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projection to the Fermi surface (Feldman and

Trubouitz 1990, 1991). Consequently, the

vanish

linearly on the Fermi surface, so that the integral over

k in [11] converges. The effect of the counter-term

function K can be described less technically: it fixes

the Fermi surface to be S, the zero set of e. Thus, K

forces S to be the Fermi surface of the interacting

system. To achieve this, K must be chosen a function

of e, k, and V. In contrast to the situation for

covariances with point singularities, the function K

will, for a nontrivial Fermi surface, be very different

from the original e. It can, however, be constructed to

all orders in perturbation theory for a large class of

Fermi surfaces. More precisely, one can prove: if e 2

, R),

v 2 C

, R), and the Fermi surface S

contains no points k with re(k) = 0 and no flat sides,

then K =



exists as a formal power series in 

and the map e 7!e þ K is locally injective on this set

of e’s (Feldman et al. 1996, 2000). With this counter-

term, the order-rm-point functions on scale j satisfy

the bounds

ðjÞ

m;r



 w

m;r

ð4mÞj=2

jjj

and

ðjÞ

m;r





m;r

½13

with constants w

m, r

and

m, r

. Here

(j)

m, r

is the

Fourier transform of v

(j)

m, r

(see [9], with the momen-

tum conservation delta function from translation

invariance removed.

Equation [13] implies that in the RG sense, the

two-point function is relevant, the four-point func-

tion is marginal, and all higher m-point functions

are irrelevant.

In one dimension, the Fermi ‘‘surface’’ reduces to

two points which are related by a symmetry, so the

counter-term function K is just a constant, that is, an

adjustment of the chemical potential , whi ch is

justified because  is only an auxiliary parameter

used to fix the average value of the particle number.

The counter-term function is a constant also in

higher dimensions in the special case e(k) = k

 :

there, rotational symmetry implies that K can be

chosen independent of k (if v is also rotationally

symmetric). However, in the generic case of non-

spherical Fermi surfaces, K depends nontrivially

on k, and an inversion problem arises: adding the

counter-term changes the model. To obtain the

Green functions of a model with a given dispersion

relation and interaction (E, V ), one needs to show

that given E in a suitable set, the equation

eðkÞþKð; e; VÞðkÞ¼EðkÞ½14

has a unique solution. If this is done, the procedure

for renormalization is as follows. For a model given

by dispersion relation and interaction (E, V), solve

[14], then add and subtract e in the kinetic term.

This automatically puts K = E  e as a counter-term,

and the expansion is now set up automatically with

the right cou nter-term. The function K describes

the shift from the Fermi surface of the free system (the

zero set of E) to that of the interacting system

(the zero set of e). Proving that K is sufficiently

regular and solving [14] is nontrivial. Uniqueness of

the solution follows from the above stated properties

of K as a function of e. Existence was shown for a

class of Fermi surfaces with strictly positive curva-

ture in Feldman et al. (1996, 2000), to every order

in perturbation theory. This implies a bijective

relation between the Fermi surfaces of the free and

the interacting model.

Positive temperature and the zero-limit temperature

One advantage of the functional-integral approach

is that the setup at positive temperatures is identical

to that at zero temperature, save for the discreteness

of the set M

at T > 0. Because 0 62 M

, the

temperature effectively pro vides an IR cutoff, so

that all term-by-term divergences are regularized in

a natural way. However, renormalization is still

necessary because the temperature is a physical

parameter and unrenormalized expansions give

disastrous bounds for the behavior of observables

as functions of the temperature. Renormalization

carries over essentially unchanged (the counter-term

function is constructed slightly differently).

Because j!j= for all ! 2 M

, [12] implies

supp 

= ; for j < J



, where



¼ log





½15

Thus, the scale decomposition is now a finite sum

over 0  j J



. This restriction is inessential for

the problem of renormalizing the Fermi surface, but

it puts a cutoff on the marginal growth of the four-

point function: [15] and [13] imply that

ðjÞ

m;r



m;r

log







½16

If one can show that

m, r

 AB

with constants A

and B, this implies that perturbation theory con-

verges for jjlog (

=) < B

1

. Such a bound has

been shown using constructive methods (Disertori

and Rivasseau 2000, Feldman et al. 2003, 2004) (see

below). The logarithm of  is due to the Cooper

instability (see Feldman and Trubowitz (1990,

1991) and Salmhofer (1998, section 4.5)).

412 Renormalization: Statistical Mechanics and Condensed Matter

The application of renormalization at positive

temperature also led to the solution of a longstanding

puzzle in solid-state physics, namely the (seeming)

discontinuity of the results of perturbation theory as a

function of the temperature claimed in the early

literature. When renormalization is done correctly,

there is no discontinuity in the temperature.

Nonperturbative Renormalization for Fermions

It is a rem arkable feature of fermio nic field theories

that for a covariance for which k

and kCk

are

both finite, the effective action defined in [7] exists

and is analyt ic in the fields and in the original

interaction V, thanks to determinant bounds. For a

V as in [6], with v weak and of short range, the

skeleton functions (where all relevant m-point

functions are projected back to their initial values

in the RG iteration) satisfy

ðjÞ

 const: k

ðm=2Þþ1

1

½17

For the many-electron covariance [10], with a

positively curved C

Fermi surface and with the

scale decomposition [12], k

is of order M

and

is of order M

j(dþ1)=2

. The right-hand side of

[17] then contains M

(dþ3m)j=2

, which agrees (up to

logarithms) with the perturbative power-counting

bounds [13] only for d = 1. In dimension d = 2, the

method has been refined by dividing the Fermi

surface into angular sectors. The corresponding

sectorized propagat ors have a better decay bound

, but the trade-off is sector sums at every

vertex. Momentum conservation restricts these

sector sums sufficiently in two dimensions to allow

for good power-counting bounds. This has allowed

for the construction of an interes ting class of

interacting fermionic models.

