lecture notes in mathematical physics

Lecture Notesin Mathematical Physics

Ivan Avramidi

New Mexico Institute of Mining and Technology

Socorro, NM 87801

October 19, 2000

Contents

1 Functional Analysis 11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 Normed Linear Spaces . . . . . . . . . . . . . . . . . . 61.1.4 Notes on Lebesgue Integral . . . . . . . . . . . . . . . . 8

1.2 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.1 Geometry of Hilbert Space . . . . . . . . . . . . . . . . 111.2.2 Examples of Hilbert Spaces . . . . . . . . . . . . . . . 121.2.3 Projection Theorem . . . . . . . . . . . . . . . . . . . . 141.2.4 Orthonormal Bases . . . . . . . . . . . . . . . . . . . . 151.2.5 Tensor Products of Hilbert Spaces . . . . . . . . . . . . 16

2 Asymptotic Expansions 192.1 Asymptotic Estimates . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 202.1.2 Properties of asymptotic estimates . . . . . . . . . . . 20

2.2 Asymptotic Sequences . . . . . . . . . . . . . . . . . . . . . . 212.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.2 Properties of asymptotic sequences . . . . . . . . . . . 21

2.3 Asymptotic Series . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Asymptotics of Integrals: Weak Singularities . . . . . . . . . . 23

2.4.1 Power Singularity on a Bounded Interval . . . . . . . . 242.4.2 Power singularity on Unbounded Interval . . . . . . . . 26

3 Laplace Method 293.1 Laplace Integrals in One Dimension . . . . . . . . . . . . . . . 29

3.1.1 Watson Lemma . . . . . . . . . . . . . . . . . . . . . . 29

I

II Contents

3.1.2 Interior Nondegenerate Maximum Point . . . . . . . . 303.1.3 Boundary Maximum Point . . . . . . . . . . . . . . . . 32

3.2 Background from Analysis . . . . . . . . . . . . . . . . . . . . 333.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.2 Morse Lemma . . . . . . . . . . . . . . . . . . . . . . . 353.2.3 Gaussian Integrals . . . . . . . . . . . . . . . . . . . . 36

3.3 Laplace Integrals in Many Dimensions . . . . . . . . . . . . . 373.3.1 Interior Maximum Point . . . . . . . . . . . . . . . . . 373.3.2 Boundary Maximum Point . . . . . . . . . . . . . . . . 393.3.3 Integral Operators with Singular Kernels . . . . . . . . 41

4 Stationary Phase Method 434.1 Stationary Phase Method in One Dimension . . . . . . . . . . 43

4.1.1 Fourier Integrals . . . . . . . . . . . . . . . . . . . . . 434.1.2 Localization Principle . . . . . . . . . . . . . . . . . . . 444.1.3 Boundary Points . . . . . . . . . . . . . . . . . . . . . 454.1.4 Standard Integrals . . . . . . . . . . . . . . . . . . . . 464.1.5 Stationary Point . . . . . . . . . . . . . . . . . . . . . 474.1.6 Principal Values of Integrals . . . . . . . . . . . . . . . 47

4.2 Stationary Phase Method in Many Dimensions . . . . . . . . . 494.2.1 Nondegenerate Stationary Point . . . . . . . . . . . . . 494.2.2 Integral Operators with Singular Kernels . . . . . . . . 50

5 Saddle Point Method 535.1 Saddle Point Method for Laplace Integrals . . . . . . . . . . . 53

5.1.1 Heuristic Ideas of the Saddle Point Method . . . . . . . 535.1.2 Level Curves of Harmonic Functions . . . . . . . . . . 565.1.3 Analytic Part of Saddle Point Method . . . . . . . . . 585.1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 60

Notation 63

Bibliography 67

Chapter 1

Functional Analysis

1.1 Preliminaries

1.1.1 Metric Spaces

Definition 1 A metric space is a set X and a mapping d : X ×X → R,called a metric, which satisfies:

i) d(x, y) ≥ 0 (1.1)

ii) d(x, y) = 0 ⇐⇒ x = y (1.2)

iii) d(x, y) = d(y, x) (1.3)

iv) d(x, y) ≤ d(x, z) + d(z, y) (1.4)

Definition 2 A sequence xn∞n=1, of elements of a metric space (X, d) issaid to converge to an element x ∈ X if d(x, xn)

n→∞−→ 0.

Definition 3 Let (X, d) be a metric space.

a) the set B(y, r) = x ∈ X | d(x, y) < r is called the open ball ofradius r about y;

the set B(y, r) = x ∈ X | d(x, y) ≤ r is called the closed ball ofradius r about y;

the set S(y, r) = x ∈ X | d(x, y) = r is called the sphere of radiusr about y;

1

2 Functional Analysis

b) a set O ⊂ X is called open if ∀y ∈ O ∃r > 0: B(y, r) ⊂ O;

c) a set N ⊂ X is called a neighborhood of y ∈ N if ∃r > 0: B(y, r) ⊂N ;

d) a point x ∈ X is a limit point of a set E ⊂ X if ∀r > 0: B(x, r) ∩(E \ x) = ∅, i.e. if E contains points other than x arbitrarily closeto x;

e) a set F ⊂ X is called closed if F contains all its limit points;

f) x ∈ G ⊂ X is called an interior point of G if G is a neighborhoodof x.

g) The intersection S of all closed sets containing S ⊂ X is called theclosure of S. The closure of S ⊂ X is the smallest set containg S.

h) The interior of S, S, is the set of interior points. It is the largestopen set contained in S.

i) The boundary of S is the set ∂S = S \ S.

Theorem 1 Let (X, d) be a metric space.

a) A set O is open iff X \O is closed

b) xnd−→ x iff ∀ neighborhood N of x ∃m: n ≥ m implies xn ∈ N ;

c) the set of interior points of a set is open;

d) the union of a set and all its limit points is closed;

e) a set is open iff it is a neighborhood of each of its points.

f) The union of any number of open sets is open.

g) The intersection of finite number of open sets is open.

h) The union of finite number of closed sets is closed.

i) The intersection of any number of closed sets is closed.

j) The empty set and the whole space are both open nd closed.

Functional Analysis 3

Theorem 2 A subset S of a metric space X is closed iff every convergentsequence in S has its limit in S, i.e.

xn∞n=1, xn ∈ S, xn → x =⇒ x ∈ S (1.5)

Theorem 3 The closure of a subset S of a metric space X is the set of limitsof all convergent sequences in S,i.e.

S = x ∈ X | ∃xn ∈ S : xn → x. (1.6)

Definition 4 A subset, Y ⊂ X, of a metric space (X, d) is called dense if

∀x ∈ X ∃ yn∞n=1, yn ∈ Y ,: ynd−→ x.

Theorem 4 Let S be a subset in a metric space X. Then the followingconditions are equivalent:

a) S is dense in X,

b) S = X,

c) every non-empty subset of X contains an element of S.

Definition 5 A metric space X is called separable if it has a countabledense set.

Definition 6 A subset S ⊂ X of a metric space X is called compactif every sequence xn in S contains a convergent subsequence whose limitbelongs to S.

Theorem 5 Compact sets are closed and bounded.

Definition 7 A sequence xn∞n=1, of elements of a metric space (X, d) iscalled a Cauchy sequence if ∀ε > 0 ∃N : n,m ≥ N implies d(xn, xm) < ε.

Proposition 1 Any convergent sequence is Cauchy.

Definition 8 A metric space in which all Cauchy sequences converge iscalled complete.


Definition 9 A mapping f : X → Y from a metric space (X, d) to a

metric space (Y, ρ) is called continuous at x if f(xn)ρ−→ f(x) ∀xn∞n=1,

xn ∈ X, xnd−→ x, i.e. the image of a convergent sequence converges to the

image of the limit.

