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

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determine adequate sets of integral equations allow-

ing one, together with regularity assumptions, to

carry out an analysis similar to above.

The Models

A Euclidean field-theoretical model can be defined

by a probability measure d(’) on the space of

tempered distributions ’ in Euclidean spacetime,

whose moments verify the Osterwalder–Schrader (or

similar) axioms. The moments of d are, for each N,

the Euclidean (Schwinger) N-point functions:

Sðx

; ...; x

Þ¼

’ðx

Þ’ðx

Þdð’Þ½5

In what follows, the meas ure d will be a

perturbed Gaussian measure which, for the massive

’

model with a volume cutoff  and an ultraviolet

cutoff , is given in d dimensions by

d

;

¼ e

ðÞ



’

ðzÞdzþaðÞ



’

ðzÞdz

d



ð’Þ=Z

;

½6

where Z

, 

is the normalization factor and where

d



(’) is the Gaussian measure of mean zero

(

’ d = 0) and covariance

Cðx  y; Þ¼

p e

ipðxyÞ

p

=

=ððÞp

þ m

where by convention m is called the bare mass.

For d = 2 or 3 one can show that, for () = 

small enough (depending on m)and() = 1, there

exists a function a()(a() = O()as !0) such

that, for any set of N distinct points, the function

S(x

, ..., x

) = lim

,  !1

, 

, ..., x

)exists,is

not Gaussian (hence does not correspond to a trivial,

free theory), and satisfies the Osterwalder–Schrader

axioms. The connected part S(x

, ..., x

)

connected

has

the following perturbative series:

lim

;!1

ð1Þ

’ðx

Þ...’ðx





½’

 aðÞ ’

ðzÞdz



d

;

ð’Þ



connected

½7

which is the (divergent) sum of the connected

renormalized (Euclidean) Feynman graphs.

The study of the perturbative series leads to the

distinction of:

1. the super-renormalizable theories, where it is

possible to take (), () not depending on .

In dimension 2, all the models where ’

replaced by

’

þc

2p1

’

2p1

þþc

’

þ’

þc

’

½8

also exist provided that c

> 0 is small enough

depending on m and on the other coefficient c’s

and ,and

2. the just renormalizabl e theories where () (and

possibly ()) depend in general on . In models

mentioned below () !0as !1; this char-

acterizes ‘‘asymptotic freedom.’’

The proof of the existence of the N-point

functions makes use of Taylor type expansions

with remainde r. The first orders are used to compute

(), (), a(). The idea is to consider the functional

integral [5] –at,  finite – as an integral over

roughly 

‘‘degrees of freedom’’ which are weakly

coupled. This corresponds to a decomposition of the

phase space (with cutoff both in x-space (the box )

and in p-space (roughly jpj <)). The coupling

between different regions in x-space comes from

the propagators C



; the coupling between different

frequencies in p-space comes from the ’

term (the

interaction vertex). The expansion is then, for each

degree of freedom, a finite expansion in the coupling

between this degree and the others so that, even if

the expansion is perturbative up to the order 

the bound on each term is qualitatively the one on a

product of 

finite order-independent expansions,

the order of which can be fixed uniformly in  (and

depending only on ). To achieve this program, the

propagator linking two points of distance of order L

must have a decrease of order e

L

1

jxyj

, that is, have

momentum larger than L

1

, so that one must

localize both in x-space and p-space ; for example,

the smallest cells of phase space correspond to fields

’ localized in x, p-spaces, the x-boxes being of side



1

and the p-localizat ion consisting of values such

that roughly (=2) jpj. More generally, a

generic cell (of index i) corresponds to fields ’ at

point x and momentum p, with x in a box of side



1

and 2

i1

<jpj < 2

i

.

These expansions are mimicking the a`laWilson

renormalization group. For just renormalizable theo-

ries (where () depends on ), one is led to introduce

the effective coupling constant (2

i

) whose pertur-

bative expansion is the value at momentum zero of

the sum of all the (connected, amputated) four-point

functions containing only propagators of momentum

(roughly) bigger than 2

i

 (plus () which in fact

tends to zero as  !1).

Then by small coupling we mean a theory where

(2

i

)=(2

i

)

is small for all i.

By convention we write 

ren

, 

ren

, a

ren

for the

effective parameters of the theory at zero

momentum.

The expansion obtained expresses S

connected

as a

sum of terms each of them being associated to a

482 Scattering, Asymptotic Completeness and Bound States

given set of phase-space cells which are ‘‘connected’’

together by ‘‘links’’ that are either propagators or

vertices. Each term decreases exponentially with the

difference i

max

 i

min

of the upper and lower indices

of the phase-space cells involved. Moreover, each set

must contain the cells associated to the fields

’(x

) ’(x

) whose indices are fixed by the order

of magnitude of the distances between the points.

On the other hand, the difference between the

theory of cutoff  and the one of cutoff 2 are

terms containing at least one cell of momentum of

order ; these terms are thus small like

cst(x

, ..., x

(cst)

, so that the limit as  !1

exists.

So far, the ‘‘construction’’ of models is possible

only at small coupling, apart from special cases. The

’

theory in dimension 4 is just renormalizable

(from the perturbative viewpoint) but the above

condition of small coupling cannot be achieved (and

it is generally believed that this model cannot be

defined as a nontrivial theory). A just renormaliz-

able model has been shown to exist, namely the

Gross–Neveu model which is a fermionic theory in

dimension 2. The elementary particl e phy sics models

are just renormalizable but their construction has

not been completed so far (in particular in view

of the confinement problem). See Constructive

Quantum Field Theory for details.