The major results obtained with the RG method

are as follows.

Luttinger liquid behavior at zero temperature was

proved for one-dimensional models with a repulsive

interaction (Benfatto and Gallavotti 1995).

Fermi liquid behavior in the region where

jjlog (

)  1 was proved for the two-

dimensional model with e(k) = k

 1, a local poten-

tial V,andaUVcutoffbothonk and the Matsubara

frequencies ! in Disertori and Rivasseau (2000).

A two-dimensional model with a band function

e(k) that is nonsymmetric under k !k and a

general short-range interaction was proved to be a

Fermi liquid at zero temperature (Feldman et al.

2003, 2004). Due to the asymmetry under k !k,

the Cooper instability can be proved to be absent. In

Feldman et al. (2003, 2004), a counter-term func-

tion as in Feldman et al. (1996, 2000) was used. The

nonperturbative proof of the corresponding inver-

sion theorem remains open.

In d = 3, the proof of Fermi liquid behavior remains

an open problem, despite some partial results.

Condensed Matter: Bosons

Recent advances in quantum optice, in particular the

trapping of ultraco ld atoms, have led to the

experimental realization of Bose–Einstein condensa-

tion (BEC), which caused a surge of theoretical and

mathematical works. For bosons, the definition of

the ensembles is similar to, but more involved than

in, the fermionic case. On a formal level, the

functional-integral representation is analogous to

fermions, except that the fields are not Grassmann

fields but complex fields, and the covariance is given

by a sum as in [10], but now the summation over !

runs over the bosonic Matsubara frequencies

= 2 TZ. The existence of even the free partition

function in finite volume restricts the chemical

potential (for free particles, <inf

"(k) must

hold). Note that C is complex and Gaussian

measures with complex covariances exist in infinite

dimensions only under rather restricted conditions,

which are not satisfied by [10]. This is inessential for

perturbative studi es, where everything can be

reduced to finite-dimensional integrals involving

the covariance, but a nonperturbative definition of

functional integrals for such systems requires again a

carefully regularized (e.g., discrete-time) definition

of the functional integral.

Bose–Einstein Condensation

The problem was treated to all orders in perturba-

tion theory at positive particle density >0by

Benfatto (Benfatto and Gallavotti 1995). The initial

interaction is again quartic, "(k) = k

, and one

considers the problem at zero temperature, in the

limit  !0



, which is the limit in which BEC occurs

for free pa rticles. The interaction is expected to

change the value of , given the density, so a

chemical potential term is included in the action, to

give the interaction

VðÞ¼

ddxdyjð; xÞj

vðx  yÞjð; yÞj

þ 

ddxjð; xÞj

½18

After writing (, x) =  þ ’(, x), where  is indepen-

dent of  and x, the density condition becomes

 = jj

.  now needs to be chosen such that the

free energy has a minimum at  =

ﬃﬃﬃ



.Thiscanbe

reformulated in terms of the self-energy of the boson.

Renormalization: Statistical Mechanics and Condensed Matter 413

Benfatto uses the RG to prove that the propagator of

the interacting system no longer has the singularity

structure (i!  k

)

1

but instead (!

þ c

)

1

, where

c is a constant. This requires a nontrivial analysis of

Ward identities in the RG flow.

BEC has been proved in the Gross–Pitaevskii limit

(Lieb et al. 2002). In the present formulation, this limit

corresponds to an infinite-volume limit L !1where

the density  is taken to zero as an inverse power of L.

A nonperturbative proof of BEC at fixed positive

particle density remains an open problem.

Superconductivity

Superconductivity (SC) occurs in fermionic systems,

but it happens at energy scales where the relevant

excitations have bosonic character: the Cooper pairs

are bosons. In the RG framework, they arise naturally

when the fermionic RG flow discussed above is

stopped before it leaves the weak-coupling region

and the dominant Cooper pairing term is rewritten by

a Hubbard–Stratonovich transformation. The fer-

mions can then be integrated over, resulting in the

typical Mexican hat potential of an O(2) nonlinear

sigma model. Effectively, one now has to deal with a

problem similar to the one for BEC, but the action is

considerably more complicated.

The Nonlinear Sigma Models

The prototypical model, into whose universality

class both examples mentioned ab ove fall, is that

of O(N) nonlinear sigma models: both BEC and SC

can be reformulated as spontaneous symmetry

breaking (SSB) in the O(2) model in dimensions

d  3. For d = 2, long-range order is possible only at

zero temperature because only then does the time

direction truly represent a third dimension, prevent-

ing the Mermin–Wagner theorem from applying.

SSB has been proved for lattice O(N) models by

reflection positivity and Gaussian domination meth-

ods (Fro¨ hlich et al. 1976). The elegance and

simplicity of this method is unsurpassed, but only

very special actions satisfy reflection positivity, so

that the method cannot be used for the effective

actions obtained in condensed matter models.

