GUE).

We fix some interval [a, b] with b < E

We define D(E):=

aE

b

(E

) as the sum of

Dirac measures at the points E

and its h-Fourier

transform as

ZðtÞ¼trac e

ðe

it

H=h

Þ :¼

expðitE

=hÞ½1

where

is the sum over E

2 [a, b].

The Duistermaat–Guillemin trick relates the

previous behavior to asymptotics of the regularized

density of eigenvalues. Let us give a function  2

S(R) so that

(t) =

itE

(E)dE is compactly

supported and

ðtÞ¼1 þ Oðt

Þ; t !0 ½2

(all moments of  vanish). We introduce, for E 2

[a, b], D



(E):=

h

(E  E

=h). D



(E) is indepen-

dent modulo O(h

)ofa, b. We have



ðEÞ¼

2h

ðtÞ Zð tÞdt

The idea is now to start from a semiclassical

approximation of U(t) = e

it

H=h

and to insert it into

eqn [1]. We need only a uniform approximation of

U(t) for t 2 Support(

). From the asymptotic expan-

sion of Z(t), we will deduce the asymptotic expan-

sion of D



, the regularized eigenvalue density.

The Smoothed Density of States

The following statement expressing the smoothed

density of eigenvalues is the main result of the

subject. Under the (ND) assumption, it gives the

existence of an asymptotic expansion for D



(E):

Theorem 2 If E is not a critical value of H and the

(ND) condition is satisfied for all p.o.’s of energy

E 2 [a, b] and period inside the support of

,



ðEÞ¼D

Weyl

ðEÞþ

WðT;EÞ

þ Oðh

Þ½3

where:

(i)

Weyl

ðEÞ¼ð2hÞ

d

j¼0

ðEÞh

with a

(E) =

H = E

dL=dH

(ii) The sum is over all the manifolds W

T, E

so that

T 2 Support(

).

(iii)

WðT;EÞ

ð2ihÞ

ððÞþ1Þ=2

iðÞ=2

 e

iAðÞ=h

j0

ðEÞh

Semiclassical Spectra and Closed Orbits 515

with

ðEÞ¼

ðTÞ

T;E

d

" ¼

1ifT

ðEÞ > 0

i if T

ðEÞ < 0



If () = 1, we get b

(T



jdet(Id  



1=2

The Weyl Expansion

If Support(

) is contained in [T

min

, T

min

], where

min

is the smallest period of a p.o.  with H() = E,

and, if E is not a critical value of H, formula [3]

reduces to



ðEÞð2hÞ

d

j¼0

ðEÞh

From the previous formula, it is possible to deduce

the following estimates:

Theorem 3 If a, b are not critical values of H:

#fjja  E

ðhÞbg

¼ð2hÞ

d

volumeða  H  bÞð1 þ OðhÞÞ

This remainder estimate is optimal and was first

shown in rather great generality by Ho¨ rmander

(1968).

Derivation from the Feynman Integral

The Feynman Integral

R Feynman (Feynman and Hibbs 1965) foun d

a geometric representation of the propagator,

that is, the kernel p(t, x, y) of the unitary group

exp (it

H=h) using an integral (FPI := Feynman path

integral) on the manifold 

t, x, y

:= { : [0, t] !

Xj(0) = x, (t) = y} of paths from x to y in the

time t;ifL(,

) is the Lagrangian, we have, for

t > 0:

pðt; x; yÞ¼



t;x;y

exp

h

LððsÞ;

ðsÞÞds



jdj

where jdj is a ‘‘Riemannian measure’’ on the

manifold 

t, x, y

with the natural Riemannian

structure.

There is no justification FPI as a useful mathema-

tical tool. Nevertheless, FPI gives good heuristics

and right formulas.

The Trace and Loop Manifolds

Let us try a formal calculation of the partition

function and its semiclassical limit. We get

ZðtÞ¼

jdxj



x;x;t

exp

h

LððsÞ;

ðsÞÞds



jdj

If we denote by 

the manifold of paths

 : R=tZ ! X, (loops) and we apply Fubini (sic !),

we get

ZðtÞ¼



exp

h

LððsÞ;

ðsÞÞds



jdj

The Semiclassical Limit

We want to apply stationary phase in order to get

the asymptotic expansion of Z(t); critical points of

: 

! R are the p.o.’s of the Euler–Lagrange flow

and hence of the Hamiltonian flow of period t.We

require the ND assumption (Morse–Bott), the Morse

index, and the determinant of the Hes sian:

1. The ND assumption is the original Morse–Bott

one in Morse theory: we have smooth manifolds

of critical points and the Hessian is transversally

ND.

