Methods of modern mathematical physics volume 4 analysis of operators michael reed, barry simon

The heart of this chapter is the second section where we shall discuss the beautiful K ato - Rellich theory of regular perturbations; this theory gives simple criteria under which one ca

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Reed, Michael.

Methods o f m o d em mathematical physics.

Vol 4 Analysis o f Operators.

Includes bibliographical references.

CONTENTS: v 1 Functional an alysis.-v 2 Fourier analysis, s e lf-a d jo in tn e s s -v 3 Scattering th e o r y -v 4

AMS (MOS) 1970 Subject Classifications: 4 7 - 0 2 , 8 1 - 0 2

PRINTED IN THE UNITED STATES OF AMERICA

87 88 89 9 8 7 6 5 4

T o David

T o R ivka and Benny

at the time of publication of Volume I We originally promised the publisher

that the entire series would be completed nine months after we submitted

Volume I Well! We have Usted the contents of future volumes below We

are not foolhardy enough to make any predictions

We were very fortúnate to have had T Kato and R Lavine read and

cnticize Chapters XII and XIII, respectively In addition, we received valuable

comments from J Avron, P Deift, H Epstein, J Ginibre, I Herbst, and

E Trubowitz We are grateful to these individuáis and others whose

comments made this book better

We would also like to th a n k :

J Avron, G Battle, C Berning, P Deift, G Hagedorn, E Harrell, II,

L Smith, and A Sokol for proofreading the galley and/or page proofs

G Anderson, F Armstrong, and B Farrell for excellent typing

The National Science Foundation, the Duke Research Council, and the

Alfred P Sloan Foundation for financia! support

Academic Press, without whose care and assistance these volumes would

have been impossible

Martha and Jackie for their encouragement and understanding

vil

The first step in the mathematical elucidation of a physical theory must

be the solution of the existence problem for the basic dynamical and kine-

matical equations of the theory Once that is accomplished, one would like

to find general qualitative features of these Solutions and also to study

in detail specific special systems of physical interest

Having discussed the general question of the existence of dynamics in Chapter X, we present methods for the study of general qualitative features

of Solutions in this volume and its companion (Volume III) on scattering

theory We concéntrate on the Hamiltonians of nonrelativistic quantum

mechanics although other systems are also treated In Volume III, the

main theme is the long-time behavior of dynamics, especially of Solutions

which are “ asymptotically free.” In this volume, the main theme involves

the five kinds of spectra defined in Sections VII.2 and VII.3: the essential

spectrum, <ress; the discrete spectrum, <xdisc; the absolutely continuous

spectrum, <xac; the puré point spectrum, <xpp; and the singular continuous

spectrum, <xsing It turns out that the study of the absolutely continuous

spectrum as well as the problem of showing that the continuous singular

spectrum is empty are intimately connected with scattering theory Thus,

the separation of the material in Volumes III and IV is somewhat artificial

For this reason, we preprinted in Volume III three sections from Volume IV

IX

These are not the only sections in which the themes of the two volumes

overlap

In these volumes specific systems are usually presented to illustrate the

application of general mathematical methods, but the detailed analysis of

the specific systems is not carried very far Mathematical physicists have to

some extent neglected the detailed study of specific systems; we believe

that this neglect is unfortunate, for there are many interesting unsolved

problems in specific systems, even in the purely Coulombic model of atomic

physics For example, it has not been shown that H " “ has no bound States

even though the analogous classical system of one positive and three nega-

tive charges has the property that its energy is lowered by moving a suitable

electrón to infinity And it is not known rigorously that the energy needed

to remove the first electrón from an atom is less than the energy needed to

remove the second, even though this is “ physically obvious.” We hope

that by collecting the general mathematical methods in Volumes II, III,

and IV, we have made the analysis of specific systems easier and more

attractive

Nonrelativistic quantum mechanics is often viewed by physicists as an

area whose qualitative structure, especially on the level treated here, is

completely known It is for this reason that a substantial fraction of the

theoretical physics community would regard these volumes as exercises

in puré mathematics On the contrary, it seems to us that much of this

material is an integral part of modern quantum theory To take a specific

example, consider the question of showing the absence of the singular

continuous spectrum and the question of proving asymptotic completeness

for the purely Coulombic model of atomic physics The former problem

was solved affirmatively by Balslev and Combes in 1970, the latter is still

open Many physicists would approach these questions with Goldberger’s

method: “ The proof is by the method of reductio ad absurdum Suppose

asymptotic completeness is false Why that’s absurd! Q.E.D.” Put more

precisely: If asymptotic completeness is not valid, would we not have dis-

covered this by observing some bizarre phenomena in atomic or molecular

physics? Since physics is primarily an experimental Science, this attitude

should not be dismissed out of hand and, in fact, we agree that it is extremely

unlikely that asymptotic completeness fails in atomic systems But, in our

opinión, theoretical physics should be a Science and not an art and, further-

more, one does not fully understand a physical fact until one can derive

it from first principies Moreover, the solution of such mathematical prob­

lems can introduce new methods of calculational interest (for example,

Faddeev’s treatment of completeness in three-body systems and the applica­

tion of his ideas in nuclear physics) and can provide important elements of

clarity (for example, the physical artificiality of “ adiabatic switching”

in nonrigorous scattering theory and the clarifying work of Cook, Jauch,

and Kato)

The general remarks about notes and problems in earlier introductions

are applicable here with one addition: the bulk of the material presented

in this volume is from advanced research literature, so many of the “ prob­

lems ” are quite substantial Some of the starred problems summarize the

contents of research papers!

Conte nts

Appendix Algébrate and geometría multiplicity o f

XIII: SPECTRA L ANALYSIS

2 Bound States o f Schródinger operators I :

A Is <rdisc (H ) finite or infinite ? 86

6 The absence o f singular continuous spectrum I :

7 The absence o f singular continuous spectrum I I :

C Local smoothness and wave operators for repulsive

8 The absence o f singular continuous spectrum I I I :

10 The absence o f singular continuous spectrum I V :

14 Compactness criteria and operators with

17 An introduction to the spectral theory o f

Contents of Other Volumes

V o lu m el: Functional Analysis

Volume II: Fouríer Analysis, Self-Adjointness

X Self-Adjointness and the Existence o f Dynamics

Volume III: Scattering Theory

X I Scattering Theory

Contents of Future Volumes: Convex Sets and Functions, Commutative

Applications o f Operator Algebras to Quantum Field Theory and Statistical

XV

XII: Perturbation of Point Spectra

In the thirties, under the demoralizing influence o f quantum-theoretic perturbation theory, the

mathematics required o f a theoretical physicist was reduced to a rudimentary knowledge o f the