Results in the direction of proving SSB in O(N)

models for d  3 by RG methods, which apply to

much more general actions, have been obtained by

Balaban (1995).

See also: Bose–Einstein Condensates; Fermionic

Systems; High T

Superconductor Theory; Holomorphic

Dynamics; Operator Product Expansion in Quantum

Field Theory; Perturbative Renormalization Theory and

BRST; Phase Transition Dynamics; Reflection Positivity

and Phase Transitions.

Further Reading

Bach V, Fro¨ hlich J, and Sigal IM (1998) Renormalization group

analysis of spectral problems in quantum field theory.

Advances in Mathematics 137: 205–298.

Balaban T (1988) Convergent renormalization expansions for

lattice gauge theories. Communications in Mathematical

Physics 119: 243–285.

Balaban T (1995) A low-temperature expansion for classical

N-vector models I. A renormalization group flow. Commu-

nications in Mathematical Physics 167: 103–154.

Benfatto G and Gallavotti G (1995) Renormalization Group.

Princeton: Princeton University Press.

Bricmont J and Kupiainen A (2001) Renormalizing the renorma-

lization group pathologies. Physics Reports 348: 5–31.

Brydges DC and Kennedy T (1987) Mayer expansions and the

Hamilton–Jacobi equation. Journal of Statistical Physics 48:

19–49.

Disertori M and Rivasseau V (2000) Interacting Fermi liquid in

two dimensions at finite temperature I. Convergent contribu-

tions. Communications in Mathematical Physics 215: 251–290.

Domb C and Green M (eds.) (1976) Phase Transitions and

Critical Phenomena, vol. 6. London: Academic Press.

Feldman J, Magnen J, Rivasseau V, and Se´ne´or R (1987)

Construction of infrared 

by a phase space expansion.

Communications in Mathematical Physics 109: 437.

Feldman J and Trubowitz E (1990) Perturbation theory for

many-fermion systems. Helvetica Physica Acta 63: 157.

Feldman J and Trubowitz E (1991) The flow of an electron-

phonon system to the superconducting state. Helvetica

Physica Acta 64: 213.

Feldman J, Salmhofer M, and Trubowitz E (1996) Perturbation

theory around non-nested Fermi surfaces I. Keeping the Fermi

surface fixed. Journal of Statistical Physics 84: 1209–1336.

Feldman J, Salmhofer M, and Trubowitz E (2000) An inversion

theorem in Fermi surface theory. Communications on Pure

and Applied Mathematics 53: 1350–1384.

Feldman J, Kno¨rrer H, and Trubowitz E (2003) A class of Fermi

liquids. Reviews in Mathematical Physics 15: 949–1169.

Feldman J, Kno¨ rrer H, and Trubowitz E (2004) Communications

in Mathematical Physics 247: 1–319.

Fro¨ hlich J, Simon B, and Spencer T (1976) Infrared bounds, phase

transitions, and continuous symmetry breaking. Communica-

tions in Mathematical Physics 50: 79.

Gallavotti G (1985) Renormalization theory and ultraviolet

stability via renormalization group methods. Reviews of

Modern Physics 57: 471–569.

Gawedzki K and Kupiainen A (1985) Massless lattice 

theory:

Rigorous control of a renormalizable asymptotically free model.

Communications in Mathematical Physics 99: 197–252.

Lieb E, Seiringer R, Solovej JP, and Yngvason J (2002) The ground

state of the Bose gas. In: Current Developments in Mathematics,

2001, pp. 131–178. Cambridge: International Press.

Polchinski J (1984) Renormalization and effective Lagrangians.

Nuclear Physics B 231: 269.

Rivasseau V (1993) From Perturbative to Constructive Renorma-

lization. Princeton, NJ: Princeton University Press.

Salmhofer M (1998) Renormalization: An Introduction, Springer

Texts and Monographs in Physics. Heidelberg: Springer.

Wiese KJ (2001) Polymerized membranes, a review. In: Domb C

and Lebowitz J (eds.) Phase Transitions and Critical Phenom-

ena, vol. 19. Academic Press.

414 Renormalization: Statistical Mechanics and Condensed Matter

Resonances

N Burq, Universite

Paris-Sud, Orsay, France

Introduction

In quantum mechanics and wave propagation,

eigenvalues (and eigenfunctions) appear naturally

as they describe the behavior of a quantum

system (or the vibration of a structure). There

are however some cases where these simple

notions do not suffice and one has to appeal to

the more subtle notion of resonances. For

example, if the vibration of a drum is well

understood in terms of eigenvalues (the audible

frequencies) and eigenfunctions (the correspond-

ing vibrating modes), the notion of resonances is

necessary to understand the propagation of waves

in the exterior of a bounded obstacle. Another

example (taken from Zworski (2002))which

allows us to understand both the similarities of

resonances with eigenvalues and their differences

is the following: consider the motion of a

classical particle submitted to a force field

deriving from the potential V

(x) on a bounded

interval as shown in Figure 1a. If the classical

momentum is denoted by , then the classical

energy is given by

E ¼jj

þ V

ðxÞ

and the classical motion is given by the relations of

Hamiltonian mechanics:

x ¼

@

¼ 2;

 ¼

¼V

ðxÞ

Since energy is conserved, if the initial energy is

smaller than the top of the barrier, then the classical

particle bounces forever in the well. Now we can

consider the same example with the potential V

(x)

on R as shown in Figure 1b. Of course, if the

particle is initially inside the well (with the same

energy as before), the classical motion remains the

same.