2. The Morse index is the Morse index of the action

functional on periodic loops: L():=

L((s),

(s))ds.

3. The Hessian is associated to a periodic Sturm–

Liouville operator for which many regulariza-

tions have already been proposed.

In this manner, we get a sum of contributions

given by the components W

j, t

of W

ðtÞ¼ðihÞ



ði=hÞLðÞ

ðhÞ

with c

(h) 

l = 0

j, l

h

and

j;0

ið=2Þ

jj

1=2

where  is the Morse index and  is a regularized

determinant.

The Integrable Case

As observed by Berry–Tabor, the trace formula in

this case comes from Poisson summation formula

using action-angle coordinates. Asymptotic of the

eigenvalues to any order can then be given in the so-

called quantum integrable case by Bohr–Sommerfeld

rules.

516 Semiclassical Spectra and Closed Orbits

The Maximally Degenerated Case

Let us assume that (X, g) is a compact Riemannian

manifold for which all geodesics have the same

smallest period T

= 2. Then we have the following

clustering property:

Theorem 4 There exists some constant C and some

integer  so that

(i) the spectrum of  is contained in the union of

the intervals

¼ k þ





C; k þ





þC



;

k ¼ 1; 2; ...

(ii) N(k) =#Spectrum() \ I

is a polynomial func-

tion of k for k large enough.

The property (ii) is consequence of the trace

formula.

Applications to the Inverse

Spectral Problem

We will now restrict ourselves to the case of the

Laplace operator on a compact Riemannian mani-

fold (X, g). The main result is as follows:

Theorem 5 (Colin de Verdie` re). If X is given, there

exists a generic subset G

, in the sense of Baire

category, of the set of smooth Riemannian metrics on

X, so that, if g 2G

, the length spectrum of (X, g) can

be recovered from the Laplace spectrum. The set G

contains all metrics with <0 sectional curvature and

(conjecturally) all metrics with <0 sectional curvature.

We can take for G

the set of metrics for which all

periodic geodesics are nondegenerate and the length

spectrum is simple.

Some cancelations may occur between the asympto-

tic expansions of two ND periodic trajectories with the

same actions if the Morse indices differ by 2 mod 4.

The Case with Boundary

If (X, g) is a smooth compact manifold with boundary,

one introduces the broken geodesic flow by extending

the trajectories by reflection on the boundary. SCTFs

have been extended to that case by Guillemin and

Melrose. Periodic geodesics which are transversal to

the boundary contribute to the density of states in the

same way as for periodic manifolds. Periodic geodesics

inside the boundary are in general accumulation of

periodic geodesics near the boundary: their contribu-

tions is therefore very complicated analytically.

Bifurcations

Let us denote by C

 R

T, E

, the set of pairs (T,E)

for which W

T, E

is not empty. The previous results

apply to the ‘‘smooth’’ part of the set C

. Among

other interesting points are points (0,E) with critical

value E of H (Brummelhuis–Paul–Uribe) and points

corresponding to bifurcation of p.o. when moving

the energy.

Detailed studies of some of these points have been

done, for example, the results of suitable applica-

tions of the theory of singularities of functions

of finitely many variables, their deformations (catas-

trophe theory), and applications to stationary-phase

method, and a significant body of knowledge on

these subjects now exists.

SCTF and Eigenvalue Statistics

One of the main open mathematical problems is:

‘‘can one really use appropriate forms of the SCTF

as quantization rules and use it in order to derive

eigenvalues statistics?’’

This problem is related to the fine-scale study of the

eigenvalue spacings ("<<h). It is one of the important

unsolved problems of the so-called ‘‘quantum chaos.’’

Many people think that progress in this field will allow

us to solve the Bohigas–Giannoni–Schmit conjecture:

‘‘if the geodesic flow is hyperbolic, eigenvalue distribu-

tion follows random matrix asymptotics.’ ’

See also: Billiards in Bounded Convex Domains;

h-Pseudodifferential Operators and Applications;

Quantum Ergodicity and Mixing of Eigenfunctions;

Random Matrix Theory in Physics; Regularization for

Dynamical Zeta Functions; Resonances.