In this chapter we shall examine the following general situation: An opera-

tor H 0 has an eigenvalue E 0 , which we usually assume is in the discrete

spectrum Suppose that H 0 is perturbed a little; that is, consider H 0 + fiV

where V is some other operator and \fi\ is small W hat eigenvalues of

H q + PV lie near E 0 and how are they related to V? W hat are their proper-

ties as functions of /?? Such a situation is familiar in quantum mechanics

where there are formal series for the perturbed eigenvalues These

Rayleigh-Schrodinger series are not special to quantum-mechanical opera­

tors but exist for many perturbations of the form H 0 + pV. The heart of this

chapter is the second section where we shall discuss the beautiful K ato -

Rellich theory of regular perturbations; this theory gives simple criteria

under which one can prove that these formal series have a nonzero radius of

convergence We then discuss what the perturbation series means in cases

where it is divergent or not directly related to eigenvalues

XII.1 Finite-dimensional perturbation theory

We first discuss finite-dimensional matrices Not only will this allow us to present explicit formulas in the simplest case, but we shall eventually treat

degenerate perturbation theory by reducing it to an essentially finite-

dimensional problem Furthermore, an important difficulty already occurs

in the finite-dimensional case, namely proving analyticity in p when there is

a degenerate eigenvalue Recall that E0 is called a degenerate eigenvalue

0, has a múltiple

E0 In an appendix to this section we review the theory of

when the characteristic equation for H 0 , det (H0

root at X

matrices with degenerate eigenvalues and, in particular, we discuss the

Jordán normal form

First consider the elementary example

1

By our definition of operator-valued analytic function in Section VI.3, T(P) is

a matrix-valued analytic function To find its eigenvalues, we need only solve

¿±(/0 ± J V ~ + ~ \are the eigenvalues This problem has several characteristic features:

Even though T(p) is entire in /?, the eigenvalues are not entire but have singularities as functions of p

(ii) The singularities are not on the real p axis where T(P) is self-adjoint but occur at nonreal /?, namely at p = ± i. Thus, while there are no singulari­

ties at “ physical” valúes, the perturbation series, i.e., the Taylor series for

0, have a finite radius of convergence due to complex

X±{P)

singularities.

(iii) “ Level Crossing” takes place at the singular valúes of p; that is, at

± i there are fewer distinct eigenvalues, namely one, than at other

points, where there are two

(iv) At the singular valúes of p the matrix T(p) is not diagonalizable

While this “Jordán artomaly” is typical, we leave a discussion of it to the

Notes; see also Problem 23

(v) The analytic continuation of an eigenvalue is an eigenvalue

For the remainder of this section, we shall suppose that T(P) is a matrix- valued analytic function in a connected región R of the complex plañe

Notice that we do not require T(P) to be linear in p. Later, we shall be able

to reduce the infinite-dimensional, linear, finitely degenerate perturbation

problem to a finite-dimensional problem, but one that is no longer linear in

p. Thus, greater generality at this point will be crucial

To find the eigenvalúes of T(P) we must solve a secular equation

det(T(/J) - A) = ( - 1)"[A” + a i (P)Xn ~ 1 + ••• + a„(P)] = 0 The basic theorem about such functions is:

Theorem XI 1.1 Let F(P, X) — A" + a¡(p)Xn~ + • • • + a j f i ) be a

polyno-mial of degree n in X whose leading coefficient is one and whose coefficients

are all analytic functions of /? Suppose that X = A0 is a simple root of

F l p o , X). Then for P near p 0 , there is exactly one root X(p) of F[fi, X) near A0 ,

and X(P) is analytic in P near P Po •

Proof This is a special case of the implicit function theorem Since F(p, X) is

analytic near p 0 and X0 , we can write F(p, A) m = 0 W ”L ( P ) with

M P o ) F(p0 , A0) = 0, an d / , (P0) = (dF/dX){p0 , A0) 0 since A0 is a simple

root Thus to find Solutions of F(P, X) = 0, we need only solve the equivalent

U P )

Because /i(/?0) ^ 0, all the coefficients /*(/í)//i(/í) are analytic near /?

We try to solve this last equation with a solution of the form X(P)

Po •

XQ -f

- 1 a*(0 for example,

m )

m m p = /»0and

It is not very hard to prove that the a’s determined recursively yield a power

series with a nonzero radius of convergence (Problem la) Uniqueness is also

fairly easy (Problem Ib) |

Corollary Let T(P) be a matrix-valued analytic function near p0 and

suppose X0 is a simple eigenvalue of T(p0). Then:

(a) For p near /?0 , T(p) has exactly one eigenvalue, X0(p), near X0

(b) X0(P) is a simple eigenvalue if p is near p 0

(c) X0(P) is analytic near p = p 0

For múltiple roots, a more complicated but still straightforward analysis

is necessary We do not prove the following basic theorem for this case

(proofs can be found in the references in the Notes)

Theorem XII.2 Let F(/?, X) = Xn + a x(p)Xn~ x + ••• + an(p) be an nth

degree polynomial in X whose leading coefficient is one and whose

coefficients are all analytic functions of p. Suppose X = X0 is a root of multi-

plicity m of F(P0 , X). Then for p near p o , there are exactly m roots (counting

multiplicity) of F(P, X) near X0 and these roots are the branches of one or

more multivalued analytic functions with at worst algebraic branch points at

P = p 0 Explicitly, there are positive integers p u , pk with x p , = m and

multivalued analytic functions Xl9 , Xk (not necessarily distinct) with con-

vergent Puiseux series (Taylor series in (/? — Po)1,p)

00

7=1

so that the m roots near X0 are given by the p x valúes of Xu the p 2 valúes of

X2, etc

Corollary If T(P) is a matrix-valued analytic function near p0 and if X0 is

an eigenvalue of T(p0) of algebraic multiplicity m, then for P near /?0 , T(p)

has exactly m eigenvalues (counting multiplicity) near X0 These eigenvalues

are all the branches of one or more multivalued functions analytic near p Q

with at worst algebraic singularities at p 0

If A and B are self-adjoint, the perturbed eigenvalues of A + PB are analy­

tic at p = 0 even if A has degenerate eigenvalues That the branch points

allowed by the last theorem do not occur in this case is a theorem of Rellich

This theorem and its sister theorem on the analyticity of the eigenvectors in

this case are the really deep results of finite-dimensional perturbation theory

The example at the beginning of this section shows that branch points can

occur for nonreal P even in the “ self-adjoint case,” T(P)* = T(p).

Theorem X I 1.3 (Rellich’s theorem) Suppose that T(p) is a matrix-

valued analytic function in a región R containing a section of the real axis,

and that T(p) is self-adjoint for /? on the real axis Let X0 be an eigenvalue of

T(p0) of multiplicity m If pQ is real, there are p < m distinct functions X^p),

, Xp(p), single-valued, and analytic in a neighborhood of p o , which are all

the eigenvalues

Proof Consider one of the functions A,(/?) given in Theorem XII.2:

00

j=i

The crucial fact that we shall use is that each branch of X(P) is an eigenvalue

so that, in particular, each branch is real for P real and near p o Thus

is real So, if p =£ 1, then a A = 0 By induction, one shows that a, = 0 if j/p is

not an integer Therefore X(P) is actually analytic at P = p Q,

We now want to consider the special case H(p) = H 0 + p v Suppose that

E0 is a nondegenerate eigenvalue of f l 0 From Theorem XII 1 we know that,

for p small, H 0 + p V has a unique eigenvalue E(P) near E0 and that E(P) is

analytic near p = 0 The coefficients of its Taylor series are called

Rayleigh-Schródinger coefficients and the Taylor series is called the

Rayleigh-Schrodinger seríes We can use the results described in the appen-

dix to find formulas for the coefficients The formulas are simpler when H 0 is

self-adjoint, so we restrict ourselves to that case E(P) is the only eigenvalue

of H 0 + p V near £ 0 , so if \E — £ 0 | < e, and e is small, £(/?) is the only

eigenvalue of H 0 4* p V in the circle {£| | £ — £ 0 1 < e} By the functional

calculus,

(H0 + p V - E ) - 1 dE 2m |£-E0|=t

is the projection onto the eigenvector with eigenvalue E(p). We shall show in

Theorem XII.9 that (H0 + fiV — E ) ~ 1 is analytic in fi near p = 0 Thus P(P)

is analytic in p at p = 0 In particular, if ÍI0 is the unperturbed eigenvector,

then P i p p o ^ 0 for j8 small since P(P)CI q -* Í20 as P 0- Since P(P)Q0 is an

unnormalized eigenvector for H(P),

(Qo , H(p)P(P)Clq) = p , f í (fío , VP(P)Qq)

This formula is very important in the development of perturbation theory

and plays a critical role in the discussions in Sections 2-4 For it says that to

find the Taylor series for £(/?), we need only find the Taylor series for ?(/?)

To do this, we need only find a Taylor series for (H0 + p V — £) 1 and

intégrate it But the Taylor series for (H0 + p v - £ ) 1 is just a geometric

series:

(Ho + p V - E ) ' 1 = (H0 - E ) ~ l - P(H0 - E ) ~ 1 V(H 0 - £ ) “ 1

+ • + ( - l ) T ( H 0 - E)~ l [V{Ho - £ ) - *]" 4- # • t

Not only is this series simple, but there is a simple form for the error term

when the series is truncated

Thus, the Rayleigh-Schródinger series for £(/?) is given by

Because of the contour integration and the división of power series, the

formulas for the Rayleigh-Schródinger coefficients are complicated To

illustrate this, let us compute E(p) up to order jS4 Since H 0 is self-adjoint, we

can choose a basis of eigenvectors, Q0 , ., Q „-i, with HQ¡ = E¡Cl¡, Let

1 (£0 - £ ) - 2(E¡ - £ ) " 1 dE = (£, - £ 0) - 2

2ni - |£-£0jThus,

b 2 = - Y J (Ei - E QY 2V0iVi0

i t oSimilarly,

terms already present in the numerator.

(iii) Most importantly, the terms in the Taylor series are quite com­

plicated, although they arise from a simple geometric series This suggests

that the simplest object to study is the resolvent: To deduce rigorous

$

theorems about £(/?) in the infinite-dimensional case, we shall generally first

prove results about the resolvent and then obtain information about the

eigenvalues by formulas that give the eigenvalue as a ratio of contour inte­

gráis of matrix elements of the resolvent

As a final result in finite-dimensional perturbation theory, we mention:

T heorem XI 1.4 Let Q0 be a nondegenerate eigenvector for T0 with

T0 Q0 — E0 Q0 , and let T(P) be a matrix-valued analytic function with

T(0) = T0 Then, for p small, there is a vector-valued analytic function Q(/?)

that obeys T(p)Cl(p) = E(p)íl(p ), where £(/?) is the eigenvalue of T(P) near

£ 0 Moreover, if T(P) is self-adjoint for p real, Í2(/?) can be chosen so that

||n(j5)|| = 1 for P real

Proof Take

¿2 m |E-Eo| = t í ™ - £ >" d £ s p ^ °

Then ^(/?) is analytic and an eigenvector Since as /I-»0, (fio,

iA(/J)) * 0 for small p. Let Í2(p) = (fio, \¡>(P)Y i,2'f(p) Then £i(p) is nor- malized when T(p) is self-adjoint for p real since then (í!0, \¡/{P)) —

(si0 , p ( p y i 0) = \ \ m v ■ i

One can also construct analytic eigenvectors in the situation covered by Rellich’s theorem; see Problems 16 and 17.

Appendix to XII.1 Algébrale and geometric multlplicity

of eigenvalues of finito matrices

We first recall some elementary defínitions about roots of algebraic equations:

Definitíon A root Ao of an algebraic equation F(A) = A" + a, A"- 1 +

• • • a„ is called nondegenerate or simple if F'(A0) ^ 0 Equivalently A0 is simple

if the decomposition F(A) = i (A — X¡) has A¡ = A© for exactly one valué

of i A0 is said to have multiplicity m if F'(A0) = = F(m_1)(A0) = 0,

F<m,(Ao) / 0 or equivalently if exactly m of the A¡ equal A0 An eigenvalue of á

matrix is called simple or nondegenerate if it is a nondegenerate root of the

secular equation In general, the algebraic multiplicity of an eigenvalue is its

multiplicity as a root of the secular equation.

The connection between algebraic multiplicity and geometric multiplicity

is explained by the following seríes of remarks:

(i) Let /i(A) be the algebraic multiplicity of A The fundamental theorem

of algebra immediately implies that ¿i(A) = n if T is an n x n matrix.

(ii) Let m(A) = dim{u ¡ Tv = Au} be the geometric multiplicity Then m(A) <, n(k).

(iii) If T is self-adjoint, m(A) = /i(A).

(iv) In general fi(X) = dim{i>|(T — Af v = 0 for some k}. This space is called the generalizad or geometric eigenspace for A.

The statements (ii) and (iv) become transparent once it is known that T

can be put in Jordán normal form, i.e., there is a basis in which T has the

• •

# #

X

where each x = 0 or 1 In this case, the generalized eigenspace

{t>|(T — A¡fv = 0} is spanned by the n(A¡) basis elements associated with the

block Tx¡ and clearly /i(A¡) is the number of times A¡ appears as a root of

det(T - A) = 0

From the fact that any matrix can be put in Jordán normal form, it is also

easy to see (Problem 2) that if e is chosen sufficiently small, then

P , = - J - < t > ( T - A ) ~ l dA

2m - |A_ a, = £

is the projection onto the generalized eigenspace associated with and

P XiP Xj = óijPXr In fact, one of the ways of establishing properties (i)-(iv) is

through the use of these P Xi (see Problems 3 and 4)

XI 1.2 Regular perturbation theory

We now turn to the main result of this chapter and prove that under very general circumstances the Rayleigh-Schródinger series has a nonzero radius

of convergence for perturbations of unbounded operators in infinite-

dimensional Hilbert spaces An example where such results are applicable is

H(p) = — A + p V on R3 where V e L2 is real-valued and p is real and posi-

tive We shall see in Section XIII.4 that a ess(H(p)) == [0, oo) and in Section

XIII 1 that inf <r{H(p)) = £(/?) is a monotonic decreasing function of /? If V

is negative in some región of IR3, E(P) will be negative for P larger than some

p0 and thus, by the result on o an eigenvalue It is reasonable to ask

if this “ ground State energy ” E(p) is analytic in P, at least in a neighborhood

of the interval (j80, oo).