On the quantum mechanics point of view, both

systems are described by the Hamiltonians

¼h

þ V

ðxÞ

acting on L

([1, 1]) (with boundary conditions) and

(R), respectively. In the first case, H

has a discrete

spectrum, 

j, h

2 R with eigenfunctions e

j, h

(x), j 2 N,

and the time evolution of the system is given by

itH

u ¼

it

j;h

 e

j;h

½1

where u

j, h

 e

j, h

is the orthogonal projection of u on

the eigenspace Ce

j, h

. In the second case, H

has no

square integrable eigenfunction, and no simple

description as [1] can consequently hold. However

as h !0, the correspondence principle tells us that

quantum mechanics should get close to classical

mechanics. Since for both quantum problems the

classical limit is the same (at least for initial states

confined in the well with energy E ), we expect that

for the second potential there should exist a

quantum state corresponding to the classical one.

In fact, this is indeed the case and one can show that

there exist resonant states e

j, h

associated to reso-

nances E

j, h

which are solution of the equation

j;h

 E

j;h

; E

j;h

 E

are not square integrable, but still have moderate

growth at infinity and are confined in the interior of

the well (see sections ‘‘Definition’’ and ‘‘Location of

resonances’’). On the other hand, the first quantum

system is confined, whereas the second one is not and

we know that even for initial states confined in the

well, tunneling effect allows the quantum particle to

escape to infinity. This fact should be described by

the theory as a main difference between eigenvalues

and resonances. This is indeed the case as the

resonances E

j, h

are not real (contrarily to eigenvalues

of self-adjoint operators) but have a nonvanishing

imaginary part (see section ‘‘Resonance-free regions’’)

Im E

j;h

 e

C=h

If we assume that a similar description as [1] still

holds for the second system, at least locally in space

(see section ‘‘Resonances and time asymptotics’’),

then, for time t >> e

C=h

, the factor e

itE

becomes

very small (the quantum particle has left the well

due to tunneling effect).

There have been several studies on resonances and

scattering theory and the presentation here cannot be

complete. For a more in-depth presentation, one can

(b)

(a)

Figure 1a, b A particle trapped in a well.

Resonances 415

consult the books by Lax and Phillips (1989) and

Hislop and Sigal (1987), or the reviews on resonances

by Vodev (2001) and Zworski (1994) for example.

Definition

There are different (equivalent) definitions of reso-

nances. The most elegant is certainly the Helffer and

Sjo¨ strand (1986) definition (see also the presentation

of complex scaling by Combes et al. (1984) and the

very general ‘‘black box’’ framework by Sjo¨strandand

Zworski (1991)). However, it requires a few prerequi-

sites and we preferred to stick to the more elementary

(but less general) resolvent point of view. The starting

point for this definition of resonances is the fact that

the eigenvalues of a (self-adjoint) operator P are the

points where P is not injective. The more general

resonances will be the points where the operator is not

invertible (on suitable spaces).

More precisely, consider a perturbation of the

Laplace operator on R

, P

(h) = h

 in the following

sense: let   R

be a (possibly empty) smooth obstacle

whose complementary, =

, is connected. Consider

a classical self-adjoint operator defined on L

():

u ¼ðh

 þ VðxÞÞu ½2

with boundary conditions (Dirichlet)

u j

@

¼ 0 ½3

(Neumann boundary conditions could be used too).

This setting contains both the Schro¨ dinger operator

= h

 þ V(x)  E on =R

) and the Helmoltz

equation with Dirichlet conditions, in the exterior of

an obstacle (waves at large frequencies: P =   

;

in this case, define h = 

1

and P

= h

), which we

shall define as acoustical scattering.

We assume that P is a perturbation of P

, that is,

V !0, jxj!þ1 sufficiently fast (see Sjo¨ strand and

Zworski (1991) for the very general black box

assumptions). For example, this perturbation

assumption is fulfilled if V has compact support.

Then the resolvent P

(z) = (P

 z)

1

is well defined

for Im z 6¼ 0 as a bounded operator from L

()to

ðÞ\H

ðÞ

(because the operator P

is self-adjoint). However, it

is not bounded for z > 0onL

() because the

essential spectrum of P

is precisely the semiaxis z >

0, but it admits a meromorphic continuation from

Im z > 0 toward the lower half-plane:

ðzÞ : L

ðÞ

comp

! L

ðÞ

loc

The poles of this resolvent R

are by definition the

semiclassical resonances, Res

Remark 1 In the case of acoustical scattering

(P =   

,  = h

1

), the introduction of the addi-

tional parameter z is pointless and one works

directly with the parameter  = h

1

ﬃﬃﬃ

. In that case

the resolvent R()(  

)

1

is well defined for

Im <0, the essential spectrum is precisely the axis

 2 R and the resolvent admits a meromorphic

continuation from Im z < 0 toward the upper half-

plane (with possibly a cut at 0):

RðÞ : L

ðÞ

comp

! L

ðÞ

loc

The acoustic resonances are by definition the poles

of this meromorphic continuation. They are related

to semiclassical resonances by the relation

Res

¼ h

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Res

It can also be shown that if z is a resonance, there

exists an associated resonant state e

such that

ðP

 zÞe

¼ 0

the function e

satisfies Sommerfeld radiation con-

ditions (in polar coordinates (r, ) 2 [0, þ1) S

n1

)

jh@

e  i

ﬃﬃﬃ

ejCje

ﬃﬃ

j=r

1þn=2

and the function

1 þ r

ð1=2Þþ

ﬃﬃ

is square integrable.