Definition 10 A bijection (one-to-one onto mapping) h : X → Y from(X, d) to (Y, ρ) is called isometry if it preserves the metric, i.e.

ρ(h(x), h(y)) = d(x, y) ∀x, y ∈ X (1.7)

Proposition 2 Any isometry is continuous.

Theorem 6 If (X, d) is an incomplete metric space, it is possible to find acomplete metric space (X, d) so that X is isometric to a dense subset of X.

1.1.2 Vector Spaces

Definition 11 A complex vector space is a nonempty set V with twooperations: + : V × V → V and · : C × V → V that satisfy the followingconditions:

∀x, y, z ∈ Vi) x+ y = y + x (1.8)

ii) (x+ y) + z = x+ (y + z) (1.9)

iii) ∃0 ∈ V : ∀x ∈ V : x+ 0 = x (1.10)

iv) ∀x ∈ V ∃(−x) ∈ V : x+ (−x) = 0 (1.11)

∀α, β ∈ C, ∀x, y ∈ Vv) α(βx) = (αβ)x (1.12)

vi) (α + β)x = αx + βx (1.13)

vii) α(x+ y) = αx + αy (1.14)

viii) 1 · x = x (1.15)

A real vector space is defined similarly.


Examples (Function Spaces). Let Ω ⊂ Rn be an open subset of Rn.

1. P (Ω) is the space of all polynomials of n variables as functions on Ω.

2. C(Ω) is the space of all continuous complex valued functions on Ω.

3. Ck(Ω) is the space of all complex valued functions with continuouspartial derivatives of order k on Ω.

4. C∞(Ω) is the space of all infinitely differentiable complex valued (smooth)functions on Ω.

Example (Sequence Spaces (lp-Spaces)). Let p ≥ 1. lp is the space ofall infinite sequences zn∞n=1 of complex numbers such that(

∞∑n=1

|zn|p) 1

p

<∞ . (1.16)

Definition 12 Let V be a complex vector space and let x1, . . . xk ∈ V andα1, . . . , αk ∈ C. A vector x = α1x1 + · · ·αkxk is called a linear combina-tion of x1, . . . xk.

Definition 13 A finite collection of vectors x1, . . . , xk is called linearlyindependent if

k∑i=1

αixi = 0 ⇐⇒ αi = 0 , i = 1, 2, . . . , k . (1.17)

An arbitrary colection of vectors B = xn∞n=1 is called linearly independentif every finite subcollection is linearly independent. A collection of vectorswhich is not linearly independent is called linearly dependent.

Definition 14 Let B ⊂ V be a subset of a vector space V . Then spanB isthe set of all finite linear combinations of vectors from B

spanB =

k∑i=1

αixi

∣∣∣ xi ∈ B, αi ∈ C, k ∈ N . (1.18)

Proposition 3 Let B ⊂ V be a subset of a vector space V . Then spanB isa subspace of V .


Definition 15 A set of vectors B ⊂ V is called a basis of V (or a baseof V ) if B is linearly independent and spanB = V . If ∃ a finite basis in V ,then V is called finite dimensional vector space. Otherwise V is calledinfinite dimensional vector space.

Proposition 4 The number of vectors in any basis of a finite dimensionalvector space is the same.

Definition 16 The number of vectors in a basis of a finite dimensional vec-tor space is called the dimension of V , denoted by dimV .

1.1.3 Normed Linear Spaces

Definition 17 A normed linear space is a vector space, V , over C (orR) and a mapping || · || : V → R, called a norm, that satisfies:

i) ||v|| ≥ 0 ∀v ∈ V (1.19)

ii) ||v|| = 0 ⇐⇒ v = 0 (1.20)

iii) ||αv|| = |α| ||v|| ∀v ∈ V, ∀α ∈ C (1.21)

iv) ||v + w|| ≤ ||v||+ ||w|| ∀v, w ∈ V (1.22)

Examples.

1. Norms in Rn:

||x||2 =

(k∑i=1

x2i

) 12

(1.23)

||x||1 =n∑i=1

|xi| (1.24)

||x||∞ = max1≤i≤n

|xi| (1.25)

2. A norm in Cn

||z|| =

(k∑i=1

|zi|2) 1

2

(1.26)


3. Let Ω ⊂ Rn be a closed bounded subset of Rn and dx = dx1 · · · dxn bea measure in Rn. Norms in C(Ω) can be defined by

||f ||∞ = supx∈Ω|f(x)| (1.27)

||f ||p =

(∫Ω

|f(x)|p dx) 1

p

(1.28)

||f ||1 =

∫Ω

|f(x)| dx (1.29)

4. A norm in lp

||z|| =

(k∑i=1

|zi|p) 1

p

(1.30)

5. A norm in l∞

||z|| = supn∈N|zn| (1.31)

Proposition 5 A normed linear space (V, || · ||) is a metric space (V, d) withthe induced metric d(v, w) = ||v − w||.

Convergence, open and closed sets, compact sets, dense sets, com-pleteness, in a normed linear space are defined as in a metric space in theinduced metric.

Definition 18 A normed linear space is complete if it is complete as a metricspace in the induced metric.

Definition 19 A complete normed linear space is called the Banach space.

Definition 20 A bounded linear transformation from a normed linearspace (V, ||·||V ) to a normed linear space (W, ||·||W ) is a mapping T : V → Wthat satisfies:

i) T (αv + βw) = αT (v) + βT (w), ∀v, w ∈ V, ∀α, β ∈ C;

ii) ||T (v)||V ≤ C||v||W for some C ≥ 0.


iii) the number

||T || = supv∈V,v 6=0

||T (v)||W||v||V

(1.32)

is called the norm of T .

Theorem 7 Any bounded linear tranformation between two normed linearspaces is continuous.

Theorem 8 A bounded linear transformation, T : V →W , from a normedlinear space (V, || · ||V ) to a complete normed linear space (W, || · ||W ) can beuniquely extended to a bounded linear transformation, T , from the completionV of V to (W, || · ||W ). The extension of T preserves the norm ||T || = ||T ||.

1.1.4 Notes on Lebesgue Integral

Definition 21 Characteristic function of a set A ⊂ X is a mappingχA : X → 0, 1 defined by

χA(x) =

1, if x ∈ A0, if x 6∈ A (1.33)

Definition 22 For a non-zero function f : Rn → R, the set, supp f , of allpoints x ∈ Rn for which f(x) 6= 0 is called the support of f , i.e.

supp f = x ∈ Rn|f(x) 6= 0 . (1.34)

Clearly, suppχA = A.

Definition 23 Let I be a semi-open interval in Rn defined by

I = x ∈ Rn | ak ≤ xk < bk, k = 1, . . . , n (1.35)

for some ak < bk. The measure of the set I is defined to be

µ(I) = (b1 − a1) · · · · (bn − an) . (1.36)

The Lebesgue integral of a characteristic function of the set I is definedby ∫

χIdx = µ(I) . (1.37)


Definition 24 A finite linear combination of characteristic functions of semi-open intervals

f =N∑k=1

αkχIk (1.38)

is called a step function.

Definition 25 The Lebesgue integral of a step function is defined bylinearity ∫ N∑

k=1

αkχIkdx =N∑k=1

αkµ(Ik) . (1.39)

Definition 26 A function f : Rn → R is Lebesgue integrable if ∃ a

sequence of step functions fk such that

f '∞∑k=1

fk , (1.40)

which means that two conditions are satisfied

a)∞∑k=1

∫|fk| dx <∞ (1.41)

b) f(x) =∞∑k=1

fk(x) ∀x ∈ Rn such that∞∑k=1

|fk(x)| <∞ . (1.42)

The Lebesgue integral of f is then defined by∫f dx =

∞∑k=1

∫fk dx (1.43)

Proposition 6 The space, L1(Rn), of all Lebesgue integrable functions onRn is a vector space and

∫is a linear functional on it.