Further Reading

Balian R and Bloch C (1970) Distribution of eigenfrequencies for the

wave equation in a finite domain I. Annals of Physics 60: 401.

Balian R and Bloch C (1971) Distribution of eigenfrequencies for the

wave equation in a finite domain II. Annals of Physics 64: 271.

Balian R and Bloch C (1972) Distribution of eigenfrequencies for the

wave equation in a finite domain III. Annals of Physics 69: 76.

Berger M (2000) Riemannian Geometry During the Second Half

of the Twentieth Century. University Lecture Series, vol. 17.

Providence, RI: American Mathematical Society.

Berger M, Gauduchon P, and Mazet E (1971) Le spectre d’une

varie´te´ riemannienne compacte. Berlin–Heidelberg–New York:

Springer LNM.

Colin de Verdie` re Y (1973) Spectre du Laplacien et longueurs des

ge´ode´siques pe´riodiques II. Compositio Mathematica 27:

159–184.

Duistermaat J (1976) On the Morse index in variational calculus.

Advances in Mathematics 21: 173–195.

Duistermaat J and Guillemin V (1975) The spectrum of positive

elliptic operators and periodic geodesics. Inventiones Mathe-

maticae 29: 39–79.

Semiclassical Spectra and Closed Orbits 517

Feynman R and Hibbs A (1965) Quantum Mechanics and Path

Integrals. New-York: McGraw-Hill.

Gutzwiller M (1971) Periodic orbits and classical quantization

conditions. Journal of Mathematical Physics 12: 343–358.

Gutzwiller M (1990) Chaos in Classical and Quantum

Mechanics. Berlin–Heidelberg–New York: Springer.

Ho¨ rmander L (1968) The spectral function of an elliptic operator.

Acta Mathematica 121: 193–218.

Marklof J (2003) Selberg’s trace formula: an introduction.

Lectures given at the International School ‘‘Quantum Chaos

on Hypebolic Manifolds,’’ Schloss Reisensburg, Gunsburg,

Germany, 4–11 October 2003. To appear in Springer LNP,

Berlin–Heidelberg, New York. http://fr.arxiv.org

Milnor J (1967) Morse Theory. Annals of Mathematics Studies

no. 51. Princeton, NJ: Princeton University Press.

Selberg A (1956) Harmonic analysis and discontinuous groups in

weakly symmetric Riemannian spaces with applications to

Dirichlet series. Journal of the Indian Mathematical Society

20: 47–87.

Semilinear Wave Equations

P D’Ancona, Universita

di Roma ‘‘La Sapienza,’’

Rome, Italy

Introduction

A semilinear wave equation is an equation of the

form

&u ¼ Fðu; u

Þ; u : R  R

! R ½ 1

where F : R

nþ2

! R is a smooth function, the

d’Alembert operator

is defined as

& ¼D

D

D

; D

½2

and u

denotes the vector of all first-order deriva-

tives of u:

¼ðD

u; D

u; ...; D

uÞðu

; u

; ...; u

Sometimes the term ‘‘semilinear’’ is used in a more

restrictive sense and refers to the special class of

equations

&u ¼f ðuÞ½3

The very particular case f (u) = mu, m > 0, corres-

ponds to the Klein–Gordon equation, used to model

relativistic particles. True nonlinear terms of the form

f (u) = mu  u

, m  0 (meson equation), or

f (u) = sin u (sine-Gordon equation) have been pro-

posed as models of self-interacting fields with a local

interaction. Notice that for the physical applications it

is natural to consider complex-valued functions u(t, x);

in the general case of eqn [1], this actually means that

we are considering a 2  2systemin<u and =u.

However, the natural physical requirement of gauge

invariance restricts the possible nonlinearities to the

functions satisfying the condition

f ðe

i

uÞ¼f ðuÞe

i

; 8 2 R ½4

Thus, in particular f (0) = 0 and we see that f must

be of the form f (u) = g(juj

)u for some g. Since the

gauge-invariant wave equation

&u ¼ gðjuj

Þu ½5

has essentially the same properties as the real-valued

equation [3], it is not too restrictive to study only

real-valued functions as we shall mostly do in the

following.