To state the result in a form convenient for our

purposes here, we introduce a splitting of the

covariance in two parts:

Cðx y; Þ¼C

ðx y ; ÞþC

ðx  y; Þ; M > m

ðp; Þ¼ðe

p

=

þm

Þðe

p

=

þM

so that C

(x  y) behaves like C at large distances but

has an ultraviolet cutoff of size M,andjC

(x  y)j

Mjxyj

decreases exponentially depending on the

(technical) choice of M. Let d

(’) be the Gaussian

measure of covariance C

One divid es also  in unit cubes and obtains for

the connected N-point function an ex pansion as a

sum over connected trees; a tree T is composed of

lines ‘ and vertices v; each line joins two vertices or

one of the external points x

, ..., x

and a vertex;

moreover, there are no loops.

To each line ‘ is associated a propagator

‘

, z

‘

) = C

(‘).

To each vertex v are associated:

1. two subsets I

, I

of {‘},

2. a connected set X

of unit cubes such that all the

‘

, ‘ 2 I

and all the z

‘

, ‘ 2 I

are contained in

; jX

j is the volume of X

, and

3. a kernel K

({z, z

}

; ’)

Finally, the external points are by convention z

‘

points; then:

;

ðx

; ...; x

connected

d

ð’Þ

jTj!

nonoverlapping





‘

not external

‘



‘2T

‘

ð‘Þ





v2T

ðfz; z

; ’Þ½9

where for coupling small enough:

d

ð’Þ

v2T

ðfz; z

; ’Þj 

v2T

Mð1ÞjX

½10

The X’s are 2 2 nonoverlapping; however, it will

suffice to sum ov er all X’s (without restriction) to

get a bound showing the convergence of the

expansion as  !1. In this formula the K(. ,’)’s

are still coupled by the measure d

(’); all the

nonperturbativity is hidden in the K’s (in particular

the contribution of momentum bigger than M).

As a consequence of [9] and if a(, ) has been

chosen such that a

ren

= 0, for M large enough and at

small coupling (depending on M, m):

jSðx; yÞ

connected

jC

ðx  yÞj þ

ðx  z

 e

Mð1Þjz

z

ðz

 yÞjþ

ðcstÞe

mð1Þjxyj

½11

More generally, the connected N-point function

satisfies

jSðx

; ...; x

connected

jcst e

mð1Þdðx

;...;x

½12

where d(x

, ..., x

) is the length of the smallest tree

joining x

, ..., x

, with possibly intermediate points.

The Irreducible Kernels

The 1PI Kernel and a Lippmann–Schwinger Equation

To then show that a theory – if the perturbation series

heuristically shows it – contains only one particle of

mass smaller than 2m(1  ), it is necessary to expand

further the coupling between the K’s in [9]. Each

perturbative step relatively to this coupling will

generate a sum of terms such that in each one there is

a ‘‘new’’ propagator C

between two K’s.

The fact that in [9] the X’s are nonoverlapping

has the consequence that an expansion where for

each pair of K

the number of propagators C

Scattering, Asymptotic Completeness and Bound States 483

remains bounded (say by n þ 1) is convergent (for

small enough couplings depending on m, n); this is

because, for a given X, the others must be farther

and farther as their number increases, and in view of

the exponential decrease (in x-space) of C

We then consider the expansion where we have

further expanded the two-point function S(x, y) such

that each term can be decomposed in the channel

x !y in C

propagators and 1PI contributions (in

the sense that any line cutting such a 1PI contribu-

tion (and outside the X

s) cuts at least two

propagators); that means that these 1PI contribu-

tions are no longer coupled by the d

(’) measure.

They are made of propagators and of K

which still

have nonoverlapping restrictions; the latter are

straightforwardly expanded using a kind of (con-

vergent) Mayer expansion; the result is finally a

Lippmann–Schwinger type equation:

Sðx; yÞ

connected

¼C

ðx  yÞþ

ðx  z

 G

ðz

; z

ÞC

ðz

 yÞþ ½13

Sðx; yÞ

connected

¼ C

p0

½G



ðx; y Þ

which is equivalent to

connected

¼ C

þ C

connected

½14

where G

is a 1PI kernel that satisfies the bound

ðt; uÞj  

ren

2mð1Þjtuj

½15

In Fourier transform, eqn [14] becomes

FðpÞ¼

ðpÞþ

ðpÞ

ðpÞG

ðpÞFðpÞ½16

Denoting by (p þ q) F(p, q) the Fourier transform of

S(x, y)

connected

, we can then compute F(p):

FðpÞ¼

ðp

þ m

Þ½

ðpÞ

ðp

þ m

Þðp

þ m

ðpÞ

½17

where (p

þ m

)

(p) !(1  m

)asp !0 and

(p)j

ren

cst(m) so that (as expected) F has no

pole in the Euclidean region at small coupling; but,

as will be seen in the next section, it has a pole

outside the Euclidean region.

The 2PI Kernel and a BS Equation

From the previous discussion, it is clear that one can

extract from [9] as many propagators as we want

between kernels K

. If one considers a splitting of

the external points in incoming x

, ..., x

and

outgoing x

pþ1

, ..., x

points, this defines a channel.

One then obtains nPI kernels (in the given channel).

In the same way as above, one obtains a relevant

structure equation; this equation makes sense only

if the kernels K

have a decrease corresponding to

n-particle irreducibility; to that purpose we take

M > nm. The expansion converges for couplings

small enough depending on m and n.

In the case n = 2 this gives a kind of BS equation

(the Lippmann–Schwinger equation corresponding

to the case n = 1); if we restrict, for simplicity, the

analysis to even theories one is led to jump directly

to the case n = 3:

Sðx

; x

connected

ð

Þðx

; x

; z

; t

 G

ðz

; t

; z

; t

Þð

Þðz

; t

; x

Þþ ½18

S ¼

p1

½G





S ¼



þ

S ½19

where

ð

Þðx

; x

Þ¼Sðx

; x

ÞSðx

; x

þ Sðx

; x

ÞSðx

; x

and where

ðt

; t

; u

Þj

 

ren

expf4mð1  Þmax

i;j

ðjt

 u

jÞg ½20

Equation [19] once amputated, and after Fourier

transformation, is eqn [2].