Resonance-Free Regions

The very first result about resonance-free regions is

based on Rellich uniqueness theorem (uniqueness for

solutions of elliptic second-order equations) and says

that there are no real resonances (except possibly 0).

The more precise determination of resonance-free

regions (originally in acoustical scattering) has been a

subject of study from the 1960s and it has motivated a

large range of works from the multiplier methods of

Morawetz (1975) to the general propagation of

singularity theorem of Melrose and Sjo¨strand (1978).

To state the main result in this direction, we need the

notion of nontrapping perturbation.

Definition 1 A generalized bicharacteristic at energy

E(x(s), (s)) is an integral curve of the Hamiltonian field

@



@

of the principal symbol p(x, ) = jj

þV(x) of the

operator P, included in the characteristic set

p(x, ) = E and which, when hitting the boundary of

the obstacle, reflects according to the laws of

geometric optics (see (Melrose and Sjo¨ strand 1978)).

416 Resonances

The operator P (or by extension the obstacle in the

case of acoustic scattering) is said to be nontrapping

at energy E if all generalized bicharacteristics go to

the infinity:

lim

s!1

jxðsÞj ¼ þ1

The operator P (or by extension the obstacle in the

case of acoustic scattering) is said to be nontrapping

near energy E if P is nontrapping at energy E

for E

in a neighborhood of E.

The following result was obtained in different

generalities by Morawetz (1975), Melrose and

Sjo¨ strand (1978), and others.

Theorem 1 Assume that the operator P is nontrap-

ping near energy E. Then for any N > 0 there exist

> 0 such that for 0 < h < h

there are no

resonances in the set

fz; jIm zjNh logðhÞg

In the case of analytic geometries (and coefficients),

this result (see Bardos et al. 1987) can be improved to

Theorem 2 Assume that the operator P is non

trapping. Then there exist >0, N

> 0 and h

> 0

such that for 0 < h < h

there are no resonances in

the set

fz; jIm zjN

1ð1=3Þ

g\fjz  Ejg

Remark 2 In the case of acoustical scattering, with

the new definition of resonances,  = h

1

ﬃﬃﬃ

, the

resonance-free zones have respectively the forms

fz; jIm zjN logðjzjÞ; jzj >> 1g

fz; jIm zjN

jzj

1=3

; jzj >> 1g

In the case of trapping perturbations, the first result

was obtained by Burq (1998).

Theorem 3 There exist C > 0 and h

> 0 such that

for 0 < h < h

there are no resonances in the set

fz; jIm zjN

C=h

g\fjz  Ejg

Resonances and Time Asymptotics

The relationship between eigenfunctions/eigen-

values and time asymptotics is straightforward.

This is no longer the case for resonances. For

nontrapping problems however, this question has

been studied in the late 1960s by Lax and Phillips

(1989) and Vainberg (1968). In particular, this

approach was decisive to study the local energy

decay in acoustical scattering. As a consequence of

Theorem 1,wehave

Theorem 4 If the acoustical problem is nontrap-

ping, then there exist C, >0 such that for any

solution of the wave equation

&u ¼0; uj

t¼0

¼u

t¼0

¼u

; uj



¼0;



¼0

with compactly supported initial data (u

)(in a

fixed compact), one has

loc

ðuÞ

\fjxjC

jruj

þj@



t

if the space dimension is even

if the space dimension is odd

½4

Trapping perturbations were investigated more

recently. In that case, the local energy decays, but the

rate cannot be uniform. The first trapping example in

acoustic scattering was studied by Ikawa (1983):the

obstacle is the union of a finite number (and at least

two) convex bodies. In that case, one has

Theorem 5 For any >0 there exists C > 0 such

that for any initial data supported in a fixed

compact set

loc

ðuÞðt ÞC e

t

kðu

; u

Þk

Dðð1Þ

ð1þÞ=2

where D((1  )

(1þ)=2

) is the domain of the

operator (1  )

(1þ)=2

. Remark that the norm in

D((1  )

1=2

) is the natural energy and consequently

the estimate above exhibits a loss of  derivatives.

For strongly trapping perturbations, the results are

worse. They are consequences of Theorem 3.

Theorem 6 For any k there exists C

> 0 such that

for any initial data supported in a fixed compact set

loc

ðuÞðt Þ

logðtÞ

kðu

; u

Þk

Dðð1Þ

ð1þkÞ=2

One can also obtain real asymptotic expansions in

terms of resonances (see the work by Tang and

Zworski (2000)).

Theorem 7 Let  2 C

) and 2 C

((0, 1))

and let chsupp = [a, b]. There exists 0 <<

c(h) < 2 such that for every M > M

there exists

L = L(M), and we have

e

itðPÞ=h

 ðPÞ¼

z2ðhÞ\ResðPÞ

Resðe

itðÞ=h

Rð; hÞ; zÞ ðPÞ

þO

H!H

ðh

Þ; for t > h

L

ðhÞ¼ða cðhÞ; b þcðhÞÞi½0; h

½5

Resonances 417

where Res(f (  ), z) denotes the residue of a mer-

omorphic family of operators, f, at z.