Theorem 9 a) If f, g ∈ L1(Rn) and f ≤ g, then∫fdx ≤

∫gdx.

b) If f ∈ L1(Rn), then |f | ∈ L1(Rn) and |∫f dx| ≤

∫|f |dx.


Theorem 10 If fk is a sequence of integrable functions and

f '∞∑k=1

fk , (1.44)

then ∫f =

∞∑k=1

∫fk , (1.45)

Definition 27 The L1-norm in L1(Rn) is defined by

||f || =∫|f |dx (1.46)

Definition 28 A function f is called a null function is it is integrableand ||f || = 0. Two functions f and g are said to be equivalent if f − g isa null function.

Definition 29 The equivalence class of f ∈ L1(Rn), denoted by [f ], isthe set of all functions equivalent to f .

Remark. Strictly speaking, to make L1(Rn) a normed space one has toconsider instead of functions the classes of equivalent functions.

Definition 30 A set X ⊂ Rn is called a null set (or a set of measurezero) if its characteristic function is a null function.

Theorem 11 a) Every countable set is a null set.

b) A countable union of null sets is a null set.

c) Every subset of a null set is a null set.

Definition 31 Two integrable functions, f, g ∈ L1(Rn), are said to be equalalmost everywhere, f = g a.e., if the set of all x ∈ Rn for which f(x) 6=g(x) is a null set.

Theorem 12

f = g a.e ⇐⇒ ||f − g|| =∫|f − g| = 0 (1.47)

Theorem 13 The space L1(Rn) is complete.


1.2 Hilbert Spaces

1.2.1 Geometry of Hilbert Space

Definition 32 A complex vector space V is called an inner product space(or a pre-Hilbert space if there is a mapping (·, ·) : V × V → C, calledan inner product, that satisfies: ∀x, y, z ∈ V, ∀α ∈ C:

i) (x, x) ≥ 0 (1.48)

ii) (x, x) = 0 ⇐⇒ x = 0 (1.49)

iii) (x, y + z) = (x, y) + (x, z) (1.50)

iv) (x, αy) = α(x, y) (1.51)

v) (x, y) = (y, x)∗ (1.52)

Definition 33 Let V be an inner product pace.

i) Two vectors x, y ∈ V are said to be orthogonal if (x, y) = 0;

ii) A collection, xiNi=1, of vectors in V is called an orthonormal setif (xi, xj) = δij, i.e. (xi, xj) = 1 if i = j and (xi, xj) = 0 if i 6= j.

Theorem 14 Every inner product space is a normed linear space with thenorm ||x|| =

√(x, x) and a metric space with the metric d(x, y) =

√(x− y, x− y).

Theorem 15 (Pythagorean Theorem) Let V be an inner product spaceand xnNn=1 be an orthonormal set in V . Then ∀x ∈ V

||x||2 =N∑n=1

|(x, xn)|2 +

∣∣∣∣∣∣∣∣∣∣x−

N∑n=1

(x, xn)xn

∣∣∣∣∣∣∣∣∣∣2

(1.53)

Theorem 16 (Bessel inequality) Let V be an inner product space andxnNn=1 be an orthonormal set in V . Then ∀x ∈ V

||x||2 ≥N∑n=1

|(x, xn)|2 (1.54)

Theorem 17 (Schwarz inequality) Let V be an inner product space. Then∀x, y ∈ V

|(x, y)| ≤ ||x|| ||y||. (1.55)


Theorem 18 (Parallelogram Law) Let V be an inner product space. Then∀x, y ∈ V

||x+ y||2 + ||x− y||2 = 2||x||2 + 2||y||2. (1.56)

Definition 34 A sequence xn of vectors in an inner product space V iscalled strongly convergent to x ∈ V , denoted by xn → x, if

||xn||n→0−→ 0 (1.57)

and weakly convergent to x ∈ V , denoted by xnw→ x, if

(xn − x, y)n→0−→ 0 ∀y ∈ V . (1.58)

Theorem 19

a) xn → x =⇒ xnw→ x (1.59)

b) xnw→ x and ||xn|| → ||x|| =⇒ xn → x (1.60)

Definition 35 A complete inner product space is called a Hilbert space.

Definition 36 A linear transformation U : H1 → H2 from a Hilbert spaceH1 onto the Hilbert space H2 is called unitary if if it preserves the innerproduct, i.e. ∀x, y ∈ H1

(Ux, Uy)H2 = (x, y)H1 . (1.61)

Definition 37 Two Hilbert spaces H1 and H2 are said to be isomorphicif there is a unitary linear transformation U from H1 onto H2.

Definition 38 Let H1 and H2 be Hilbert spaces. The direct sum H1⊕H2

of Hilbert spaces H1 and H2 is the set of ordered pairs z = (x, y) with x ∈ H1

and y ∈ H2 with inner product

(z1, z2)H1⊕H2 = (x1, x2)H1 + (y1, y2)H2 (1.62)

1.2.2 Examples of Hilbert Spaces

1. Finite Dimensional Vectors. CN is the space of N -tuples x =

(x1, . . . , xN) of complex numbers. It is a Hilbert space with the inner product

(x, y) =N∑n=1

x∗nyn . (1.63)


2. Square Summable Sequences of Complex Numbers. l2 is thespace of sequences of complex numbers x = xn∞n=1 such that

∞∑n=1

|xn|2 <∞ . (1.64)

It is a Hilbert space with the inner product

(x, y) =∞∑n=1

x∗nyn . (1.65)

3. Square Integrable Functions on R. L2(R) is the space of complexvalued functions such that ∫

R

|f(x)|2 dx <∞ . (1.66)


(f, g) =

∫R

f ∗(x)g(x) dx (1.67)

4. Square Integrable Functions on Rn. Let Ω be an open set in Rn (inparticular, Ω can be the whole Rn). The space L2(Ω) is the set of complexvalued functions such that ∫

Ω

|f(x)|2dx <∞ , (1.68)

where x = (x1, . . . , xn) ∈ Ω and dx = dx1 · · · dxn. It is a Hilbert space withthe inner product

(f, g) =

∫Ω

f ∗(x)g(x) dx (1.69)

5. Square Integrable Vector Valued Functions. Let Ω be an open setin Rn (in particular, Ω can be the whole Rn) and V be a finite-dimensionalvector space. The space L2(V,Ω) is the set of vector valued functions f =(f1, . . . , fN) on Ω such that

N∑i=1

∫Ω

|fi(x)|2dx <∞ . (1.70)



(f, g) =N∑i=1

∫Ω

f ∗i (x)gi(x) dx (1.71)

6. Sobolev Spaces. Let Ω be an open set in Rn (in particular, Ω canbe the whole Rn) and V a finite-dimensional complex vector space. LetCm(V,Ω) be the space of complex vector valued functions that have partialderivatives of all orders less or equal to m. Let α = (α1, . . . , αn), α ∈ N, bea multiindex of nonnegative integers, αi ≥ 0, and let |α| = α1 + · · · + αn.Define

Dαf =∂|α|

∂xα11 · · · ∂xαnn

f . (1.72)

Then f ∈ Cm(V,Ω) iff

|Dαfi(x)| <∞ ∀α, |α| ≤ m, ∀i = 1, . . . , N, ∀x ∈ Ω . (1.73)

The space Hm(V,Ω) is the space of complex vector valued functions suchthat Dαf ∈ L2(V,Ω) ∀α, |α| ≤ m, i.e. such that

N∑i=1

∫Ω

|Dαfi(x)|2dx <∞ ∀α, |α| ≤ m. (1.74)


(f, g) =∑

α, |α|≤m

N∑i=1

∫Ω

(Dαfi(x))∗Dαgi(x) dx (1.75)

Remark. More precisely, the Sobolev space Hm(V,Ω) is the completion ofthe space defined above.