More General Irreducible Kernels

and Structure Equations

Irreducible kernels with various degrees of irreduci-

bility in various channels can be defined in a similar

way. Corresponding expansions of N-point func-

tions follow , in terms of integrals involving these

kernels and two-point functions. These kernels are

again convergent at small coupling (! 0 as their

irreducibility !1) as well as the corresponding

structure equations (which generalize eqn [18]).

Analyticity, AC, and Bound States

As explained in the introduction, we now proceed

by analytic conti nuation away from the Euclidean

region in complex energy–momentum space.

484 Scattering, Asymptotic Completeness and Bound States

First, it is easily seen that the two-point function

is analytic in the region s < (2m)

  apart from a

pole at s = m

which defines the physical mass

is the zero in p

of the denominator in

formula [17]). In view of the bounds of the previous

two sections, m

is close to the ‘‘bare’’ mass m.

The 2PI kernel, for even theories, is shown, again by

Laplace transform theorem, to be analytic and bounded

in domains around and away from the Euclidean region

up to s = (4m)

 , and is of the order of 

ren

As we ha ve seen in the section ‘‘AC an d

analyt icity,’’ the analytic ity of G

entails the analytic

structure of F (two-sheeted or multisheeted at the

threshold). On the other hand, further poles of F can

be generated by the BS integral equation [2] in the

physical or unphysical sheets. If a pole in the

physical sheet occurs at s < (2m

)

real, it will

correspond to a new particle in the theory, namely a

two-particle bound state.

AC in the Low-Energy Region

The analysis of possible bound states, which will be

presented in the following, will show that there

might be at most one two-particle bound stat e of

mass m

< 2m

which tends to 2m

as the

couplings tends to zero.

On the other hand, for even theories, in view of

the analyticity properties of the two-point function

and of the 2PI kernel G

, equation [1] holds in the

region (2m

) < s < (4m

)

 , where  is on-shell

convolution with particles of mass m

If there is no two-particle bound state, this

characterizes the AC of the theory for s < (4m

)

 .

If there is a bound state of mass m

,ACis

established only in the region s < (3m

)

 .

For non-even theories , the analysis is similar but

requires the introduction of new irreducible kernels

in view of the fact that the non-evenness opens new

channels. AC in all cases can be established, for

small couplings, up to s < (3m

)

 .

Analysis of Possible Two-Particle Bound States

for Even Theories at Small Coupling

It can be checked that such poles of F, if there are,

either lie far away in the unphysical sheet(s) or are

close to the two-particle threshold (s = (2 m

)

This is due to the convergence, at small coupling, of

the Neumann series F = G

þ G



þ. Indi-

vidual terms G





are, in fact, defined

away from the Euclidean region by analytic con-

tinuation in a two-sheeted (d even) or multisheeted

(d odd) domain around the threshold: to that

purpose locally distorted integration co ntours (initi-

ally the Euclidean region) are introduced as in the

section ‘‘A C and analytic ity,’’ so as to avo id the pole

singularities of the two-point functions involved in



, the threshold singularities being due to the

pinching of this contour between the two poles as

s !(2m

)

. If a fixed neighborhood of the thresh-

old is excluded, one does obtain uniform bounds of

the form (cst 

ren

)

(for a term with q factors G

)in

any bounded domain, which ensures the conver-

gence of the Neumann series.

It remains to study the neighborhood of the

threshold. To that purpose, the follow ing method

is convenient. One shows that the convolution

operator 

can be written in the form



¼ gð s Þþr½ 21

where  is, as in the section ‘‘AC and analyticit y,’’

on-shell convolution for s > (2m

)

or is obtained

by analytic continuation for complex value of s

around the threshold; g(s) = 1=2 for d even and, if d

is odd, g(s) = (i=2) log , where  = 4m

 s.In

view of this definition of g(s), the operator r is

regular: it is an analytic one-sheeted operation

around the threshold (this is equivalent to [4]), and

it has no pole singularities. This property of r can

be established by geometric methods or by an

explicit evaluation.

It is then useful to introduce a new kernel U

linked to G

by the integral equation

U ¼ G

þ UrG

½22

In view of the regularity and bounds of r and G

one sees (e.g., by a series expansion) that U,likeG

is an alytic in a neighborhood of the threshold and

behaves in the same way at small 

ren

By a simple algebraic argument F and U are

related by the integral equations

F ¼ U þ gðsÞU  F ¼ U þ gðsÞF  U ½23

Two-dimensional models We start the analysis with

the case d = 2. The mass shell is trivial in this case; let f

be the restriction of F to the mass shell; it depends only

on s = (p

þ p

)

due to the mass shell and e.m.c.

constraints (as also Lorentz invariance). On the mass

shell, the operation  becomes a mere multiplication

and the integral equation [23] becomes

f ðsÞ¼uðsÞþ

aðsÞ

f ðsÞuðsÞ½24

where u is the mass shell restriction of U and the

factor a(s) arising from  is of the form

a(s) = cst s

1=2



1=2

,  = (2m

)

 s, which gives

f ðsÞ¼

aðsÞuðsÞ

aðsÞuðsÞ

½25

Scattering, Asymptotic Completeness and Bound States 485

In turn one obtains

F ¼ U þ

aðsÞuðsÞ

½26

where U

(resp.,

U)isU with p

, p

(resp., p

, p

)

restricted to the mass shell. Equation [26] comple-

tely characterizes the local structure of F in view of

the local analyticity of U.