The function c(h) depends on the distribution of

resonances: roughly speaking we cannot ‘‘cut’’

through a dense cloud of resonances. Even in the

very well understood case of the modular surface

there is, currently at least, a need for some

nonexplicit grouping of terms. The same ideas can

be applied to acoustic scattering.

Trace Formulas

Trace formulas provide a description of the classical/

quantum correspondence: one side is given by the trace

of a certain function of the operator f (P

), whereas the

other side is described in terms of classical objects

(closed orbits of the classical flow). In the case of

discrete eigenvalues, the question is relatively simple

and can be solved by using the spectral theorem. In the

case of continuous spectrum, the problem is much more

subtle (self-adjoint operators with continuous spectrum

behave in some ways as non-normal operators). It has

been studied by Lax and Phillips (1989), Bardos et al.

(1982),andMelrose (1982). More recently, Sjo¨strand

(1997) introduced a local notion of trace formulas.

Let W   be an open precompact subsets of

i[2

,0]

]0, þ1[. Assume that the intersections I

and J of W and  with the real axis are intervals and

that  is simply connected.

Theorem 8 Let f (z, h) be a family of holomorphic

functions on z 2  such that jfj

nW

j1. Let  2

(R) equal to 1 on a neighborhood of I. Then

Trace fðÞP

ðÞfðÞh





 a resonance of P

\

f ; hðÞþOh

n

ðÞ

The use of this result with a clever choice of functions f

allows Sjo¨ strand to show that an analytic singularity of

the function E 7!Vol({x; V(x)  E}) (observe that if V

is bounded, this function vanishes for large E and

consequently it has analytic singularities) gives a lower

bound for  a neighborhood of E

]ResðP

Þ\  ch

n

which coincides with the upper bound (see Zworski

(2002) and the references given there).

Location of Resonances

In some particular cases, one can expect to have a

precise description of the location of resonances.

This is the case in Ikawa’s example in acoustic

scattering where the obstacle is the union of two

disjoint convex bodies. In this case, the line

minimizing the distance, d, between the bodies is

trapped. However, this trapped trajectory is isolated

and of hyperbolic type (unstable). Ikawa (1983) and

Ge´rard (1988) have obtained:

Theorem 9 There exist geometric positive constants

!þ1 as p !þ1 such that all resonances

located above the line Im z C (C arbitrary large

but fixed) have an asymptotic expansion

  

j;p

l;p



l=2

j;p

þOð

1

j;p

Þ; j !þ1

where the approximate resonanc es



j;p

¼ j



 ik

are located on horizontal lines.

Another example is when the obstacle is convex.

This example is nontrapping and Sjo¨ strand and

Zworski (1999) are able to prove that the resonances

in any region Im z  Njzj

1=3

(N arbitrary large) are

asymptotically distributed near cubic curves

¼fz 2 C; Im z ¼c

jzj

1=3

Finally, the last main example where one can give a

precise asymptotic for resonances is when there

exists a stable (elliptic) periodic trajectory for the

Hamiltonian flow. In that case it had been known

from the 1960s (see the works by Babic

(1968)) that

one can construct quasimodes, that is, compactly

supported approximate solutions of the eigenfunc-

tions equation:

ðP

 E

Þe

¼Oðh

It is only recently that Tang and Zworski (1998) and

Stefanov (1999) proved that these quasimodes

constructions imply the existence of resonances

asymptotic to E

, h !0.

See also: h-Pseudodifferential Operators and

Applications; Semi-Classical Spectra and Closed Orbits.

Further Reading

Babic

VM (1968) Eigenfunctions which are concentrated in the

neighborhood of a closed geodesic. Zapiski Nauchnykh

Seminanov Leningradkogo Otdeleniya Matematicheskogo

Instituta Imeni V.A. Steklova 9: 15–63.

Bardos C, Guillot JC, and Ralston J (1982) La relation de Poisson

pour l’e´quation des ondes dans un ouvert non borne´.

Application a` la the´orie de la diffusion. Communications in

Partial Differential Equations 7: 905–958.

Bardos C, Lebeau G, and Rauch J (1987) Scattering frequencies and

Gevrey 3 singularities. Inventiones Mathematicae 90: 77–114.

Burq N (1998) De´croissance de l’e´nergie locale de l’e´quation des

ondes pour le proble` me exte´rieur et absence de re´sonance au

voisinage du re´el. Acta Mathematica 180: 1–29.

418 Resonances

Combes J-M, Duclos P, and Seiler R (1984) On the shape

resonance. In: Resonances – Models and Phenomena (Biele-

feld, 1984), Lecture Notes in Physics, vol. 211, pp. 64–77.

Berlin: Springer.

Ge´rard C (1988) Asymptotique des poˆ les de la matrice de scattering

pour deux obstacles strictement convexes. Supple´ment au

Bulletin de la Socie´te´ Mathe´matique de France 116: 146 pp.

Helffer B and Sjo¨ strand J (1986) Resonances en limite semi-

classique. Me´moire de la S.M.F 114(24–25): 228 pp.