The more general equations of the form [1],

involving the derivatives of u, are encountered in

several physical theories, including the nonlinear

-models and general relativity.

However, beyond the concrete physical applica-

tions, eqn [1] is important since it is a simplified but

relevant model of much more general equations and

systems of mathematical physics; despite its simp le

structure, the semilinear wave equation presents

already all the main difficulties and phenomena of

nonlinear wave interaction, and it represents an

ideal laboratory for such problems.

In this article we plan to give a concise but, as far

as possible, comprehensive review of the main

research directions concerning eqn [1], and in

particular we shall focus on the global existence of

both large and small nonlinear waves, and the

problem of local existence for low-regularity solu-

tions. A large part of the theory extends to nonlinear

perturbations of the form &u = F(u, u

, u

)andto

the fully nonlinear case; we have no space here to

give an account of these developments and we must

refer the reader to the books and papers cited in the

‘‘Further reading’’ section.

Classical Results

Equations [1] and [3] are hyperbolic with respect to

the variable t. This is a precise way of stating that

the ‘‘correct’’ problem for it is an initial-value

problem (IVP) with data at some fixed time, or

518 Semilinear Wave Equations

more generally on some spacelike surface: this

means that we assign two functions u

(x), u

(x),

called the ‘‘initial data,’’ and we look for a function

u(t, x) satisfying the IVP:

&u ¼Fðu;u

Þ; uð0;xÞ¼u

ðxÞ; u

ð0;x Þ¼u

ðxÞ½6

This setting is in agreement with the physical picture

of an evolution problem: the data represent the

complete state of a system at a fixed time, and they

uniquely determine the evolution of the system,

which is described by the differen tial equation.

This rough statement of the problem is sufficient

when working with smooth functions, as in the

classical approach. By purely classical methods, that

is, energy inequalities and nonlinear estimates, it is

not difficult to prove the following local existence

result, where H

= H

) denotes the Sobo lev

space of functions with k derivatives in L

Theorem 1 Assume F is C

. Let ( u

, u

) 2 H



k1

for some k > 1 þ n=2. Then there exists a time

T = T(ku

þku

k1

) > 0 such that problem

[6] has a unique solution belongi ng to (u, u

) 2

C([T, T]; H

)  C([T, T]; H

k1

If F = F(u) depends only on u, the result holds for

all k > n=2.

Proof We decided to include a sketchy but com-

plete proof of this result since it shows the ba sic

approach to nonlinear wave equations: many results

of the theory, even some of the most delicate ones,

are obtained by suitable vari ations of the contrac-

tion method, and are similar in spirit to this classical

theorem.

Assume for a moment that the equation is linear

so that F = F(t, x) is a given smooth funct ion of (t, x).

For the linear equation &u = F, we can construct a

solution u using explicit formulas. Moreover, u

satisfies the energy inequality

ðtÞE

ð0Þþ

kFð s; Þk

k1

ds ½7

where the energy E

(t) is defined as

ðtÞ¼kuðt; Þk

þku

ðt; Þk

k1

½8

Now we introduce the space X

= C([T, T]; H

) \

([T, T]; H

k1

), the space Y

= C([T, T]; H

k1

the mapping  : F !u that takes the function F(t, x)

into the solution of &u = F (with fixed data u

, u

and the mapping (u) = F(u, u

) which is the original

right-hand side of the equation.

The energy inequality tells us that  is bounded

from Y

to X

. Actually, for M large enough with

respect to E

(0) (the H

norm of the data),  takes

any ball B

(0, N)ofY

into the ball B

(0, M þ NT)

of X

. Moreover, if we apply [7] to the difference of

two equations

u = F and

v = G, we also see that

 is Lipschitz continuous from Y

to X

,witha

Lipschitz constant CT.

On the other hand, (u) = F(u, u

) takes X

to Y

provided k > 1 þ n=2; we can even say that it is

Lipschitz continuous from B

(0, M)toB

(0, C(M ))

for some funct ion C(M), with a Lipschitz constant

(M) also depending on M. This follows easily

from Moser type estimates like

kFð u; u

Þk

k1

 ðkuk

Þkuk

; k >

þ 1

kFð uÞk

 ðkuk

Þkuk

; k >

Now it is easy to conclude: the composition   

maps X

into itself, and actually is a contraction of

(0, M) into itself provided M is large enough with

respect to the data, and T is small enough with

respect to M. The unique fixed point is the required

solution.