The an alysis of the possible poles follows from the

fact that U is equal to G

up to higher order in 

ren

;

on the other hand, G

is equal to a first known term

plus higher-order corrections in 

ren

(if we expand in



ren

the expression for G

obtained in the previous

section), so that the leading contribution of u(s)is

known and the results follow.

For a theory (see [8]) containing a 

ren

’

term there is

exactly one pole, which corresponds to the zero of a(s) 

u(s), lying in the region (2m

)

 <s < (2m

)

This pole is either in the physical sheet for 

ren

< 0orin

the second sheet if 

ren

> 0. In the case 

ren

< 0, this

pole corresponds to a two-particle bound state of

physical mass m

which tends to 2m

as 

ren

!0.

In a model without ’

term (

ren

= 0) the lowest-

order contribution to G

, hence to U, is in general of

the order of the square of the leading coupling, in

which case there is always one bound state.

The treatment of the fermionic Gross–Neveu

model, which involves spin and color indices, is

analogous, with minor modifications. Equations now

involve, in the two-particle region, 4  4matrices;

poles of F are now the zeros of det (a(s)I  m(s)u(s)),

where m(s)isthe4 4 matrix obtained from 2  2

residue matrices (whose leading matrix elements are

explicitly computable). The detailed analysis, which

requires the consideration of different channels

(various color and spin indices) is omitted.

Three-dimensional models The results are similar:

F is decomposed as F

þ F

, where F

is the ‘ = 0

‘‘partial wave component’’ of F, namely F

= (1=2)

F d, where  is the ‘‘scattering angle’’ of the

channel; its complement F

is shown to be locally

bounded in view of a further factor . The analysis

is then analogous to the case d = 2 with a(s) now

behaving like cst= log  as  !0. There is, a priori,

either no pole, or one pole in the physical sheet at

s = m

< (2m

)

with m

= 2m

þ O(e

cst=

ren

depending again on the signs of the couplings. For

the existing even models such as the ’

model, there

is no pole, hence no two-particle bound state.

Four-dimensional models The existence of the ’

model in dimension 4 is doubtful. If a four-

dimensional model were defined, and if the 2PI

kernel G

of a massive channel could be defined and

shown to satisfy analyticity properties analogous to

above, there would be no two-particle bound state at

small coupling. In fact, the kinematical factor 

(d3)=2

(for d even) generated by the mass shell convolution

is no longer equal to 

1=2

as in the d = 2 case but

now to 

1=2

. As a consequence, the Neumann series

giving F in terms of G

is convergent also in the

neighborhood of the two-particle threshold.

Non-even theories The analysis for the non-even

theories follows similar lines. As already mentioned,

the analysis requires the introduction of new irredu-

cible kernels. For the models ’

þ c

’

, which do

exist at small couplings in dimensions 2 and 3, there

will be either exactly one or no two-particle bound

state, depending on the respective values of , c

Structure Equations and AC in

Higher-Energy Regions

The structure equations of the previous section provide,

after analytical continuation away from the Euclidean

region, a rigorous version of the analysis presented at

the end of the section ‘‘AC a nd analyticity .’’ The

irreducible kernels can here be defined in a direct way

following the previous section, together with their

analyticity properties. One has then to derive the

discontinuity formulas that in turn characterize AC.

This program has been carried out in the 3 !3 particle

region, and partly in the general case. It seems possible

to complete general proofs up to some technical

(difficult) problems. As already mentioned, in this

approach, the coupling should be taken smaller and

smaller as the energy region considered increases.

See also: Axiomatic Quantum Field Theory; Constructive

Quantum Field Theory; Dispersion Relations; Dynamical

Systems in Mathematical Physics: An Illustration from

Water Waves; Perturbation Theory and its Techniques;

Quantum Chromodynamics; Scattering in Relativistic

Quantum Field Theory: Fundamental Concepts and

Tools; Scattering in Relativistic Quantum Field Theory:

the Analytic Program; Schro

dinger operators.

Further Reading

Bros J (1984) r-particle irreducible kernels, asymptotic complete-

ness and analyticity properties of several particle collision

amplitudes. Physica A 124: 145.

Bros J, Epstein H, and Glaser V (1965) A proof of the crossing

property for two-particle amplitudes in general quantum field

theory. Communications in Mathematical Physics 1: 240.

Bros J, Epstein H, and Glaser V (1972) Local analyticity

properties of the n-particle scattering amplitude. Helvetica

Physica Acta 43: 149.

Epstein H, Glaser V, and Iagolnitzer D (1981) Some analyticity

properties arising from asymptotic completeness in quantum

field theory. Communications in Mathematical Physics 80: 99.

Glimm J and Jaffe A (1981, 1987) Quantum Physics:

A Functionnal Integral Point of View. Heidelberg: Springer.

486 Scattering, Asymptotic Completeness and Bound States

Iagolnitzer D (1992) Scattering in Quantum Field Theories: The

Axiomatic and Constructive Approaches, Princeton Series in

Physics. Princeton University Press.

Iagolnitzer D and Magnen J (1987a) Asymptotic completeness

and multiparticle structure in field theories. Communications

in Mathematical Physics 110: 51.

Iagolnitzer D and Magnen J (1987b) Asymptotic completeness

and multiparticle structure in field theories. II. Theories with

renormalization: the Gross–Neveu model. Communications in

Mathematical Physics 111: 89.

Martin A (1970) Scattering Theory: Unitarity, Analyticity and

Crossing. Heidelberg: Springer.

Rivasseau V (1991) From Perturbative to Constructive Renorma-

lization. Princeton: Princeton University Press.