This section is divided into four parts: (1) A brief discussion of the discrete

spectra of not necessarily self-adjoint operators (2) A proof of the analyticity

of discrete eigenvalues in the nondegenerate case for “ analytic families of

operators.” This is the general theory of regular perturbations This theory

has many applications in quantum mechanics where eigenvalues are pos-

sible valúes of the energy For this reason, we shall sometimes use the words

energy level in place of eigenvalue Another term we borrow from quantum

mechanics is coupling constant, which we shall use for the variable p. (3) Two

simple criteria (type (A) and type (B)) for H 0 -I- PV to be an analytic family;

these techniques enable one to apply the general theory to specific cases (4)

A brief discussion of degenerate perturbation theory.

We defined the discrete spectrum of a self-adjoint operator A in Section

VII.3 For such operators, A e aaiK(A) means that A is an isolated point of

o(A) and dim P w < °o where Pa is the projection-valued measure asso-

ciated with A. In the case of a general operator, we obviously should keep

the requirement that A be an isolated point of a(A). To replace the spectral

projection, we use the projection which we introduced in Section XII 1:

Theorem X I 1.5 Suppose that A is a closed operator and let A be an

isolated point of <r( A). Explicitly, suppose that {/¿| |¿í — A | < e} n a (A) —

{A} Then,

(a) For any r with 0 < r < e,

Px = ~ ¿ $ ( A - n ) ~ l dn

exists and is independent of r.

(b) Px — Px - Thus P x is a (not necessarily orthogonal) projection.

(c) If Gx = Ran P x and F x = Ker P x , then Gx and F x are complementary

(not necessarily orthogonal) closed subspaces; that is, Gx + F x =

and Gx n F x = {0} Moreover, A leaves Gx and F x invariant in the following precise sense: Gx <= D(A), AGX c Gx , Fx n D(A) is dense in

Fx , and A[FX n D(A)] c Fx

(d) If ^ e Gx and Gx is finite dimensional, then (A — X)ni¡/ — 0 for some n. If

B = A t Fx , then A 4 <*(&)■

Proof (a) We already know that p) 1 is an analytic function on

integral That it is independent of r is a consequence of the Cauchy integral

and the cj"' and /í;"1 are chosen so th a t the sums converge to

(27i/)-1 f (A — fx)~ l i¡/(¡¡i. A simple com putation using the form ula

Al A 1 + u(A - 1 proves that and th at {Aijjn} is

Cauchy Since A is closed, we conclude that \¡/ e D(A) and the above

approx-imation procedure proves that Aij/ AP x P X(A\¡/). Thus A\¡/ e Gx The

statements about D(A) and F x are left to the reader

(d) Suppose that A í ¡ j = vi// Then

It follows that the only eigenvalue of A \ Gx is X. If Gx is finite dimensional,

the Jordán normal form of C = A f Gx has only X along the diagonal and

some Ts above the diagonal Thus (C

all ip e Gx

(dim G a)

0, i.e., (A m 0 for

Finally, let

\ H - X \ = r

By doing computations similar to those in (b), one finds that R XP X = P¿Rx

and that (A — X)Rx = R X(A — X) = 1 — P x R X(A — A) = 1 — P x indicates

an operator equality applied to vectors in D(A). Thus R x takes F x into itself

and (B - X)Rx « R X(B - X) = / r F x |

We are now in a position to define discrete spectrum:

Definition A point X e a(A) is called discrete if X is isolated and P x (given

by Theorem XII.5) is finite dimensional; if P x is one dimensional, we say X is

a nondegenerate eigenvalue

The reader should check that this definition of discrete spectrum agrees

with the definition given in Chapters VII and VIII when A is self-adjoint

Note that if A is a nondegenerate eigenvalue, any \¡/ e Ran P x obeys

A\¡/ = Xi¡/. To complete our discussion of the discrete spectrum, we prove a

converse to Theorem XII.5

Theorem XII.6 Let A be an operator with —A| = r} c p(A).

Then P = (~ -2m )~ x f ^ - X\ ^ r(A - ¡j)~l dfi is a projection If P has dimen­

sión n < oo, then A has at most n points of its spectrum in {/i 11 fx — X | < r]

and each is discrete If n = 1, there is exactly one spectral point in {/i 11/¿

< r} and it is nondegenerate

prove that P is a projection and the proof of (c) implies that G = Ran P and

F = Ker P are closed complementary invariant subspaces Let A x = A f G

and A 2 = A \ F. As in the proof of Theorem XII.5d, v 4 <7( ^ 2) ^

v — X I < r Thus (A — v)~1 exists for such v if and only if ( A x — v)~1 exists

If G is finite dimensional, A x has eigenvalues vx , vk (k < n), so <r(A)n

{v| |v — X\ < r} is a finite set To see that each spectral point in the circle is

discrete, we note that if P v is the spectral projection of Theorem XII.5 and if

v is in the circle, then P v P = P P V = P v TTius Ran P v c Ran P, which com­

pletes the proof |

Having completed our brief discussion of discrete spectra, we can get down to the real object of study:

D efinition A (possibly unbounded) operator-valued function T(f}) on a

complex domain R is called an analytic family or an analytic family in the

sense of Kato if and only if:

(i) For each p e R, T(p) is closed and has a nonempty resolvent set

(ii) For every p 0 6 R, there is a A0 6 p (T (p 0)) so that A0 e p{T(f})) for p

near p0 and (T(P) — Ao)” 1 is an analytic operator-valued function of P

near p 0

If T(P) is a family of bounded operators, this definition is equivalent to the

definition of bounded operator-valued analytic function (Problem 8) The

number A0 in the above definition does not play a special role:

T heorem XI 1.7 Let T(P) be an analytic family on a domain R Then

r - K A A>| p e R, X e p(T(P))}

is open and the function (T(P) — A ) '1 defined on T is an analytic function of

two variables

Proof Let </?0 , Xx) e T and suppose that (T(P) — A0)” 1 exists and is analy­

tic in p for p near p 0 By the first resolvent identity, 1 — (Xx — A0)x

(T(p0) — ¿ o )" 1 has an inverse equal to (T(p0) — X0)(T(p0) — Aj)” 1 Since

the set of invertible operators in is open, [1 — (A — X0)(T(p) — A0) ~ l]

is invertible if A is near Xx and P is near p 0 , For such </?, A>, T(p) — A has an

inverse equal to

so </?, A> e T Thus T is open To prove the analyticity of (T(P) — A ) ~ w e

note that 1 — (A — X0)(T(P) — A0) ~ 1 is analytic for A near A0 and P near p0

with valúes in the invertible operators By a general theorem (Problem 9), it

follows that (1 — (A — X0)(T(P) — A0)“*1 1 and therefore {T(P) — A)” 1 is

analytic |

Only a simple technical lemma remains to complete the machinery for an effortless proof of the Kato-Rellich theorem :