1.2.3 Projection Theorem

Definition 39 Let M be a closed subspace of a Hilbert space H. The set,M⊥, of vectors in H which are orthogonal to M is called the othogonalcomplement of M .

Theorem 20 A closed subspace of a Hilbert space and its orthogonal com-plement are Hilbert spaces.


Theorem 21 Let M be a closed subspace of a Hilbert space H. Then ∀x ∈ H∃ a unique element z ∈M closest to x.

Theorem 22 (Projection Theorem) Let M be a closed subspace of a Hilbertspace H. Then ∀x ∈ H ∃z ∈M and ∃w ∈M⊥ such that x = z +w. That is

H = M ⊕M⊥ (1.76)

Remark. The set, L(H,H ′), of linear transformations from a Hilbert spaceH to H ′ is a Banach space under the norm

||T || = sup||x||H=1

||Tx||H′ (1.77)

Definition 40 The space H∗ = L(H,C) of linear transformations from aHilbert space H to C is called the dual space of H. The elements of H∗

are called continuous linear functionals.

Theorem 23 (Riesz Lemma) Let H be a Hilbert pace. Then ∀T ∈ H∗

∃yT ∈ H such that ∀x ∈ H

T (x) = (yT , x), and ||T ||H∗ = ||yT ||H (1.78)

1.2.4 Orthonormal Bases

Definition 41 Let S be an orthonormal set in a Hilbert pace H. If there isno other orthonormal set that contains S as a proper subset, then S is calledorthonormal basis (or complete orthonormal system) for H.

Theorem 24 Every Hilbert space has an othonormal basis.

Theorem 25 Let S = xαα∈A be an orthonormal basis for a Hilbert spaceH. Then ∀y ∈ H

y =∑α∈A

(xα, y)xα (1.79)

||y||2 =∑α∈A

|(xα, y)|2 (1.80)


Definition 42 Let S = xαα∈A be an orthonormal basis for a Hilbert spaceH. The coefficients (xα, y) are called the Fourier coefficients of y ∈ Hwith respect to the basis S.

Definition 43 A metric space which has a countable dense subset is said tobe separable.

Theorem 26 A Hilbert space H is separable iff it has a countable orthonor-mal basis S. If S contains finite number, N , of elements, then H is isomor-phic to CN . If S contains countably many elements, then H is isomorphicto l2.

1.2.5 Tensor Products of Hilbert Spaces

Let H1 and H2 b Hilbert spaces. For each ϕ1 ∈ H1 and ϕ2 ∈ H2 let ϕ1 ⊗ ϕ2

denote the conjugate bilinear form on H1 ×H2 defined by

(ϕ1 ⊗ ϕ2)(ψ1, ψ2) = (ψ1, ϕ1)H1(ψ2, ϕ2)H2 (1.81)

where ψ1 ∈ H1 and ψ2 ∈ H2. Let E be the set of finite linear combinationsof such bilinear forms. An inner product on E can be defined by

(ϕ⊗ ψ, η ⊗ µ)E = (ϕ, η)H1(ψ, µ)H2 (1.82)

(with ϕ, η ∈ H1 and ψ, µ ∈ H2) and extending by linearity on E.

Definition 44 The tensor product H1 ⊗H2 of the Hilbert paces H1 and H2

is defined to be the completion of E under the inner product defined above.

Theorem 27 Let H1 and H2 be Hilbert spaces. If ϕk and ψl are or-thonormal bases for H1 and H2 respectively, then ϕk⊗ψl is anorthonormalbasis for the tensor product H1 ⊗H2.

Fock Spaces. Let H be a Hilbert space and let

H0 = C (1.83)

Hn = H ⊗ · · · ⊗H︸︷︷︸n

(1.84)


denote the n-fold tensor product of H. The space

F (H) = ⊕∞n=0Hn (1.85)

is called the Fock space over H. Fock space F (H) is separable if H isseparable. For example, if H = L2(R), then an element ψ ∈ F (H) is asequence of functions

ψ = ψo, ψ1(x), ψ(x1, x2), ψ3(x1, x2, x3), . . . (1.86)

so that

||ψ|| = |ψ0|2 +∞∑n=1

∫Rn

|ψn(x1, . . . , xn)|2dx1 . . . dxn <∞ (1.87)

Let Pn be the permutation group of n elements and let ϕk be a basisfor H. Each σ ∈ Pn defines a permutation

σ(ϕk1 ⊗ · · · ⊗ ϕkn) = ϕkσ(1)⊗ · · · ⊗ ϕkσ(n)

. (1.88)

By linearity this can be extended to a bounded operator on Hn, so one candefine

Sn =1

n!

∑σ∈Pn

σ (1.89)

An =1

n!

∑σ∈Pn

ε(σ)σ (1.90)

where

ε(σ) =

1, if σ is even−1 if σ is odd

(1.91)

Finally, the Boson (symmetric) Fock space is defined by

Fs(H) = ⊕∞n=0SnHn (1.92)

and the Fermion (antisymmetric) Fock space is defined by

Fa(H) = ⊕∞n=0AnHn (1.93)

In the case H = L2(R), ψn ∈ SnHn is a function of n variables symmetricunder any permutations of variables, and ψn ∈ AnH

n is a function of nvariables that is odd function under interchanges of any two variables.

Chapter 2

Asymptotic Expansions

2.1 Asymptotic Estimates

Let M be a set of real or complex numbers with a limit point a. Let f, g :M → R (or f, g : M → C) be some functions on M .

Definition 45 The following are asymptotic estimates

i) f(x) ∼ g(x) (x→ a, x ∈M)

if limx→a, x∈M

f(x)

g(x)= 1 (2.1)

ii) f(x) = o(g(x)) (x→ a, x ∈M)

if limx→a, x∈M

f(x)

g(x)= 0 (2.2)

iii) f(x) = O(g(x)) (x ∈M)

if ∃C : |f(x)| ≤ C|g(x)| ∀x ∈M (2.3)

iv) f(x) = O(g(x)) (x→ a, x ∈M)

if ∃C and a neighborhood U of a such that :

|f(x)| ≤ C|g(x)| ∀x ∈M ∩ U (2.4)

19

20 Asymptotic Expansions

2.1.1 Examples

1.lnx = o(x−α) (x→ 0+), α > 0 . (2.5)

2.lnx = o(xα) (x→∞), α > 0 . (2.6)

3.sin z ∼ z (z → 0) . (2.7)

4.sinx = O(1) (x ∈ R) . (2.8)

5.n! ∼

√2πne−nnn (n→∞) . (2.9)

Remarks. The relation f(x) = o(g(x)) means that f(x) is infinitesimalwith respect to g(x) as x→ a. Similarly, f(x) = O(g(x)) means that f(x) isbounded with respect to g(x) as x→ a. In particular, f(x) = o(1), (x→ a)means that f(x) is infinitesimal as x→ a and f(x) = O(1), (x→ a) meansthat f(x) is bounded as x→ a.