Schro¨ dinger Operators

V Bach, Johannes Gutenberg-Universita

Mainz, Germany

Schro¨ dinger operators are linear partial differential

operators of the form

¼ þ VðxÞ½1

acting on a suitable dense domain dom(H

)  L

()

in the Hilbert space of square-integrable functions

on a spatial domain   R

, where d 2 N. Here,

= =

 = 1

=@x



is (minus) the Laplacian

on , and the potential V :  ! R acts as a multi-

plication operator, [V ](x):= V(x) (x).

Historical Origin and Relation

to Theoretical Physics

In 1926, Schro¨ dinger formulated quantum theory as

wave mechanics and proved later that it is equiva-

lent to Heisenberg’s matrix mechanics. He proposed

that the state of a physica l system at time t 2 R is

given by a normalized wave function

2 L

()

whose dynamics is determined by a linear Cauchy

problem:

is the state at time t = 0, and for t > 0,

it evolves according to

¼ H

½2

the Schro¨ dinger equation. More generally,

is a

normalized element of a Hilbert space H,and

the Hamiltoni an H

is a self-adjoint operator,

that is, dom(H

) = dom(H



) Hand H

= H



dom(H

). Formally, eqn [2] is solved by the

evolution operat or or propagator exp(itH

)in

the form

= exp(itH

)

. The self-adjointness

of H

insures the existence and unitarity of

the propagator exp(itH

), for all t 2 R,so

k= k

k= 1. For physics, this unitarity is crucial,

because k

is interpreted as the total probability

of the system to be at time t in some state in H. The

general validity of eqn [2] as the fundamental

dynamical law of all physical theories, including,

for example, nonrelativistic and (special) relativistic

quantum mechanics, quantum field theory, and

string theory, deserves appreciation.

If the physical system under consideration is a

nonrelativistic point particle of mass m > 0ina

potential

V : R

! R, then, according to the princi-

ples of classical (Newtonian) mechanics, its state is

determined by its momentum p 2 R

and its posi-

tion x 2 R

, its kinetic energy is (1=2m)p

, its

potential energy is

V(x), and the dynamics is given

by the Hamiltonian flow generated by the

Hamiltonian function H

class

(p, x) = (1=2m)p

V(x).

Schro¨ dinger derived the Hamiltonian (operator)

H = (h

=2m) þ

V(x)in[2] from the replace-

ment of the momentum p 2 R

by the momentum

operator ihr

. This prescription is called quanti-

zation and is further discussed in the section

‘‘Quantization and semiclassical limit.’’ The

Schro¨ dinger operator H

in [1] is then obtained after

an additional unitary rescaling, (x) 7!

d=2

(x),

by  :=h(2m)

1=2

, and a redefinition V(x):=

V(x=)

of the potential.

For more details, we refer the reader to

Schro¨ dinger (1926) and Messiah (1962).

Self-Adjointness

Led by the requirement of unitarity of the propa-

gator, the domain dom(H

)in[1] is usually chosen

such that H

is self-adjoint, which, in turn, is most

often established by means of the Kato–Rellich

perturbation theory, briefly described below. If

V  0, then H

equals the Laplacian , which

is a positive self-adjoint operator, provided

dom(H

) = W

b.c.

() is the second Sobolev space

with suitable conditions on the boundary @ of .

Typical examples are dom(H

) = W

), for

=R

, and W

Dir

() and W

Neu

() with Dirichlet

or Neumann boundary conditions on @, respec-

tively, in case that  is a bounded, open domain in

Schro¨ dinger Operators 487

with smooth boundary @. Starting from this

situation, V is required to be relatively H

-bounded,

that is, that M(V, r):= V( þ r1)

1

defines

(extends to) a bounded operator on L

(), for any

r > 0. If lim

r!1

kM(V, r)k< 1, then H

is self-

adjoint on dom(H

) and semibounded, that is, the

infimum inf (H

) of its spectrum (H

) is finite; in

other words, H

 c1, for some c 2 R,asa

quadratic form. (The semiboundedness corresponds

to quasidissipativity, as a generator of the semigroup

exp(H

).)

A fairly large class of potentials fulfilling these

requirements is defined by

lim

&0

sup

x2

jxyj

jx  yj

4d

VðyÞ

()

¼ 0 ½3

for d 6¼ 4, and with jx  yj

4d

replaced by (ln jx 

yj)

1

, for d = 4. For d  3, [3] is equivalent to the

uniform local square integrability of V, that is,

sup

x2

jxyj1

V(y)

y < 1. Note that [3] allows

for local singularities of V, provided they are not too

severe; in this respect, quantum mechanics is more

general than classical mechanics. Equation [3] is a

sufficient condition for H

=  þ V to be self-

adjoint on dom() because lim

r!1

kM(V, r)k= 0.

Moreover, as eqn [3] only misses some borderline

cases, it is also almost necessary for the self-

adjointness of H

. By means of Kato’s inequality, the

conditions on V, especially on its positive part

:= maxfV,0g, can be further relaxed. Also, if one

realizes H

as the Friedrichs extension of a semi-

bounded quadratic form, the conditions to impose on

V are milder. One possibly loses, however, control

over the operator domain dom(H

), and typically

dom() is only a core for H

For further details on self-adjointness, we refer the

reader to Reed and Simon (1980a, b), Kato (1976),

and Cycon et al. (1987).