The wave operator has an additional important

property called the finite speed of propagation,

which can be stated as follows: given the IVP

&u ¼ 0; uð0; xÞ¼u

ðxÞ; u

ð0; xÞ¼u

ðxÞ

if we modify the data ‘‘outside’’ a ball B(x

, R)  R

the values of the solution inside the cone

Kðx

; RÞ¼fðt; xÞ: t  0; jx  x

j < R  tg

do not change. Notice that K(x

, R) is the cone with

basis B(x

, R) and tip (R, x

); the slope of its mantle

represents the speed of propagation of the signals,

which for the wave operator

is equal to 1. The

property extends without modification to the semi-

linear problem [6], at least for the smooth solutions

given by Theorem 1. Actually, it is not difficu lt to

modify the proof of the theorem to work on cones

instead of bands [T, T]  R

; in other words, given

a ball B = B(x

, R), we can assign tw o data

2 H

(B), u

2 H

k1

(B)(k > n=2 þ 1) and prove

the existence of a local solution on the cone

K(x

, R) for some time interval t 2 [0, T].

In general, the finite speed of propagation allows

us to localize in space most of the results and the

estimates; as a rule of thumb, we expect that what is

true on a band [0, T]  R

should also be true on

any truncated cone K(x

, R) \ {0  t  T}.

Semilinear Wave Equations 519

Symmetries

The linear wave equation can be written as the

Euler–Lagrange equation of a suitable Lagrangian.

This is still true for the semilinear perturbations of

the form

&u þ f ðuÞ¼0 ½9

Indeed, denoting with F(s) =

f ()d the primitive

of f, the Lagrangian of [9] is

LðuÞ¼



þFðuÞ



dtdx ½10

The functional L is not positive definite; hence, the

variational approach gives only weak results. How-

ever, this point of view allows us to apply Noether’s

principle: any invarianc e of the functional is related

to a conservation law of the equation. These

conserved quanti ties can also be obtained by taki ng

the product of the equation by a suitable multiplier,

although this method is far from obvious in many

cases. We describe here this circle of ideas briefly.

The functional L is invariant under the Poincare´

group, generated by time and space translations and

the Lorentz transformations (>1, c 6¼ 0):

t 7!

t  x

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



 1

; x

x

 ct

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



 1

½11

The infinitesimal generators of the translations are

simply the partial derivatives D

and D

.TheLorentz

transformations can be decomposed as a rotation

followed by a boost, and indeed a corresponding

complete set of infinitesimal generators are the operators



¼ x

 x

; 

¼ x

þ tD

½12

All the operators in the Poincare´ group commute

with

exactly.

The conservation law related to time translations

(time derivative) is the fundamental ‘‘conservation of

energy’’

EðtÞ¼

þ FðuÞ



dx ¼ Eð0Þ½13

while spatial translations (spatial derivatives) lead to

the conservation of momenta

dx ¼ const:; j ¼ 1; ...; n

On the other hand, infinitesimal rotations and

boost [12] are connected to the conservation of

angular momenta

u  x



 D

u dx ¼ const:;

j; k ¼ 1; ...; n ½14

and

eðuÞþD

u½dx ¼const:;

k ¼1;...;n

½15

where

eðuÞ¼

þ FðuÞ½16

is the energy density.

The Poincare´ group does not exhaust the invar-

iance properties of the free wave equation. Among

the other transformations which commute or almost

commute with

, we mention the spacetime dilations

and inversions (which together with translations and

Lorentz transformations generate the larger confor-

mal group), the scaling u 7!u, the spatial dilations,

and, in the complex- valued case, the gauge transfor-

mation u 7!e

i

u. In this way several useful conserva-

tion laws can be obtained, including the conformal

energy identities of K Morawetz.

Strichartz Estimates

Energy estimates are very useful tools but they have

some major shortcomings. The main one is clearly

the large num ber of derivatives necessary to estimate

the nonlinear term. This is why the modern theory

of semilinear wave equations relies mainly on

different tools, which go under the umbrella name

of Strichartz estimates and express the decay

properties of solutions when measured in L

related norms. In this sect ion we summarize these

estimates in their most general form, and try to give

a feeling of the techniques involved.