2.1.2 Properties of asymptotic estimates

There holds (as x→ a, x ∈M):

o(f(x)) + o(f(x)) = o(f(x)) (2.10)

o(f(x))o(g(x)) = o(f(x)g(x)) (2.11)

o(o(f(x))) = o(f(x)) (2.12)

O(f(x)) +O(f(x)) = O(f(x)) (2.13)

O(f(x))O(g(x)) = O(f(x)g(x)) (2.14)

O(O(f(x))) = O(f(x)) (2.15)

o(f(x)) +O(f(x)) = O(f(x)) (2.16)

o(f(x))O(g(x)) = o(f(x)g(x)) (2.17)

O(o(f(x))) = o(f(x)) (2.18)

o(O(f(x))) = o(f(x)) (2.19)

Asymptotic Expansions 21

2.2 Asymptotic Sequences

Definition 46 Let ϕn : M → R, n ∈ N, and a be a limit point of M .Let ϕn(x) 6= 0 in a neighborhood Un of a. The the sequence ϕn is calledasymptotic sequence at x→ a, x ∈M if ∀n ∈ N

ϕn+1(x) = o(ϕn(x)) (x→ a, x ∈M) (2.20)

2.2.1 Examples

1. Power asymptotic sequences

(a)

(x− a)n, x→ a . (2.21)

(b)

x−n, x→∞ . (2.22)

2. Let λn be a decreasing sequence of real numbers, i.e. λn < λn+1, andlet 0 < ε ≤ π/2. Then the sequence

eλnz, z →∞, | arg z| ≤ π

2− ε (2.23)

is an asymptotic sequence.

2.2.2 Properties of asymptotic sequences

1. Any subsequence of an asymptotic sequence is an asymptotic sequence.

2. Let f(x) 6= 0 for x ∈ M in some neighborhood of a and ϕn be anasymptotic sequence at x→ a, x ∈M . Then the sequence f(x)ϕn(x)is an asymptotic sequence as x→ a, x ∈M .

3. Let ϕn(x), ψn(x) be asymptotic sequences as x→ a, x ∈M . Thenthe sequence ϕn(x)ψn(x) is an asymptotic sequence as x→ a, x ∈M .


2.3 Asymptotic Series

Let f : M → R and a be a limit point of M .

Definition 47 Let ϕn be an asymptotic sequence as x → a, x ∈ M . Wesay that the function f is expanded in an asymptotic series

f(x) ∼∞∑n=0

anϕn(x), (x→ a, x ∈M), (2.24)

where an are constants, if ∀N ≥ 0

RN(x) ≡ f(x)−N∑n=0

anϕn(x) = o(ϕN(x)), (x→ a, x ∈M) . (2.25)

This series is called asymptotic expansion of the function f with respectto the asympotic sequence ϕn. RN(x) is called the rest term of theasymptotic series.

Remarks

1. The condition RN(x) = o(ϕN(x)) means, in particular, that

limx→a

RN(x) = 0 for any fixed N (2.26)

2. Asymptotic series could diverge. This happens if

limN→∞

RN(x) 6= 0 for some fixed x (2.27)

3. There are three possibilities:

(a) asymptotic series converges to f(x);

(b) asymptotic series converges to a function g(x) 6= f(x);

(c) asymptotic series diverges.

Theorem 28 Asymptotic expansion of a function with respect to an asymp-totic sequence is unique.


Remark. Two different functions can have the same asymptotic expansion.For example, f(x) = ex and g(x) = ex + e−1/x have the same asymptoticexpansion with respect to the asymptotic sequence xn:

ex ∼ ex + e−1/x ∼∞∑n=0

xn

n!, x→ 0+ (2.28)

Theorem 29 Asymptotic series can be added and multiplied by numbers,but, cannot be multipied by asymptotic series.

Theorem 30 One can multiply and divide power asymptotic series.

Definition 48 Let f : M ×S → R be a function of two variables and a be alimit point of M and ϕn be an asymptotic sequence as x→ a. Let for anyfixed y ∈ S the function f is expanded in an asymptotic series

f(x, y) ∼∞∑n=0

an(y)ϕn(x), (x→ a, x ∈M) . (2.29)

This asymptotic expansion is called uniform with respect to the parametery ∈ S, if the relation

RN(x, y) ≡ f(x, y)−N∑n=0

an(y)ϕn(x) = o(ϕN(x)), (x→ a, x ∈M) (2.30)

is valid uniformly with respect to y ∈ S.

Theorem 31 A uniform asymptotic expansion can be integrated with respectto the parameter term by term.

Remark. One cannot, in general, differentiate asymptotic series, neitherwith respect to x nor with respect to a parameter.

2.4 Asymptotics of Integrals: Weak Singu-

larities

Let us consider the integrals of the form

F (ε) =

∫ a

0

f(x, ε) dx (2.31)


where a > 0 and ε > 0 is a small positive parameter. Here f ∈ C∞([0, a] ×(0, ε0]) is a smooth function for 0 ≤ x ≤ a, 0 < ε ≤ ε0, with some ε0. Thenthe integral converges for ε > 0. Let f have a singularity when ε = 0, i.e.g(x) = f(x, 0) has a singularity at some 0 ≤ x ≤ a. If this singularity is ofpower or logarithmic type then we say that the integral F (ε) has a weaksingularity.

This definition is obviously extended for unbounded intervals. Let

F (ε) =

∫ ∞a

f(x, ε) dx (2.32)

where a > 0 and ε > 0 is a small parameter. Here f ∈ C∞([a,∞)× (0, ε0]) isa smooth function for a ≤ x ≤ ∞, 0 < ε ≤ ε0, with some ε0. Let the integralconverge for ε > 0 and diverge for ε = 0. If the function g(x) = f(x, 0) isof power or logarithmic order at x→∞, then we say that the integral F (ε)has a weak singularity.

2.4.1 Power Singularity on a Bounded Interval

Let a, α, β ∈ R, a, β > 0, be some real numbers and ε > 0 be a small positiveprameter. Let ϕ ∈ C∞[0, a] be a smooth function on [0, a]. We will studythe asymptotics as ε→ 0+ of the integrals of the form

F (ε) =

∫ a

0

tβ−1(t+ ε)αϕ(t) dt . (2.33)

Remarks. The function F is holomorphic in complex plane ε with a cutalong the negative half-axis. At the point ε = 0 this function has a singularity(if α > 0 is not integer). The type of this singularity is determined by thebehavior of the function ϕ at small t ≥ 0.

Standard Integral. One needs the following result. Let α and β be twocomplex numbers such that Re β > 0, Reα < 0 and Re (β + α) < 0. Then∫ ∞

0

tβ−1(t+ 1)α dt =Γ(β)Γ(−α− β)

Γ(−α), (0 < Re β < −Reα) . (2.34)

Theorem 32 Let ϕ ∈ C∞[0, a]. Let r, δ > 0 and Sδ = ε ∈ C|0 < |ε| ≤r, | arg ε| ≤ π − δ be a sector in the complex plane of ε.


1. If α + β is not integer, then

F (ε) ∼∞∑n=0

Γ(β + n)Γ(−α− β − n)

Γ(−α)

ϕ(n)(0)

n!εα+β+n +

∞∑n=0

anεn

(ε→ 0, ε ∈ Sδ) (2.35)

2. If α + β = N is an integer, then

F (ε) ∼ −∞∑

n≥max0,−N

Γ(N + n)

Γ(α)Γ(N + n− α)

ϕ(n)(0)

n!εn+N ln ε+

∞∑n=0

bnεn

(ε→ 0, ε ∈ Sδ) (2.36)

The coefficients an and bn depend on the values ϕ(t) for 0 ≤ t ≤ a. Thebranch for the functions εγ and ln ε is choosen in such a way that εγ > 0 andln ε is real for ε > 0.

Examples. In all examples ϕ ∈ C∞([0, a]) is a smooth function boundedwith all its derivatives.