Spectral Analysis

The self-adjointness of H

establishes a functional

calculus, generalizing the notion of diagonalizability of

finite-dimensional self-adjoint matrices: there exists a

unitary transformation W : L

() ! L

((H

), d)

such that H

acts on elements ’ of L

((H

), d

)

as a multiplication operator, [H

’](!) = !’(!). The

spectral measure 

decomposes into an absolutely

continuous (ac) part 

,ac

, a pure point (pp) part



,pp

, and a singular continuous (sc) part 

,sc

mutual disjointly supported on the ac spectrum



), the pp spectrum 

), and the sc

spectrum 

)  R, respectively, whose union is

the spectrum (H

)ofH

. There is an additional

decomposition of the spectrum of H

into the discrete

spectrum 

disc

), which consists of all isolated

eigenvalues of H

of finite multiplicity, and its

complement 

ess

) = Rn

disc

), the essential

spectrum of H

, as its residual spectrum is void. One

of the main goals of the spectral analysis is to

determine the spectral measure for a given potential

V as precisely as possible.

In many applications, =R

and the potential V in

is not only relatively H

-bounded, but even

relatively H

-compact, that is, M(V,1)iscompact.In

this case, lim

r!1

kM(V, r)k= 0, insuring self-

adjointness on dom(H

) and semiboundedness of H

Moreover, a theorem of Weyl implies that its essential

spectrum agrees with the one of H

, that is, with the

positive half-axis R

, and the discrete spectrum is

contained in the negative half-axis R



. If, furthermore,

þ 1)

1

[x rV(x)](H

þ 1)

1

is compact, then the

essential spectrum on the positive half-axis is purely

absolutely continuous, 

ess

) \ R

= 

) \

,andhence

disc

)  

disc

) [

{0}; the singular continuous spectrum is void.

We remark that the ab sence of singular contin-

uous spectrum is not understood. Indeed, it is

possible to explicitly construct potentials V such

that H(V) has singular continuous spectrum. In

terms of the Ba ire cate gory, singular continuous

spectrum is even typical. The appearance of singular

continuous spectrum can, perhaps, be easier

understood in term s of the dynamical properties of

exp [ itH

], rather than the spectral analysis of its

generator H

: Singular continuous spectrum occurs

when initially localized states are not bound states,

but move out to infinity very slowly.

The reader is referred to Simon (2000), Reed and

Simon (1980a, b) and Cycon et al. (1987) for further

detail.

Properties of Eigenfunctions

Let us assume =R

, that V  0 is nonpositive,

fulfills [3], and that lim

jxj!1

V(x) = 0. From the

statements in the last section we conclude that

=  þ V(x) is semibounded, that the essential

spectrum is the positive half-axis and that all

eigenvalues are negative and of finite multiplicity,

possibly accumulating only at 0. We collect some

properties of the eigenfunctions

2 L

) with

corresponding eigenvalue e

< 0, that is, H

. The smallest eigenvalue e

:= inf (H

) (coin-

ciding with the bottom of the spectrum) is simple,

and the correspon ding eigenfunction

(x) > 0is

strictly positive a.e. Elliptic regularity implies that at

a given point x 2 R

, the eigenfunction

is almost

2  d/2 degrees more regular than V. For example,

488 Schro¨ dinger Operators

if V 2 C

2

(x)], for some >0, then

kþ‘

(x)], for all ‘<2  d=2. Agmon estimates

(originally obtained by S’nol and also known in

mathematical physics as Combes–Thomas argu-

ment) furthermore show that, for unbounded ,

the eigenfunction

decays exponentially: j

(x)j



jxj

, for any 0 <<e

For more de tails, see Reed and Simon (1978,

1980a, b) and Cycon et al. (1987).

One Dimension and Sturm–Liouville

Theory

For d = 1, the stationary Schro¨ dinger equation

reduces to a second-order ordinary differential

equation known as a Sturm–Liouvi lle problem,



ðxÞþVðxÞ ðxÞ¼E ðxÞ½4

on L

([a,b]), with V 2L

([a,b)] and independent

boundary conditions at 1a < b 1, say. Equa-

tion [4] admits an almost explicit solution by means of

the Pru¨ fer transformation defined by ’(x):=

arctan [ (x)=

(x)] and R (x):= ln



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

(x)



The key point about the Pru¨ fer transformation is that it

effectively reduces the second-order differential equa-

tion [4] into a (nonlinear) first-order equation for ’,

’

ðxÞ¼ E  VðxÞðÞsin

½’ðxÞ þ cos

½’ðxÞ ½5

Note that [5] does not involve R and that the

boundary conditions on and

at a and b can be

easily expressed in terms of ’(a)and’(b). More-

over, having determined ’ on [a, b] from [5], the

function R is immediately obtained by integrating

(x) = [1 þ V(x)  E] sin [’(x)] cos [’(x)]. In case of

a bounded interval, 1 < a < b < 1, or a confin-

ing potential, lim

x!1

V(x) = 1, it is not difficult to

derive from [5] the following basic facts: the

spectrum of H(V) consists only of simple eigenva-

lues E

< E

< with lim

n!1

= 1. More-

over, the corresponding eigenfunction

6¼ 0,

n 2 N

, with H(V)

= E

, has precisely n zeros,

and Sturm’s oscillation theore m holds.

See Amrein et al. (2005) for more detai ls.

Quantization and Semiclassical Limit

The quantization procedure postulated by Schro¨dinger

is the replacement of the classical momentum p 2 R

by the quantum-mechanical momentum operator

ihr

. It is known (and, in fact, easy to see,

cf. Messiah (1962)) that the classical Hamiltonian

equation of motions is invariant under symplectic

transformations, but Schro¨ dinger’s quantization

procedure does not commute with symplectic

changes of the classical variables. The question of

the geometrically sound definition of quantization,

with a general d-dimensional manifold replacing

the spatial domain , has attracted many mathe-

maticians and has led to the mathematical fields

of geometric quantization an d deformation

quantization.