Lemma If P and Q are two (not necessarily orthogonal) projections and

dim(Ran P) ± dim(Ran Q), then ||P — Q\\ > 1 In particular, if P(x) is a

continuous projection-valued function of x on a connected topological

space, then dim(Ran P(x)) is a constant

Proof Without loss of generality suppose dim(Ran P) < dim(Ran Q). Let

F = Ker P and let E — Ran Q. Then dim fF1) = dim(Ran P) < dim E As a

result, F n £ ^ { 0} (see Problem 4 of Chapter X) Let >p f 0, <p e F r\ E

Then Pip = 0, Qtp = ip, so \\(P — Q)ip\\ = ||^|| This implies that

P - eil > 1 The final statement follows from an elementary connectedness

argument |

Theorem XII.8 (Kato-Rellich theorem) Let T(P) be an analytic family

in the sense of Kato Let E0 be a nondegenerate discrete eigenvalue of T((i0)

Then, for P near p 0 , there is exactly one poiht £(/?) of ff (£(/?)) near E0 and

this point is isoiated and nondegenerate £(/?) is an analytic function of /? for

P near p 0 , and there is an analytic eigenvector Cl(p) for /? near p 0 If T(P) is

self-adjoint for P — Po real, then £!(/?) can be chosen to be normalized for

o real

Proof Pick e so that the only point of o(T(P0)) within {£ 11 £ — £ 0 1 < e} is

£ 0 • Since the circle {£ 11 £ — £ 0 1 = s} is compact and the set T of the last

theorem is open, we can find ó so that £ 4 o(T(P)) if | £ — £ 0 1 = i and

Po \ <$■ Let N = {p 11 p — p0 1 < 5} Then

P(P) = - (2ni)~' j ( T ( p ) - E Y 1 dE

| E - E o | = t

exists and is analytic for p e N. The nondegeneracy of £ 0 as an eigenvalue of

P(p) is one dimensional for all P e N. Thus, by Theorem XII.6, there is

exactly one eigenvalue E(p) of T(P) with | E{p) — £ 0 1 < e when p e N and

this eigenvalue is nondegenerate The analyticity of £(/?) follows from the

formula

( n 0 , ¿ W o )

We obtain an analytic eigenvector by choosing Q(P) = P(/?)£20 or

Q(P) = (fi0 » P(P)&o)~ ll 2 P(P)&o

in the real case, where Q0 is the unperturbed eigenvector

We thus see how easy it is to prove that energy levels are analytic in the

coupling constant i f we know that T(P) is an analytic family This would not

be very useful if we did not have convenient criteria for T(P) to be analytic

Fortunately, there are two simple ones reflecting the usual operator/form

dualism We shall discuss the operator criterion in detail and the form

criterion briefly

Definition Let R be a connected domain in the complex plañe and let

r(/J), a closed operator with nonempty resolvent set, be given for each P e R

We say that T(P) is an analytic family of type (A) if and only if

(i) The operator domain of T(p) is some set D independent of p.

(ii) For each i¡/ e Z), T((¡)\j/ is a vector-valued analytic function of /?

Of course, every family of type (A) is an analytic family in the sense of Kato We leave the general case of this theorem to the problems and con-

sider only the linear case T(P) = H 0 + fiV. We first prove a lemma that is of

interest in itself since it is a convenient criterion for a family to be type (A)

Lemma Let H 0 be a closed operator with nonempty resolvent set Define

H 0 + p v on D(H0) n D(V). Then H 0 + p v is an analytic family of type (A)

near p = 0 if and only if:

Proof Suppose first that H 0 4- PV is an analytic family of type (A) Then

D(Hq) - D{H0 + PV) = D(H0) n D(V) so (a) holds Since H 0 is^ closed,

D(H0) with the norm |||^||| = || H 0 \j/1| + || $ || is a Banach space D. Fix P

small and positive so that p and — P are both in the domain of analyticity

Ho + PV: D -► is every where defined and has a closed graph in Ó x j f

since the graph is closed in x with a weaker topology Thus, by the

closed graph theorem,

||(H0 + / W | | sí a, HWH

and

||(H0 - W | | < a 2 | W |Thus,

l l ^ l l < ( 2 ^ ) - 1 [ ||(//0 + W | | + \\(H0 - flV)ip\\]

< ( 2 P ) ' 1{al + a 2)|||^|||

so that condition (b) holds

Conversely, let (a) and (b) hold Then, for i¡/ e D(H0),

I I H o < II(tfo + W I I + IPWWW

< ||(tfo + W l l + IP I« IIH 0 1 1| + \ P \ b M

Thus, if | P | < a~ *, we have

||H0^|| < (1 - \ P \ a ) - l \\(H0 4- pV)ip\\ + (1 - \fi\a)~lb \ p \ m

Therefore, H 0 + PV is closed on D(H0) for if \¡/„ -* \p in with \p„ e D(H0)

and (H0 + PV)\¡/His Cauchy, then H 0 \¡/„ is Cauchy by the above inequality

and thus ip e D(H0). That (H0 + ¡BV)i]/ is analytic for ip e D(H0) is

obvious |

It is a corollary of the above proof that if V is infinitesimally small with

respect to H 0 , then H 0 + p V is an entire family of type (A).

Example 1 Let V e L2(R3) 4- L“ (R3) and let H 0 = - A on L2(IR3) More

generally, let V = ^ V¡j with Vi} e L2 + L® and H 0 = — A on R3n Then

H0 + PV is an entire analytic family of type (A).

Example 2 It can be shown that if V < < H 0 and W < < H 0 , then

W < < H 0 4- V (Problem 11) Thus, letting H 0 == — A, — A2 — 2¡rx — l / r 2

on L2(R6) and V = | r t — r 2 1" *, we see H 0 4- PV is an analytic family of type

(A) In the approximation of infinite nuclear mass, H 4- V is the helium atom

llamiltonian (see Section XI.5 for the kinematics)