Hislop PD and Sigal IM (1987) Shape resonances in quantum

mechanics. In: Differential Equations and Mathematical

Physics (Birmingham, Ala., 1986), Lecture Notes in Mathe-

matics, vol. 1285, pp. 180–196. Berlin: Springer.

Ikawa M (1983) On the poles of the scattering matrix for two

convex obstacles. Journal of Mathematics of the Kyoto

University 23: 127–194.

Lax PD and Phillips RS (1989) Scattering Theory, Pure and Applied

Mathematics, 2nd edn., vol. 26. Boston: Academic Press.

Melrose RB (1982) Scattering theory and the trace of the wave

group. Journal of Functional Analysis 45: 429–440.

Melrose RB and Sjo¨ strand J (1978) Singularities of boundary

value problems. I. Communications in Pure and Applied

Mathematics 31: 593–617.

Melrose RB and Sjo¨ strand J (1982) Singularities of boundary

value problems. II. Communications in Pure and Applied

Mathematics 35: 129–168.

Morawetz CS (1975) Decay for solutions of the exterior problem

for the wave equation. Communication in Pure Applied

Mathematics 28: 229–264.

Sjo¨ strand J (1997) A trace formula and review of some estimates

for resonances. In: Rodino L (ed.) Microlocal Analysis and

Spectral Theory (Lucca, 1996), NATO Adv. Sci. Inst. Ser. C

Math. Phys. Sci., vol. 490, pp. 377–437. Dordrecht: Kluwer

Academic.

Sjo¨ strand J and Zworski M (1991) Complex scaling and the

distribution of scattering poles. Journal of the American

Mathematical Society 4(4): 729–769.

Sjo¨ strand J and Zworski M (1999) Asymptotic distribution of

resonances for convex obstacles. Acta Mathematica 183(2):

191–253.

Stefanov P (1999) Quasimodes and resonances: sharp lower

bounds. Duke Mathematical Journal 99(1): 75–92.

Tang S-H and Zworski M (1998) From quasimodes to reason-

ances. Mathematical Research Letters 5(3): 261–272.

Tang SH and Zworski M (2000) Resonance expansions of

scattered waves. Communication in Pure and Applied Mathe-

matics 53(10): 1305–1334.

Vainberg BR (1968) On the analytical properties of the resolvent

for a certain class of operator pencils. Mathematics of the

USSR Sbornik 6(2): 241–273.

Vodev G (2001) Resonances in euclidean scattering. Cubo Mate-

matica Educacional 3: 317–360. http://www.math.sciences.

univ-nantes.fr.

Zworski M (1994) Counting scattering poles. In: Ikawa M (ed.)

Spectral and Scattering Theory (Sanda, 1992), Lecture Notes

in Pure and Applied Mathematics, vol. 161, pp. 301–331.

New York: Dekker.

Zworski M (2002) Quantum resonances and partial differential

equations. In: Li Ta Tsien (ed.) Proceedings of the Interna-

tional Congress of Mathematicians (Beijing, 2002), vol. III,

pp. 243–252. Beijing: Higher Education Press.

Ricci Flow see Singularities of the Ricci Flow

Riemann Surfaces

K Hulek, Universita

t Hannover, Hannover, Germany

Introduction

Riemann surfaces were first studied as the natural

domain of definition of (multivalued) holomorphic

or meromorphic functions. They were the starting

point for the development of the theory of

real and complex manifolds (see Weyl (1997)).

Nowadays, Riemann surfaces are simply defined

as one-dimensional complex manifolds (see the

next section). Compact Riemann surfaces can

be embedded into projective spaces and are thus,

by virtue of Chow’s theorem, algebraic curves. By

uniformization theory, the universal cover of

a connected Riemann surface is either the unit

disk, the complex plane, or the Riemann sphere

(see the section ‘‘Uniformization’’).

This article discusses the basic theory of compact

Riemann surfaces, such as their topology, their

periods, and the definition of the Jacobian variety.

Studying the zeros and poles of meromorphic

functions leads to the notion of divisors and linear

systems. In modern language this can be rephrased

in terms of line bundles, resp. locally free sheaves

(see the section ‘‘Divisors, linear systems, and line

bundles’’). One of the fundamental results is the

Riemann–Roch theorem which expresses the

difference between the dimension of a linear system

and that of its adjoint system in terms of the degree

of the linear system and the genus of the curve. This

theorem has been vastly generalized and is truly one

of the cornerstones of algebraic geometry.

A formulation of this result and a discussion of

some of its applications are also discussed.

Riemann Surfaces 419

A study of the subsets of the Jacob ians parame-

trizing linear systems of given degree and dimension

leads to Brill–Noether theory, which is discussed in

the section ‘‘Brill–N oether theory .’’ This is follow ed

by a brief introduction to the theory of equations

and syzygies of canonical curves .

Moduli spaces play a central role in the theory

of com plex variables and in algebraic geometry.

Arguably, the most important of these is the

moduli space of curves of genus g. This and

related moduli problems are treated in the section

‘‘Moduli of compact Riemann s urfaces.’’ I n parti-

cular, the space of stable maps is closely related to

quantum cohomology. Finally, we present a brief

discussion of the Verlinde formula and conformal

blocks.