It is remarkable, however, that Schro¨dinger himself

discovered already in his early paper the fact that

classical dynamics derives as the scaling limit h ! 0

from quantum mechanics. The systematic study of

the convergence of wave functions and of operators

and their spectral properties is known as semiclassical

analysis, which is nowadays considered to be part of

microlocal analysis. We illustrate the type of results

one obtains by the following example on =R

Let F 2 C

(R; R) be a smooth characteristic

function, compactly supported in an interval I  R



away from the essential spectrum of the semiclassi-

cal Schro¨ dinger operator H

h

= h

 þ V with a

smooth potential V 2 C

) of compact support.

We define the operator F[H

] by functional calculus

(note that I  

) and F[H

] is of trace class).

Let, furthermore, A

h

jjM



(x)@



be a differ-

ential operator representing an observable. Then

tr{A

F[H

]}, which exists because the eigenfunctions

of H

h

are smooth and decay exponentially, is, up to

normalization, interpreted to be the expectation of the

observable A

h

in the state represented by the spectral

projection of H

h

in I, approximated by F[H

Semiclassical analysis then yields an asymptotic

expansion of the form

tr{A

F½ H

} =h

d

þ c

h þþc

h

þ oðh

ÞðÞ

for arbitrarily large integers n 2 N. The leading-

order coefficient c

is determined by Bohr’s corre-

spondence principle,

tr A

F½H

h



a½x; pF½p

þ VðxÞ

dp dy

ð2hÞ

þ o ð2hÞ

d



½6

Semiclassical analysis thus provides the mathemati-

cal link between quantum and classical mechanics.

The proof of [6] usually involves pseudodifferential

and/or Fourier integral operators, depending on the

method. Advanced topics in semiclassical analysis

studied more recently are the construction of

quasimodes, that is, wave functions

E, h, n

which

solve the eigenvalue problem (H

h

 E)

E, h, n

= O(h

)

up to errors of order h

, for arbitrarily large n 2 N,

and the relation between semiclass ical asymptotics

Schro¨ dinger Operators 489

and the KAM (Kolmogorov–Arnold–Moser) theory

from classical mechanics.

For more details, see Dimassi and Sjo¨ strand

(1999), and Robert (1987). See also Stability Theory

and KAM, KAM Theory and Celestial Mechanics in

this encyclopedia.

Lieb–Thirring Inequalities

Lieb–Thirring inequalities are estimates on eigenva-

lue sums of H

V

=    V(x), where V  0is

assumed to be non-negative (note that we changed

the sign of V) and vanishing at 1; the most

important examples for these sums are the number

of eigenvalues below a given E  0 and the sum of

its negative eigenvalues, counting multiplicities.

More generally, denoting by []

:= max {, 0} the

positive part of  2 R, Lieb–Thirring inequalities are

estimates on tr{[E  H

V

]



}, for   0. The num-

ber of eigenvalues below E is then obtained in the

limit  ! 0, and the sum of the negative eigenvalues

corresponds to E = 0and = 1. We henceforth

assume E = 0, for simplicity. A guess inspired by

[6] with F[]:= []



, A = 1, and h = 1 then is that

tr{[H

V

]



} is approximately given by

VðxÞp





x d

ð2Þ

¼ C

ð; dÞ

VðxÞ

ðd=2Þþ

x ½7

for a suitable constant C

(, d) > 0 depen ding only

on  and d (but not on V). While this guess is

wrong, it is nevertheless a useful guiding principle.

Namely, in a rather large range of  and d, there

exist constants C

(, d) > 0 such that

trf½H

V





 C

ð; dÞ

VðxÞ

ðd=2Þþ

x ½8

for all V  0, for which the right-hand side is finite

(with the understanding that this finiteness also

insure that [H

V

]



is trace class, in the first place).

Of course, C

(, d)  C

(, d), by [6]. The

Lieb–Thirring conjecture, which is still open today,

says that the best possible choice of C

(1, 3) equals

(1, 3) in the physically most relevant case  = 1

and d = 3. It is known that C

(, d) > C

(, d), for

<1ord < 3.

Lieb–Thirring estimates have been derived for

various modifications of the original model, depend-

ing on the application. One of these are pseudor-

elativistic Hamiltonians of the form H = T(p)  V,

where T(p) =

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

þ m

, with m  0, another one

includes an external magnetic field, for example,

H = (p  A)

 V (see the next and the last section).

The reader is referred to Thirring (1997), Reed and

Simon (1978),andSimon (1979) for further details.

Magnetic Schro¨ dinger Operators

Magnetic Schro¨ dinger operators are Hamiltonians

of the form

ðA; VÞ¼ p  AðxÞðÞ

VðxÞ

on L

ðR

Þ½9

Pauli

ðA; VÞ¼ s  p  AðxÞðÞ½

VðxÞ

on L

ðR

ÞC

½10

where V is the (electrostatic) potential; as before,

A : R

! R

is the vector potential of the magnetic

field B =  ^A, and s = (

, 

) are the Pauli

matrices. H

(A, V)andH

pauli

(A, V) generate the

dynamics of a particle moving in an external electro-

magnetic field of spin s = 0andspins = 1=2, respec-

tively. The operator H

Pauli

(A, V) is usually called Pauli

Hamiltonian, and we refer to H

(A, V)asthe

magnetic Hamiltonian. To keep the exposition simple,

we assume henceforth that A



and @





are uniformly

bounded, which suffices to prove the self-adjointness

of both Hamiltonians.

At a first glance, the magnetic and the Pauli

Hamiltonians may seem to differ only marginally,

but in fact, some of their spectral properties are

fundamentally different.

1. The magnetic Hamiltonian fulfills the diamagnetic

inequality, je

H

(A, V)

(x, y)je

H

(0, V)

(x, y), for

almost all x, y 2 R

, where m(x, y) denotes the

integral kernel of an operator m. As a consequence,

inf [H

(A, V)]  inf [H

(0, V)] = inf [H(V)],

and the quadratic form of the magnetic Hamilto-

nian is semibounded, for all choices of A, provided

H(V)is.