1. Let 0 < a < 1 and

F (ε) =

∫ a

0

ϕ(t)

t+ εdt . (2.37)

ThenF (ε) = −ϕ(0) ln ε+O(1), (ε→ 0+) (2.38)

2. Let 0 < a < 1 and

F (ε) =

∫ a

0

ϕ(t)

t2 + ε2dt . (2.39)

By using1

t2 + ε2=

1

2iε

(1

t− iε− 1

t+ iε

)(2.40)

we obtain from the previous example

F (ε) = ϕ(0)π

2ε+O(1) (ε→ 0+) (2.41)


3. Let α > 1/2 and

F (ε) =

∫ a

0

ϕ(t)

(t2 + ε2)αdt . (2.42)

Then

F (ε) = ϕ(0)

√π

2

Γ(α− 1/2)

Γ(α)ε1−2α+O(ε3−2α)+O(1), (ε→ 0+) (2.43)

2.4.2 Power singularity on Unbounded Interval

Standard Integral. To compute the following asymptotics one needs thefollowing standard integral. Let Re β > 0 and Reα > −1. Then∫ ∞

0

tαe−tβ

dt =1

βΓ

(α + 1

β

), (Re β > 0, Reα > −1) . (2.44)

Examples. Let ϕ ∈ C∞([a,∞)) be a smooth function on [a,∞) that hasasymptotic expansion as x→∞

ϕ(x) ∼∞∑k=0

akx−k . (2.45)

1. Let a, β > 0, and

F (ε) =

∫ ∞a

ϕ(x)xαe−εxβ

dx . (2.46)

If α < −1, then the integral is not singular as ε→ 0+. Its asymptoticexpansion can be obtained either by integration by parts or by a changeof variables.

So, let now α + 1 > 0 and let N = [α + 1] ≥ 0 be the integer partof α + 1. Let us single out the first N + 1 terms of the asymptoticexpansion in ϕ, i.e.

ϕ(x) =N∑k=0

akx−k +RN(x) . (2.47)

Since RN(x) = O(x−(N+1)) as x→∞, we have

F (ε) =N∑k=0

ak

∫ ∞a

xα−ke−εxβ

dx+O(1) . (2.48)


Now by changing the variables and extending the interval to [0,∞) weobtain

F (ε) =N∑k=0

akβε−

α−k+1β

[Γ

(α− k + 1

β

)+O(1)

]+O(1)

=a0

βε−

α+1β Γ

(α + 1

β

)+O(ε−

αβ ) (2.49)

If α = −1, then

F (ε) = −a0

βln ε+O(1), (ε→ 0+) . (2.50)

2. Let a > 0 and let

P (x) = xn + · · ·+ a1x n ≥ 1 . (2.51)

Consider the integral

F (ε) =

∫ ∞a

ϕ(x)xαe−εP (x) dx . (2.52)

If α > −1, then the main term of the asymptotics is

F (ε) =a0

nΓ

(α + 1

n

)ε−

α+1n +O(ε−

αn ), (ε→ 0+) (2.53)

If α = −1, then

F (ε) = −a0

nln ε+O(1), (ε→ 0+) (2.54)

If α < −1, then the integral is not singular as ε→ 0+. By integrationby parts the integral can be reduced to the cases considered above.

Chapter 3

Laplace Method

3.1 Laplace Integrals in One Dimension

Let M = [a, b] be a closed bounded interval, S : M → R be a real valuedfunction, ϕ : M → C a complex valued function and λ be a large positiveparameter. Consider the integrals of the form

F (λ) =

∫ b

a

ϕ(x) exp[λS(x)] dx . (3.1)

Such integrals are called Laplace integrals. We will study the asymptoticsof the Laplace integrals as λ→∞.

Lemma 1 Let supa<x<b S(x) = L < ∞ and the integral (3.1) convergesabsolutely for some λ0 > 0. Then

1.|F (λ)| ≤ C|eλL| (Reλ ≥ λ0) . (3.2)

2. if f, S ∈ C(a, b), then F (λ) is holomorphic in the halfplane Reλ > λ0.

3.1.1 Watson Lemma

Lemma 2 (Watson) Let 0 < a <∞, α > 0, β > 0 and let Sε be the sectorSε = λ ∈ C | | arg λ| ≤ π/2− ε in the complex plane λ. Let ϕ ∈ C∞([0, a])and let

Φ(λ) =

∫ a

0

ϕ(x)xβ−1 exp(−λxα) dx (3.3)

29

30 Laplace Method

Then, there is an asymptotic expansion as λ→∞, λ ∈ Sε,

Φ(λ) ∼ 1

α

∞∑k=0

λ−(β+k)/αΓ

(β + k

α

)ϕ(k)(0)

k!(3.4)

Laplace Transform. Let ϕ ∈ C∞(R+) be a smooth function on the posi-tive real axis such that its Laplace transform

L(ϕ)(λ) =

∫ ∞0

ϕ(x)e−λx dx (3.5)

converges absolutely for some λ0. Then

L(ϕ)(λ) ∼∞∑k=0

λ−kϕ(k)(0) (|λ| → ∞, λ ∈ Sε) (3.6)

3.1.2 Interior Nondegenerate Maximum Point

Let now S and ϕ be smooth functions and the function S have a maximumat an interior point x0 of the interval [a, b], i.e. a < x0 < b. Then S ′(x0) = 0.Assume, for simplicity, that S ′′(x0) 6= 0. Then S ′′(x0) < 0. In other words,in a neighborhood of x0 the function S has the following Taylor expansion

S(x) = S(x0) + S ′′(x0)(x− x0)2

2+O((x− x0)3) . (3.7)

Such a point is called nondegenerate critical point.Then, as λ→∞ the main contribution to the integral comes from a small

neighborhood of x0. In this neighborhood the function ϕ is almost constantand can be replaced by its value at x0. The terms of order (x− x0)3 can beneglected in the exponent and the remaining integral can be extended to thewhole real line. By using the standard Gaussian integral∫ ∞

−∞exp

(−α

2y2)dy =

√2π

α, (Reα > 0) , (3.8)

one obtains finally the main term of the asymptotics

F (λ) ∼ λ−1/2

√2π

−S ′′(x0)ϕ(x0)eλS(x0), (λ→∞) (3.9)

One can prove the general theorem.

Laplace Method 31

Theorem 33 Let M = [a, b] and ϕ, S ∈ C∞(M), S has a maximum only atone point x0, a < x0 < b and S ′′(x0) 6= 0. Then as λ → ∞, λ ∈ Sε there isasymptotic expansion

F (λ) ∼ eλS(x0)

∞∑k=0

ckλ−1/2−k . (3.10)

The coefficients ck are expressed in terms of derivatives of ϕ and S at x0.

The theorem can be proved as follows. First, we change the integrationvariable

x = x0 + λ−1/2y . (3.11)

So, y is the scaled fluctuation from the maximum point x0. The intervalof integration should be changed accordingly, so that the maximum point isnow y = 0. Then, we expand both functions S and ϕ in Taylor series at x0

getting

λS(x0 + λ−1/2y) = λS(x0) +1

2S ′′(x0)y2 +

∞∑n=3

S(n)(x0)

n!ynλ−(n−2)/2 , (3.12)

ϕ(x0 + λ−1/2y) =∞∑n=0

ϕ(n)(x0)

n!ynλ−n/2 . (3.13)

Since the quadratic terms are of order O(1) we leave it in the exponentand expand the exponent of the rest in a power series. Next, we extend theintegration interval to the whole real line and compute the standard Gaussianintegrals of the form ∫ ∞

−∞exp

(−α

2y2)y2k+1 dy = 0 , (3.14)

∫ ∞−∞

exp(−α

2y2)y2k dy = Γ

(k +

1

2

)(α2

)−k−1

, (3.15)

where k is a nonnegative integer and α has a positive real part, Reα > 0.Finally, we get a power series in inverse powers of λ. The coefficients ck ofthe asymptotic expansion are polynomials in the higher derivatives S(k)(x0),k ≥ 3, and derivatives ϕ(l)(x0), l ≥ 0, and involve inverse powers of S ′′(x0).