fheorem XI 1.9 Let H 0 + p V be an analytic family of type (A) in a

sgion R, Then H 0 + p V is an analytic family in the sense of Kato In

particular, if 0 6 R and if E 0 is an isolated nondegenerate eigenvalue of H 0 ,

hen there is a unique point E(p) of a ( H 0 4- p V ) near E0 when | p | is small

vhich is an isolated nondegenerate eigenvalue Moreover, E(P) is analytic

tear p = 0

troof Since analyticity is a local property, we suppose that 0 e R and

irove analyticity in the sense of Kato near p = 0 Choose X 4 <r(//0) Then

|Ff0 - A)- 1 and H 0(H0 — A)"1 = 1 4- X(H0 — A)- 1 are bounded Thus, for

ny <p g ,

|| V(H0 - A)“ V I < a\\H0(H0 - X)~ V || + b\\(H0 - X)~ V ||

Thus V(H0 ~ X)~l is bounded; so for P small, [1 + P V (H 0 — l] ~ 1 exists

and is analytic in /? (being given by a geometric series) Direct computation

(Problem 12) shows that (H0 — X)~ *[1 + PV(H0 — X)" *]“ 1 is an inverse for

(fí0 + 0 V - X), so for p small, X 4 <r(H0 + p v ) and (H0 + PV - X)~ 1 is

analytic in p, This proves that H 0 + PV is an analytic family in the sense of

Kato near P = 0 By writing H 0 + pv = (H0 + P0 V) + (P - P0)V, we prove

analyticity at P = p 0 |

Example 1, re v is ite d By Theorem X.15 and Theorem XII.9, it follows

that E0(P), the lowest eigenvalue of — A + p v is an analytic function of p in

a neighborhood of (/?0 , oo) where p 0 = inf[P > 0 | E 0(P) < 0) In applying

Theorem XII.9, we are assuming the nondegeneracy of the ground State

which we shall prove in Section XIII 12

Example 2, revisited h = —A x — 2/rl is an operator with an exactly

solvable eigenvalue problem Its lowest eigenvalue is E = — 1 H 0 is of the

form h ® 1 + l ® h o n L2(IR3) ® L2(R3) = Í?(R6), so its ground state energy

is —2 For \p\ small, the ground state energy E(P) is analytic with the

Taylor coefficients at p = 0 given by the Rayleigh-Schródinger formula dis-

cussed in Section 1

Physically, one is interested in the ground state energy £(1) of the helium

atom The question immediately arises as to whether the Taylor series for

E(P) about p = 0 has a radius of convergence bigger than 1 In Theorem

XII 11 below, we shall obtain explicit lower bounds on the radius of conver­

gence of the Rayleigh-Schródinger series, but our bounds will be crude and

we shall not be able to use them directly to prove that P = 1 is within the

circle of convergence By hard work one might be able to show P = 1 is

actually in the circle of convergence (we expect it is true), but the question is

really academic! For when P = 1, even if the series is convergent, a large

number of terms of the Taylor series are necessary to approximate £(1) well,

and the higher order Rayleigh-Schródinger coefficients are hard to compute

For example, the first-order approximate valué (Q0 , VQ0) for £(1) - £(0)

disagrees with experiment by about 15% It turns out that other methods,

which we shall discuss in Section XIII.2, can be used to obtain an accuracy

of better than 1 % with experiment (and if various relativistic corrections

are taken into account, of one part in 106) However, if p is small, perturba­

tion theory is more accurate It turns out that the ground state energy of Li +

is directly related to £ ( |) and it is given by the first-order approximant to

within 5% £(i), which is related to the ground state energy of B e+ + , is given

within 2%

Example 3 (hyperfine structure in hydrogen) Perturbation theory is con-

nected with one of the more spectacular agreements between theory and

experiment in quantum physics In the usual model for the hydrogen atom

there is one energy level near —13 eV, the ground State energy The physical

atom has two levels; this splitting is due to interactions between the mag-

netic moments of the electrón and proton It is the transition between these

levels that radio astronomers observe when looking for intergalactic gas

clouds, and it is this transition that is the dominant one in a hydrogen maser

For the latter reason, the energy difference is very well measured; in fact, in

units with ft = 1, so that AE has units of hertz (Hz) s cycles per second,

A £ (ls1/2) = 1,420,405,751.800 Hz

There is an oíd theory of the magnetic interaction due to Fermi and Segré

which is suggested by classical models of interacting magnets The Ferm i-

Segré potential has a coupling constant P made up of fundamental constants

(magnetic moments of the electrón and proton, the electric charge), a spin-

spin interaction and multiplicaron by p(r), the effective charge distribution

of the proton In practice p(r) is approximated by a 5 function, which means

that it is technically outside the mathematical theory we have discussed, but

a peaked smooth function p(r) is within our theory, and leads to approxi-

mately the same lowest order contributions to perturbation theory

In comparing theory and experiment an interesting problem arises The

physical constants needed to compute p are known to only about one part in

105 or 106 and A £ (ls1/2) = Pax + P2a2 + •••, where P 10~4 and a u a 2 ,

measured in units of the ground state energy of hydrogen, are about 1 For a

truly accurate comparison with experiment, one also looks at the hyperfine

splitting in the first excited State A£(2s1/2) = p b t + ¡92b 2 If we look at the

ratio A£(2s1/2)/A £(ls1/2), then since P is already approximately 10~4, an

error of P in its sixth place only affects (ax + Pa2)l(bx + p b 2) in its tenth

place! Experimentally:

(1.000034495)

The Fermi-Segré theory (with relativistic corrections) and lower order per­

turbation theory predicts

A £(2 s 1/2) = 1 00003445)

A £ ( 1s 1/2) 8 v VAAA' ^ - ’''

Better agreement than this would actually be embarrassing since the above

calculations ignore the finite size of the nucleus, corrections due to strong

interactions, etc

Let us return to general criteria for a linear function H 0 + flV to be an analytic family in the sense of Kato There is a form notion analogous to the

operator notion of family of type (A) An analytic family of type (b) is a

family of closed, strictly m-sectorial forms, q(P), one for each P in a región R

of the complex plañe, so that:

(i) The form domain of q(P) is some subspace F independent of /?

(ii) (4>, q(P)\l/) is an analytic function in R for each {¡/ e F.

If q(P) is an analytic family of type (b), then, for each P e R, there is asso-

ciated a unique closed operator T(P) by Theorem VIII 16 T(P) is called an

analytic family of type (B) As in the type (A) case, any analytic family of type

(B) is an analytic family in the sense of Kato, and H 0 + p v defined as a form

on Q{H0) n Q(V) is an analytic family of type (B) near p = 0 if and only if V

is H o form bounded

Type (B) methods can be used to extend the results discussed under

Example 1 above to potentials in the Rollnik class R + L°° Type (B)

techniques imply strong analyticity properties for H 0 + PV if H 0 and V are

positive:

Theorem XI 1.10 Let H 0 be positive and self-adjoint and let V be self-

adjoint Let F+ = j ( V + I V \ ); VL = | V — V). Suppose that:

Q(K+) n Q(H0) is dense

(ii) K is H 0 form bounded with relative bound zero

Then H 0 + p V is an analytic family of type (B) in the cut plañe

{P \ H ( —00, 01}.

A reference for the proof of this theorem can be found in the Notes

Example 4 From our discussion in Section XIII 12, it will follow that the

ground state of —d2/ d x 2 + x 2 + Px4 is nondegenerate if p > 0 Thus,

Theorem XII 10 says that its ground state energy E(P) is analytic in a neigh-

borhood of the positive real axis

There are examples of analytic families that are neither type (A) ñor type

(B) For example, let T(P) be an analytic family of type (A) and let C be any

bounded self-adjoint operator Then U(P) = exp (¡PC) is an en tire analytic

function It is not hard to see that T ( P ) = U(P)T(P)U(P)~l , defined on

D(T(P)) ñor Q(T(P)) is constant

We would like to make a few remarks, some of which are warnings about pitfalls First, we note that as in Section 1 one has explicit formulas for the

coefficients of the Taylor series for £(/?) given as contour integráis of resol-

vents If H 0 has purely discrete spectrum, we can do the integráis explicitly

and obtain formulas identical to those of the preceding section If H 0 is

self-adjoint, we can still do these contour integráis, obtaining spectral inte­

gráis in place of sums; for example, if £ 0 is an isolated nondegenerate

eigenvalue of H 0 so that dist(£0 , v ( H 0)\Eo) > ^ then

a 2 = - I (A - E 0)~ 1 d(VQ0 , P x Víl0)

| A - E 0 I > £

Secondly, we warn the reader that it may happen that the power series for

£(/?) has a circle of convergence larger than the circle in which H(P) has £(/?)

as an eigenvalue

Example 5 Let H 0 = — A — 1/r and V = 1/r Then, the eigenvalúes of

H 0 + PV for P small are — ¿n - 2 (l — p )2, h = 1, 2, In particular, the

ground State energy (n = 1) E 0(p) = — ¿ + \P - \ P l is given by a function

with an analytic continuation to the entire complex plañe But for p > 1, H 0

has no eigen valúes at a ll!