2. If inf [H

(A, V)] is an eigenvalue, the diamag-

netic inequality reflects the fact that the corre-

sponding eigenvector is not positive or of

constant phase. The determination of the nodal

set of eigenfunctions is a difficult task on its own.

3. For V = 0, the diamagnetic inequality and the

minimax principle imply that p  A has no zero

eigenvalue.

4. The diam agnetic inequality fails to hold for the

Pauli Hamiltonian. On the contrary, if A is

carefully adjusted in H

(A, Zjxj

1

), and Z is

sufficiently large, then the corresponding

490 Schro¨ dinger Operators

quadratic form may assume arbitrarily small

values (even if the corresponding field energy is

added).

5. For many choices of A, the (Dirac) operator

s (p  A) has a nontrivial kernel .

From (1)–(4) it is clear that the proof of stability of

matter (see the next section) in presence of a

magnetic field is more difficult than in absence of it.

This can be illustrated by the fact that magnetic Lieb–

Thirring inequalities, being the natural analog of eqn

[8], are more involved to derive than the original

estimate [8]. The currently best bound is of the form

trf½H

V





 C

mLT

½VðxÞ

5=2

þjBðxÞj½VðxÞ

3=2

þjBðxÞj þ L

ðxÞ

2



ðxÞ

1

½VðxÞ

x ½11

for some universal C

mLT

< 1, where L

(x) is a local

length scale associated with B. It is nonlocal in x

and somewhat reminiscent of a maxi mal function.

We further remark that if restricted to two

dimensions, d = 2, both the magnetic and the Pauli

Hamiltonians play an important role in the theory of

the (integer) quantum Hall effect.

For more details, see Simon (1979), Cycon et al.

(1987), Rauch and Simon (1997), and Erdo¨ s and

Solovej (2004). See also the article Quantum Hall

Effect in this encyclopedia.

N-Body Schro¨ dinger Operators

The origin of quantum mechanics is atomic (K = 1

below) or molecular (K  2) physics. If we regard

the nuclei of the molecule as fixed point charges

Z := (Z

, ..., Z

) > 0 at respective positions

R := (R

, ..., R

) 2 R

, then the Hamiltonian (in

convenient units) of this molecule with N 2 N

electrons is the following Schro¨ dinger operator:

ðZ; RÞ¼

n¼1





k¼1

 R

()

1m<nN

 x

½12

defined on H

(N)

n = 1

 Z

]  L

[(R



)

], the space of totally antisymmetric, square-

integrable wave functions in N space–spin variables

, 

), ...,(x

, 

) 2 R

 Z

. The antisymmetry

of the wave function accounts for the fact that

electrons are fermions and is of crucial importance.

Note that the number N of electrons is possibly very

large. It is clear that we cannot expect to carry out

the spectral analysis of this Schro¨ dinger operator

directly, but rather only suitable approximations.

In spite of the fact that H

(Z, R)wasoneofthe

basic operators of quantum mechanics from its very

beginning in the late 1920s, H

(Z, R)was,strictly

speaking, not known to be self-adjoint before Kato

developed the perturbation theory (described in the

section ‘‘Self-adjointness’’) some 20 years later, which

then also yielded the semiboundedness of H

(Z, R).

So, the ground-state energy E

(Z, R):= inf [ H

(Z, R)] > 1 is finite. From the HVZ (Hunziker–

van Winter–Zishlin) theorem follows that inf 

ess

(Z, R)] = E

N1

(Z, R), which particularly implies that

(Z, R) is monotonically decreasing in N and

negative (because E

(Z, R) < 0).

It is known that E

(Z, R) = E

Nþ1

(Z, R) and that

(Z, R) has no eigenvalue, for N  2Z

tot

þ 1,

where Z

tot

k = 1

is the total nuclear charge

of the atom. On the other hand, it is known that

(Z, R) is an eigenvalue, provided N < Z

tot

. Thus,

defining N

crit

to be the smallest number such that

(Z, R) is not an eigenvalue, for all N  N

crit

, that

is, N

crit

is the maximal number of electrons the

molecule can bind, we have that Z

tot

 N

crit



tot

þ 1. In increasing precision, asymptotic neu-

trality, N

crit

= Z

tot

þ R(Z

tot

), with R(Z

tot

) = o(Z

tot

)

and R(Z) = o(Z

5=7

), was shown for atoms and for

molecules, respectively. The ionization conjecture

states that N

crit

 Z

tot

þ C, for some universal

constant C. It is still open for the full model

represented by H

(Z, R), but has been proved in

the Hartree–Fock approximation. It has been proved

in the Hartree–Fock approximation by Solovej.

The semiboundedness of H

(Z, R), for fixed Z, R,

and N, alone does not rule out a physical collapse of

the matter described by H

(Z, R), but the stronger

property of stability of matter does. It holds if there

exists a constant C, possibly depending on

Z,suchthat

ðZ; RÞþ

1k<‘K

‘

 R

‘

CðN þ KÞ½13

that is, if the ground-state energy plus the repul sive

electrostatic energy of the nuclei is bounded below

by a consta nt times the total number N þ K of

particles in the system. Equation [13] was shown to

hold for H

(Z, R).

In connection with stability of matter, Thomas–

Fermi theory and the question of the limit of large

nuclear charge came into the focus of research. For

simplicity, we restrict ourselves to atoms, K = 1, that

is, there is one nucleus of charge Z := Z

at the

origin, R

= 0, and we consider E(Z):= min

N2N

(Z, 0) (which amounts to fixing N := N

crit

). An

asymptotic expansion for E(Z) of increasing

Schro¨ dinger Operators 491