Consider the following IVP for a homogeneous

linear wave equation:

&u ¼ 0; uð0; xÞ¼0; u

ð0; x Þ¼f ðxÞ½17

The conservation of energy states that

ðt; Þk

þkr

uðt; Þk

kf k

½18

for all times t. Thus, we see that L

-type norms of

the solution do not decay. The interesting fact is that

if we measure the solution u in a different L

-norm,

p > 2, the norm decays as t !1, and the decay is

fastest for the L

-norm.

To appreciate the dispersive phenomena at their

best, let us assume that the Fourier transform of the

data is localized in an annulus of order 1:

supp

f ðÞf1=2 jj2g½19

520 Semilinear Wave Equations

Then the corresponding solut ion u(t, x) has the same

property, and we see that

kuk

¼k

 2kjj

 2kr uk

 4kuk

We condense the last line in the shorthand notation

kuk

’kruk

We shall also write

kvk



kwk

() k vk

 Ckw k

for some C

We can now rewrite the conservation of energy

[20] in a very simple form; for localized data (and

hence a localized solution) as in [19], we have

kuðt; Þk



kf k

½20

The basic L

-estimate for a solution of [17] with

localized data as in [19] is simply

kuðt; Þk



ðn1Þ=2

kf k

½21

This estimate is well known since the 1960s; it can

be proved easily by several techniques, notably by

the stationary-phase method. Property [21] mea-

sures the fact that as time increases, the total

energy of the solution remains constant but spreads

over a region of increasing volume, due to the

propagation of waves. If we interpolate between

[20] and [21], we obtain the full set of dispersive

estimates

kuðt; Þk



ðn1Þð1=21=qÞ

kf k

¼ 1; 2  q 1

½22

Recall that we are working with localized solutions

on the annulus jj1; it is easy to extend the

above estimates to general solutions by a rescaling

argument, exploiting the fact that, if u(t, x)isa

solution of the homogeneous wave equation,

u(t , x) is also a solution for any constant .

Indeed, if

f (and hence

u) is supported in the

annulus 2

j1

jj2

jþ1

, j 2 Z, by rescaling [21],

we obtain

kuðt; Þk



ðn1Þ=2

jðn1Þ=2

kf k

½23

If f is any smooth function, not localized in

frequency, we can still write it as a series

f ¼

j 2Z

where supp

 {2

j1

jj2

jþ1

}. The quantity

kf k

1;1

j 2Z

is by definition the

1,1

Besov norm of f. Thus,

summing the estim ates [23] over j, we conclude that

a general solution of [17] satisfies the dispersive

estimate

kuðt; Þj



ðn1Þ=2

kf k

ðn1Þ=2

1;1

½24

The Strichartz estimates can be obtai ned as a

consequence of the above dispersive estimates, plus

some subtle functional analytic arguments. In the

general form we give here, they were proved by

J Ginibre and G Velo, and in the most difficult

endpoint cases by Keel and T Tao. The solution of

the homogeneous problem [17] studied above can be

written as

uðt; xÞ¼

sinðtjDjÞ

jDj

f ; jD jF

1

jjF

(here F denotes the Fourier transform). On the

other hand, the solution of the complete nonhomo-

geneous problem

&u ¼ Fðt; xÞ; uð0; xÞ¼u

; u

ð0; xÞ¼u

ðxÞ½25

can be written by Duhamel’s formula as

uðt; xÞ¼

sinðtjDjÞ

jDj

sinðtjDjÞ

jDj

sinððt  sÞjDjÞ

jDj

f ds

and we see that the above estimates [22] apply to all

the operators appearing here. If we consider problem

[25] and we assume that the data F( t, x), u

, u

are

localized in frequency so that

F( t, ),

have

support in the annulus jj1, the Strichartz estimate

takes the following form:

kuk



þku

þkFk

½26

Here the dimension is n  2; L

denotes the space

with norm

kuk

kuðt; Þk

ðR



1=p

; I ¼½0; T

or I ¼ R

the indices p, q satisfy the conditions

n  1



n  1

;

p 2½2; 1; ðn; p; qÞ 6¼ð3; 2; 1Þ

½27

while

q satisfy an identical condition (and p

denotes the conjugate index to p). The constant in

inequality [26] is unifor m with respect to the

interval I.

Semilinear Wave Equations 521