32 Laplace Method

Stirling Formula

Γ(x+ 1) =√

2πxx+1/2e−x[1 +O(x−1)], (x→∞) (3.16)

is obtained by applying the Laplace method to the integral

Γ(x+ 1) = xx+1

∫ ∞0

exp[x(ln t− t)] dt (3.17)

Stieltjes Transform Let ϕ : R+ → C have finite moments

mn(ϕ) =

∫ ∞0

tnϕ(t) dt <∞ ∀n ∈ N. (3.18)

Then the Stieltjes transform of ϕ

S(ϕ)(x) =

∫ ∞0

ϕ(t)

t+ xdt (3.19)

has asymptotic expansion as x→∞

S(ϕ)(x) ∼∞∑k=0

(−1)kmk(ϕ)x−1−k (3.20)

3.1.3 Boundary Maximum Point

Let the function S have a maximum at a boundary point x0 = a. Let, forsimplicity, S ′(a) 6= 0, i.e. S ′(a) < 0, and ϕ(a) 6= 0. Then, as λ → ∞, themain contribution to the integral comes from the interval [a, a+ ε], where

S(x) = S(a) + (x− a)S ′(a) +O((x− a)2) . (3.21)

Now, by replacing the function ϕ by its value at a and neglecting nonlinearterms in S(x), we get

F (λ) ∼ λ−1 ϕ(a)

−S ′(a)eλS(a), (λ→∞) (3.22)

In this way one can prove the following theorem.

Theorem 34 Let M = [a, b], ϕ, S ∈ C∞(M), S has a maximum only at thepoint x = a and S ′(a) 6= 0. Then, as λ → ∞, λ ∈ Sε, there is asymptoticexpansion

F (λ) ∼ eλS(a)

∞∑k=0

ckλ−1−k . (3.23)

The coefficients ck are expressed in terms of derivatives of ϕ and S at x = a.

Laplace Method 33

Error Function The asymptotic expansion of the (complementary) errorfunction as x→∞ has the form

Erfcx =

∫ ∞x

e−t2

dt ∼ e−x2

2x

∞∑k=0

(−1)k(2k − 1)!!

2kx−2k (x→∞) . (3.24)

Incomplete Gamma Function The incomplete gamma-function

γ(a, x) =

∫ x

0

ta−1e−t dt, (0 < a <∞, x > 0) (3.25)

has the following asympotic expansion as x→∞

γ(a, x) = Γ(a) + e−xxa−1

∞∑k=0

Γ(a)

Γ(a− k)x−k (3.26)

3.2 Background from Analysis

3.2.1 Definitions

1. Let xj, j = 1, . . . , n be real numbers and let x be the n-tuple x =(x1, . . . xn). The set of all n-tuples of real numbers is denoted by Rn.A connected open subset Ω of Rn is called a domain. The set ∂Ω ofboundary points of Ω is called the boundary of Ω. The union Ω∪∂Ωis called the closure of Ω and is denoted by Ω.

2. Let x, ξ ∈ Rn. Then the scalar product of x and ξ is defined byx · ξ = x1ξ1 + · · ·+ xnξn.

3. On Rn there is standard Lebesgue measure dx = dx1 · · · dxn.

4. Let αj, j = 1, . . . , n, be non-negative integers, αj ≥ 0. The n-tupleα = (α1, . . . , αn) is called a multi-index. Further, let

|α| = α1 + · · ·+ αn (3.27)

α! = α1! · · ·αn! (3.28)

∂αf(x) =∂|α|f(x)

∂(x1)α1 · · · ∂(xn)αn(3.29)

34 Laplace Method

Dαf(x) =

(1

i

∂

∂x1

)α1

· · ·(

1

i

∂

∂xn

)αnf(x) (3.30)

Then the Taylor expansion of a smooth function ϕ at x0 can be writtenin the form

ϕ(x) =∞∑|α|=0

1

α![∂αϕ(x0)] (x− x0)α . (3.31)

5. The boundary ∂Ω is said to be smooth, denoted ∂Ω ∈ C∞, if in aneighborhood of any boundary point x0 ∈ ∂Ω it can be locally definedby an equation xj = ϕ(x′) with a smooth function ϕ.

6. The set of all continuous functions on Ω is denoted by C(Ω).

7. The set of all functions with continuous partial derivatives up to orderk on Ω is denoted by Ck(Ω).

8. The set of all functions with continuous partial derivatives up to orderk on Ω is denoted by Ck(Ω).

9. The set of all functions with continuous partial derivatives up to orderk on Ω that vanish in a neighborhood of the boundary ∂Ω is denotedby Ck

0 (Ω).

10. The closure of the set where a function is not equal to zero is calledthe support of the function, denoted by

supp f = x ∈ Ω | f(x) 6= 0 . (3.32)

11. A mapping ϕ : Ω→ Ω, is said to be of class Ck if ϕ ∈ Ck(Ω).

12. A one-to-one mapping ϕ : Ω → Ω of Ω onto Ω is called diffeomor-phism of class Ck if ϕ ∈ Ck(Ω) and ϕ−1 ∈ Ck(Ω).

13. Let ϕj, j = 1, . . . , k be some scalar functions on Rn and ϕ be a k-tupleϕ = (ϕ1, . . . , ϕk). In other words, ϕ : Rn → R

k. The matrix

∂xϕ(x) =

(∂ϕi(x)

∂xj

), i = 1, . . . , k; j = 1, . . . , n (3.33)

is called the Jacobi matrix.

Laplace Method 35

Theorem 35 (Inverse Function Theorem) Let ϕ : Rn → Rn is of class

Ck, k ≥ 1, in a neighborhood of a point x0 and det ∂xϕ(x0) 6= 0. Then ϕ islocal diffeomorphism of class Ck in a neighborhood of the point x0.

Theorem 36 (Implicit Function Theorem) Let Ω be a domain in R2n,let F : Ω → R

n be a mapping of class Ck(Ω) and let (x0, y0) ∈ Ω be a pointin Ω such that

F (x0, y0) = 0, det ∂yF (x0, y0) 6= 0 . (3.34)

Then in a neighborhood of the point x0 there is a mapping y = f(x) of classCk such that y0 = f(x0) and

F (x, f(x)) ≡ 0 . (3.35)

3.2.2 Morse Lemma

Let S : Ω → R be a real valued function of class Ck on a domain Ω in Rn

with k ≥ 2. Let

∂2xS(x) =

(∂2S(x)

∂xi∂xj

), i, j = 1, . . . , n . (3.36)

Definition 49 1. The point x0 is called a critical point of the functionS if ∂S(x0) = 0

2. A critical point x0 is called non-degenerate if det ∂2xS(x0) 6= 0.

3. The determinant det ∂2xS(x0) is called the Hessian of the function S

at the point x0.

Lemma 3 (Morse) Let S : Rn → R and x0 ∈ Rn be a non-degeneratecritical point of the function S. Let S ∈ C∞ in a neighborhood of the pointx0 and let µj 6= 0, j = 1, . . . , n be the eigenvalues of the matrix ∂2

xS(x0).Then there are neighborhoods U and V of the points x0 and 0 and a smoothlocal diffeomorphism ϕ : V → U of class C∞ such that det ∂yϕ(0) = 1 and

S(ϕ(y)) = S(x0) +1

2

n∑j=1

µj(yj)2 . (3.37)

36 Laplace Method

Remark. Nondegenerate critical points are isolated.

3.2.3 Gaussian Integrals

Proposition 7 Let A = (aij be a complex symmetric nondegenerate n × nmatrix with the eigenvalues µj(A), j = 1, . . . , n,. Let ReA ≥ 0, which meansthat x ·ReAx ≥ 0 ∀x ∈ Rn, x 6= 0, or Reµj(A) ≥ 0, j = 1, . . . , n. Then forλ > 0, ξ ∈ Rn there holds∫

Rn

exp

(−λ

2x · Ax− iξ · x

)dx

=

(2π

λ

)n/2(detA)−1/2 exp

(− 1

2λξ · A−1 ξ

). (3.38)

The branch of√

detA is choosen as follows

(detA)−1/2 = | detA|−1/2 exp (−i IndA) , (3.39)

where

IndA =1

2

n∑j=1

arg µj(A), | arg µj(A)| ≤ π

2. (3.40)

By expanding both sides of this equation in Taylor series in ξ we obtainthe following result.