Thus, one vestige of the finite-dimensional theory is not present: In

general, the analytic continuation of an eigenvalue need not be an

eigenvalue However, in one important special case it can be proven that the

analytic continuation of an eigenvalue is an eigenvalue (see Problem 13)

«

Finally, we note that one can obtain explicit lower bounds on the radius of

convergence of the Taylor series:

Theorem Xll.11 Suppose that || V<p\\ < a\\H0 <p|| + b\\(p\\. Let H 0 be self-

adjoint with an unperturbed isolated, nondegenerate eigenvalue £ 0 , and let

e = i dist(£0 , <t(H 0 )\{Eq}). Define

r(a, by E 0 , s) = [a + l [b + u ( |£ 0 1 + fi)]]’ 1

Then the eigenvalue E(P) of H 0 + p v near £ 0 is analytic in the circle of

radius r(a, b, £ 0 , e)

The reader is asked to provide a proof in Problem 14

♦ * *

As a final subject in regular perturbation theory, we shall discuss the case

where E 0 is an isolated degenerate eigenvalue of finite multiplicity of T(0O)

Because of our experience with the finite-dimensional case, we shall suppose

that T(P) is self-adjoint for 0 real If T(P) is a K ato family, we have no

trouble in proving that P(0) = ( — 2ni )~ 1 j> (T(¡3) — E )~ 1 dE is analytic in 0

for 0 near p o We are thus faced with finding the eigenvalues of H (0) res-

tricted to the variable finite-dimensional subspace Ran P(P). To reduce this

to a truly finite-dimensional problem, we need the following technical result

of Kato, which has several other applications (see Problems 15 and 17)

T heorem XI 1.12 Let R be a connected, simply connected región of the

complex plañe containing 0 Let P(P) be a projection-valued analytic func­

tion in R. Then, there is an analytic family U(P) of invertible operators with

U(fi)P(0)U(P)- 1 = P(P)

Moreover, if P(p) is self-adjoint for P real and in P, then we can choose U(P)

unitary for 0 real

We defer the proof to the conclusión of this section

T heorem XI 1.13 Let T(P) be an analytic family in the sense of K ato for

P near 0 that is self-adjoint for P real Let E 0 be a discrete eigenvalue of

multiplicity m Then, there are m not necessarily distinct single-valued

functions, analytic near P = 0, E{1)(P) , , £ (m)(0), with E{k)(0) = E 0 , so that

£ (1)(0), ., £ (m)(0) are eigenvalues of T(P) for p near 0 (with a repeated

entry in £ (1), £ (w) indicating a degenerate eigenvalue) Further, these are

the only eigenvalues near E 0

Proof Since E 0 is an isolated point of cr(T(0)), and T(P) is an analytic

family, P(P) = ( — 2ni )~ 1 j> (T(P) — E ) ~ 1 dE exists and is analytic for p

small By the proof of Theorem XII.6,

o(T(P) r Ran P(p)) = o(T(P)) n { E \ \ E - E 0 \ < s}

From Theorem XII 12 we know that there exists a family U(P\ analytic near

0 = 0, unitary for p real, so that U(P)P(Q)U(P) ~ 1 = P(0) Let f (0) =

U(P)~ l T(P)U(P). Then Ran P(0) is an invariant subspace for all the T(0)

Thus S(P) s f ( P ) \ Ran P(0) is a finite-dimensional analytic family, self-

adjoint for p real The theorem now follows from Rellich’s theorem

(Theorem XII.3) |

Since we have reduced the infinite-dimensional problem to a finite-

dimensional one, the existence of analytic eigenvectors in the finite-

dimensional case implies their existence in the infinite-dimensional case

Example 1, revisited If H 0 + p 0 V has n eigenvalues in ( — o o , 0 ) , then

H 0 + p V has at least n eigenvalues for \P - p 0 \ small, and the ones near

those of H 0 + po V are analytic in P near p 0

Finally, we shall prove Theorem XII 12 The idea of the proof comes from diflerentiating U(P)P(0)U(P) ' 1 = P(jS), finding P'(p) = [U,(p)U(P)' \ P(p)]

where [A , B] = A B — BA. Thus, one seeks an operator Q(P) that satisfies

Lemma Let R be a connected, simply connected subset of C with 0 e R

and let A(P) be an analytic function on R with valúes in the bounded opera-

tors on some Banach space X Then for any x0 e X , there is a unique

function / (/?), analytic in R , with valúes in X obeying

d

d P

Proof By standard methods of analytic continuation, it is enough to sup­

pose that R is a circle of radius r0 and to show that there is an analytic

solution in the circle of radius r0 — 2e for any e We first note that unique-

ness follows from (2): By supposing f ( P ) = £*= o /« P" and knowing

A(f¡) = ^ ° =0 A„pn, one finds that

We now show that if the f n are defined by (3), then f„ p n converges if

\P\ < r0 — 2e Let M = max{l -f \\A(P)\\ I \P\ < rQ - e} By the Cauchy

integral formula, ¡ A J < M (r0 — s) ".A simple inductive argument using (3)

shows that \\fn II < (M(r0 — e)~ ^"IIxq II It follows that f ( P ) is analytic in a

circle of radius (r0 — e)/M By repeating this argument at points j3 with \p\

near (r0 — e)/M, we can prove that / is analytic in a circle of radius

(r0 ~ e ) M ~ l + M l (r0 — e)(l — M ”"1) See Figure XII 1 After a finite

number of repetitions we get analyticity in a circle of radius r0 - 2e |

[£(/?), P(£)] = P'(/2)P(/?) + P(/?)P'(0) - 2P(jí)P'0í)P(j8) = P'O?)

by (4)

(ii) Using the lemma with X = y y( ), solve dU/dft = Q(P)U(P) and

U(P)V{P) - V(P)U(P) = 1; in particular, U(P) is invertible For

and thus F(7 = 1 On the other hand, if F(P) = U(p)V(p), then F(P) solves

the differential equation d F / d f = Q(P)F(P) - F(fi)Q(P); F(0) = 1 Since

F(P) = 1 solves this equation with initial condition, we conclude that

F(f) — 1 by the uniqueness of Solutions proven in the lemma

(iii) U(P)P(0)V(P) = P(P). For let P(fi) = U(P)P( O)V(P). Then

dP(P)/dp = [Q(P\ P{P)] with initial condition P(0) = P(0) On the other

hand, by step (i), P(p) also solves dP(P)/dp = [Q(p), P(P)] with P(P)\p=0

P(0) By the uniqueness of Solutions of differential equations, P(P) = P(P).