Basic Definitions

Riemann surfaces are one-dimensional complex

manifolds. An n-dimensional c omplex manifold

M is a topological Hausdorff space (i.e., for any

two points x 6¼ y on M, there are disjoint open

neighborhoods containing x and y), which h as a

countable basis for its t opology, together with a

complex atlas A. The latter is an open covering



)

2A

together with homeomorphisms f



: U



 C

,wheretheU



are open subsets of M and

the V



are open sets in C

. The main requirement

is that these charts are holomorphically compati-

ble, that is, for U



\ U



6¼;, the map shown in

Figure 1,



 f

1



ðU





: f



ðU



\ U



Þ!f



ðU



\ U



ÞC

is biholomorphic. A map h : M ! N between two

complex manifolds is holomorphic if it is so with

respect to the local charts. This means the following:

for each point x 2 M, there are charts



: U



 C

near x and f



: U



! V





near h(x) with h(U



)  U



such that the map

shown in Figure 2



 h ðf



1

: V



! V



 C

is holomorphic (one checks easily that this does not

depend on the choice of the charts).

A Riemann surface i s a one-dimensional com-

plex manifold. Trivial examples are given by open

sets in C (where one chart suffices). Another

example is the Riemann sphere

C = C [ {1},

which can be covered by the two charts given by

z 6¼1and z 6¼ 0. Both of these charts are home-

omorphic to C with the transition function given

by z 7!1=z. Historically, Riemann surfaces were

viewed as (branched) coverings of C or of the

sphere, where they appear as the natural domain

of definition of multivalued holomorphic or

meromorphic functions.

Uniformization

If M is a Riemann surface, then its universal

covering

M is again a Riemann surface. The

connected and simply connected Riemann surfaces

can be fully classified. Let

E ¼fz 2 C; jzj < 1g

be the unit disk and

C = C [ {1} the Riemann

sphere. The latter can be identified with the complex

projective line P

Theorem 1 (Generalized Riemann mapping

theorem). Every connected and simply connected

Riemann surface is biholomorphically equivalent

⊂

–1

Figure 1 Charts of a complex manifold.

h(x)

β °

)

–1

⊂ C

Figure 2 Holomorphic map between manifolds.

420 Riemann Surfaces

to the unit disk E, the complex plane C, or the

Riemann sphere

This theore m was proved rigor ously by Koebe

and Poincare´ at the beginning of the twentieth

century.

Compact Riemann Surfaces

The topological structure of a compact Riemann

surface C is determined by its genus g (Figure 3).

Topologically, a Riemann surface of genus g is a

sphere with g handles or, equivalently, a torus with

g holes.

Analytically, the genus can be characterized as the

maximal number of linearly independent holo-

morphi c form s on C (see also the section ‘‘The

Riem ann–Roc h theore m an d ap plications’’).

There exists a very close link with algebraic

geometry: every compact Riemann surface C can

be embedded into some projective space P

(in

fact already into P

). By Chow’s theorem, C is

then a (projective) a lgebraic variety, that is, it can

be described by finitely many homogeneous equa-

tions. It should be noted that such a phenomenon

is special to complex dimension 1. The crucial

point is that one can always construct a non-

constant meromorphic function on a Riemann

surface ( e.g., by Dirichlet’s principle). Given such

a function, it is not difficult to find a projective

embedding of a compact Riemann surface C.On

the other hand, it is easy to construct a compact

two-dimensional torus T = C

=L for some suitably

chosen lattice L, which cannot be embedded into

any projective space P

The dichotomy Riemann surface/algebraic curve

arises from different points of view: analysts think

of a real two-dimensional surface with a Rieman-

nian metric which, via isothermal coordinates,

defines a holomorphic structure, whereas algebraic

geometers think of a complex one-dimensional

object.

In this article, the expressions compact Riemann

surface and (projective) algebraic curve are both

used interchangeably. The choice depends on

which expression is more commonly used in the

part of the theory w hich is discussed in the

relevant section.

Periods and the Jacobian

On a compact Riem ann surface C of genus g, there

exist 2g homologically indepe ndent paths, that is,

(C, Z) ﬃ Z

Let 

, ...,

be a basis of H

(C, Z)and

let !

, ..., !

be a basis of the space of holom orphic

1-forms on C. Integrating these forms over the paths



, ..., 

defines the period matrix

 ¼











If Q = (

, 

) is the intersection matrix of the paths



, ..., 

, then  satisfies the Riemann bilinear

relations

Q

¼ 0;

ﬃﬃﬃﬃﬃﬃﬃ

1

Q

> 0 ½1

where the latter condition means positive definite.

One can choose (see Figure 4) 

, ..., 

such that

Q ¼ J ¼

0 1

1



where 1

is the g  g unit matrix. Moreover,

, ..., !

can be chosen such that

 ¼

1  0 

 

0  1 

 

Let



¼ð

1i; jg

Then the Riemann bilinear relations [1] become



¼ 

; Im 

> 0

that is, 

is an element of the Sieg el upper half-

space

¼  2 Matðg  g; CÞ;  ¼ 

; Im >0

The matrix 

is defined by the Riemann surface C

only up to the action of the symplectic group

Spð2g; Z Þ¼ M 2 Matð2g  2g; ZÞ; MJM

¼ J

g = 0 g = 1 genus g

Figure 3 Genus of Riemann surfaces.

Figure 4 Homology of a compact Riemann surface.

Riemann Surfaces 421