(iv) Finally, we must prove that U(p) is unitary for ¡3 real if P(P) is self-adjoint for p real Thus, let us suppose that P(P)* = P(P) if P = p. By the

Schwarz reflection principie, it follows that P(P)* = P(j8) for all p. By the

definition of Q, Q(P)* = —Q(P)> Let V (P )= U (p )* Then V obeys

dV/dp = — V(P)Q(P)\ F(0) = / By the uniqueness of Solutions of differen tial

equations, V(P) = V(p). Thus, for /? real, U(p)* = V(P) = V(P) = U(p)~ l, so

U is unitary |

XII.3 Asymptotic perturbation theory

The elegant regular perturbation theory developed in the preceding sec- tion is not always applicable, even to simple-looking examples, Consider

the family of Hamiltonians H(P) = H 0 4- p V where .W = L2(íR),

H 0 = —d 2/ d x 2 4* x 2, and V = x4 We discussed the self-adjointness of this

family from several points of view in Chapter X For any p > 0, H(P) is

self-adjoint on D(H0) n D(V) = D(p2) n Z)(x4); see Problem 23 of Chapter

X Since D(H0) = D(p2) n D(x2), we see that the domain changes as soon as

the perturbation is tum ed on Thus the analyticity criterion of Theorem

XII.9 is not applicable A similar change occurs in the form domain Q(H{p))

In fact, no analyticity criterion can hold since the perturbation series about

p = 0 diverges.

The following argument has been made by various authors to predict this divergence of the perturbation series for the eigenvalues of H(P) for P ^ 0: If

P is negative, then x 2 + /?x4 — oo as x ± oo, so H 0 + PV is qualitatively

very difierent from H 0 (it is actually not even essentially self-adjoint) For

this reason, one expects that the perturbation series should diverge for p

%

negative Since power series converge in circles, the series should not converge

for any /? Whether or not one wants to accept this heuristic argument, its

conclusión is correct A detailed analysis allows one to prove that the

Rayleigh-Schródinger coefficients a„ for £ 0(/J), the ground State energy, obey

|a„ | > A B nr(n/2) for suitable constants A and B.

Thus, one is faced with deciding whether the perturbation series makes any sense in this case It is precisely this question that we consider in this

section and the next The meaning of divergent perturbation series is of

interest in a wider area than the realm of nonrelativistic quantum mechanics

The most useful calculational tool in certain (at present ill-defined) quantum

field theories is another perturbation series known as the Gell’M ann-Low

series or Feynman series In some cases these series have been proven to

diverge and they are believed to be divergent in others For this reason and

especially since there are formal similarities between some field theory H am ­

iltonians and p2 + x 2 + fix4 (see Section X.7), the problems we study in

these two sections are relevant to quantum field theory

The simplest interpretation of a formal series is as an asymptotic series:

Definition Let / be a function defined on the positive real axis We say

that anz” is asymptotic to / as z J, 0 if and only if, for each fixed N,

N

lim \ f ( z ) ~ Y a nzn\ z N = 0

: 1 0 \ n = 0

I f / is defined in a sectorial región of the complex plañe {z|0 < |z | < B\

arg z | < 0}, we say that £ an zn is asymptotic to / as | z | -* 0 uniformly in

the sector if, for each N,

Example 1 Consider the function / ( z ) = exp( —z “ for z > 0 Then

z ~ nf (z) -> 0 as z | 0, s o / h a s zero asymptotic series In fact, the zero series is

asymptotic uniformly in any sector |arg z\ < 6 with 6 < 7ü/2

This example illustrates an important fact about asymptotic series: Two dijferent functions may have the same asymptotic series. Saying that / has a

certain asymptotic series gives us no information about the valué of/ (z) for

some fixed nonzero valué of z We know th a t/( z ) is well approximated by

aQ + z as z gets “ small,” but the definition says nothing about how small

is “ small.” If an asymptotic series ^ ° « 0 a«z" *s not convergent, the typical

behavior is the following: If z is “ small,” the first few partial sums are a fairly

good approximation to / (z), but as N -► oo, the sums oscillate wildly and no

longer approximate / very well For example, we shall presently show that

the Rayleigh-Schródinger series for E0(P), the ground state energy of the

Hamiltonian, p2 + x 2 + / x 4 (/? > 0), is asymptotic to E 0(P) as fi ¡ 0 For

/? = 0.2, variational methods (see Section XIII.2) show that E 0(P) =

1.118292 The first 15 partial sums are given in the accompanying

Thus, we see the typical behavior of wandering near the right answer for a

while (and not even that near!) and then going wild And as N gets larger,

things get worse: The 50th partial sum is about 1045 in magnitude and the

lOOOth about ÍO2000 in magnitude

Example 2 L e t / b e C 00 on [ — 1, 1] Then ( / <w)(0)/n!)x” is asympto­

tic as x l 0 or x | 0 to / For Taylor’s theorem with a remainder says that

/( * )

n = O n!

< x

7V + 1 ( N + 1)! | a | £ | x |s u p [ | /<"+1)(a)|]

Since C 00 functions can be nonanalytic, this example shows that asymptotic

series may not converge; and even if they do, the sum may not have anything

to do with the function / (see Example 1)

We defined analytic families on open sets We will abuse this terminology

by saying that a family is analytic on a closed set if it is norm continuous

on the set and analytic in the interior

Theorem XI 1.14 Let H 0 be a self-adjoint operator Suppose that //(/?) is

an analytic family in the región {/? 10 < | /? | < JB; | arg /? | < 0} and that the

following conditions are obeyed:

(a) lim ||(H(P) — X)~x — (H0 - X)~ 11| = 0 for some X 4 <r(H0).

IPI-o

| a r g / ? | < 0

(b) There is a closed symmetric operator V so that C°°(/í0) c D(V) and

(c)

V[C*(H0)] c C°°(Ho)

for all in the sector and for \¡/ e C°°(//0),

H 0 il/ + pV\lt.

Let £ 0 be an isolated nondegenerate eigenvalue of H 0 Then if \{l\ is small

and |arg f$\ < 0, there is exactly one eigenvalue £(/?) of H(¡i) near E 0

Moreover, the formal Rayleigh-Schródinger series for the

eigenvalue of H 0 + p v is finite term by term and is an asymptotic series for

£(/?) uniformly in the sector Explicitly,

This is the main result of this section The conclusión makes several

distinct statements, and for this reason we divide the proof into several

lemmas The first conclusión States that there is an eigenvalue E(fi) of

H 0 + p v near E 0 if /? is small We shall emphasize this property by giving it

a ñame (stability) In Sections 5 and 6 we discuss situations where such

stability does not hold The second part of the conclusión concerns the

asymptotic nature of the perturbation series for £(/?) A similar result holds

for the eigenvector associated with £(/?) (see Problem 24) The main tool

used in establishing the asymptotic property is familiar from Section 2: We

use the formulas

and

P(P) = - ¿ f (H(P) - E ) ~ 1 dE

As we have seen, these formulas allow one to unravel the complicated struc-

ture of the Rayleigh-Schródinger coefficients into several simple operations

on a geometric series O ur main tool will be the well-known error term for

this series

Definition Let A(fi) be a family of operators in the set {/?|0 < |/?| < B ,

arg P\ < 0} Suppose that an operator A 0 exists so that for some X $ a ( A 0\

s-lim (A(P) — X)~ 1 = (A0 — X)~ 1

101-0

|arg f i \ £ 0