Corollary 1∫Rn

exp

(−λ

2x · Ax

)xi1 · · ·xi2k+1dx = 0 (3.41)

∫Rn

exp

(−λ

2x · Ax

)xi1 · · ·xi2k dx

=

(2π

λ

)n/2(detA)−1/2 (2λ)−k

(2k)!

k!G(i1 i2 · · ·Gi2k−1 i2k) . (3.42)

Here k is a non-negative integer, G = A−1, and the round brackets denotecomplete symmetrization over all indices included.

An important particular case of the previous formula is when the matrixA is real.

Laplace Method 37

Proposition 8 Let A be a real symmetric nondegenerate n× n matrix. Letν+(A) and ν−(A) be the numbers of positive and negative eigenvalues of Aand

sgnA = ν+ − ν− (3.43)

be the signature of the matrix A. Then for λ > 0, ξ ∈ Rn there holds∫Rn

exp

(iλ

2x · Ax− iξ · x

)dx

=

(2π

λ

)n/2| detA|−1/2 exp

(− i

2λξ · A−1 ξ +

iπ

4sgn (A)

). (3.44)

3.3 Laplace Integrals in Many Dimensions

3.3.1 Interior Maximum Point

Let Ω be a bounded domain in Rn, S : Ω→ R, f : Ω→ C are some functionson Ω and λ > 0 be a large positive parameter. We will study the asymptoticsas λ→∞ of the multidimensional Laplace integrals

F (λ) =

∫Ω

f(x) exp[λS(x)] dx . (3.45)

Let S and f be smooth functions and the function S have a maximumonly at one interior nondegenerate critical point x0 ∈ Ω. Then ∂xS(x0) = 0and [det ∂2

xS(x0)] < 0. Then in a neighborhood of x0 the function S has thefollowing Taylor expansion

S(x) = S(x0) +1

2(x− x0) · [∂2

xS(x0)](x− x0) +O((x− x0)3) . (3.46)

One could also use the Morse Lemma to replace the function S by a quadraticform. Then as λ → ∞ the main contribution to the integral comes from asmall neghborhood of x0. In this neighborhood the terms of the third orderin the Taylor expansion of S can be neglected. Also, since the function fis continuous at x0, it can be replaced by its value at x0. Then the regionof integration can be extended to the whole Rn. By using the formula forthe standard Gaussian integral one gets then the leading asymptotics of theintegral F (λ) as λ→∞

F (λ) ∼ exp[λS(x0)]

(2π

λ

)n/2 [− det ∂2

xS(x0)]−1/2

f(x0) . (3.47)

38 Laplace Method

One can prove the general theorem.

Theorem 37 Let f, S ∈ C∞(Ω) and let x0 be a nondegenerate critical pointof the function S where it has the only maximum in Ω. Let 0 < ε < π/2.Then there is asymptotic expansion as λ → ∞ in the sector Sε = λ ∈C | | arg λ| ≤ π/2− ε

F (λ) ∼ exp[λS(x0)]λ−n/2∞∑k=0

akλ−k . (3.48)

The coefficients ak are expressed in terms of the derivatives of the functionsf and S at the point x0.

The idea of the proof is the same as in the one-dimensional case and goesas follows. First, we change the integration variables

xi = xi0 + λ−1/2yi . (3.49)

So, y is the scaled fluctuation from the maximum point x0. The intervalof integration should be changed accordingly, so that the maximum point isnow y = 0. Then, we expand both functions S and ϕ in Taylor series at x0

getting

λS(x0 + λ−1/2y) = λS(x0) +1

2y · [∂2

xS(x0)]y +∞∑|α|=3

λ−(|α|−2)/2

α![∂αS(x0)] yα ,

(3.50)

ϕ(x0 + λ−1/2y) =∞∑|α|=0

λ−|α|/2

α!∂αϕ(x0) yα . (3.51)

Since the quadratic terms are of order O(1) we leave them in the exponentand expand the exponent of the rest in a power series. Next, we extendthe integration domain to the whole Rn and compute the standard Gaus-sian integrals. Finally, we get a power series in inverse powers of λ. Thecoefficients ak of the asymptotic expansion are polynomials in the higherderivatives ∂αS(x0), |α| ≥ 3, and derivatives ∂αϕ(x0), |α| ≥ 0, and involveinverse matrices G = [∂2

xS(x0)]−1.

Laplace Method 39

Remark. If x0 is a degenerate maximum point of the function S, thenthe asymptotic expansion as λ→∞ has the form

F (λ) ∼ exp[λS(x0)]λ−n/2∞∑k=0

N∑l=0

aklλ−rk(lnλ)l , (3.52)

where N is some positive integer and rk, rk ≥ n/2, k ∈ N, is a increasingsequence of nonnegative rational numbers.

The coefficients ak (and akl) of the asymptotic expansion of the integralF (λ) are invariants under smooth local diffeomorphisms in a neighborhoodof x0 and play very important role in various applications.

3.3.2 Boundary Maximum Point

Let now S has maximum at the boundary point x0 ∈ ∂Ω. We assume,for simplicity, that both the boundary and the function S are smooth, i.e.S ∈ C∞ and ∂Ω ∈ C∞.

Since the boundary is smooth we can smoothly parametrize it in a neigh-borhood of x0 by (n − 1) parameters ξ = (ξa), (a = 1, . . . , n − 1). Let thethe parametric equations of the boundary be

xi = xi(ξ), i = 1, . . . , n . (3.53)

Then

Ta = (T ia) =

(∂xi

∂ξa

)(3.54)

are tangent vectors to the boundary.Let r = r(x) be the normal distance to the boundary. Then the equation

of the boundary can be written as

r(x) = 0 . (3.55)

and for x ∈ Ω we have r > 0. Obviously, r(x(ξ)) ≡ 0. From this equationwe obtain that the vector

N = (Ni) =

(∂r

∂xi

)(3.56)

is orthogonal to all tangent vectors and is therefore normal to the boundary.It can be certainly normalized, since it is nowhere zero. We choose it to bethe inward normal.

40 Laplace Method

The normal and tangential derivatives are defined as usual

∂r =n∑i=1

∂xi

∂r

∂

∂xi,

∂

∂ξa=

n∑i=1

∂xi

∂ξa∂

∂xi(3.57)

The point x0 is not, in general, a critical point of S, since the normalderivative of S at x0 does not have to be equal to zero.

Definition 50 The point x0 is said to be a nondegenerate boundarymaximum point of S if

∂rS(x0) 6= 0 (3.58)

and the (n− 1)× (n− 1) matrix ∂2ξS(x(ξ)) is negative definite.

In a neighborhood of a nondegenerate boundary maximum point the func-tion S has the following Taylor expansion

S(x) = S(x0) + [∂rS(x0)]r +1

2[∂2rS(x0)]r2

+[∂r∂ξS(x0)] · (ξ − ξ0)r +1

2(ξ − ξ0) · [∂2

ξS(x0)](ξ − ξ0)

+ · · · , (3.59)

up to third order terms in r and (ξ − ξ0).Now we replace the integral F (λ) by an integral over a small neighborhood

of x0. We change the variables of integration from xi, i = 1, . . . , n, to (ξa, r),a = 1, . . . , n − 1, and neglect the terms of third order in the Taylor series.We also replace the function f by its value at the point x0. In the remainingintegral we extend the integration to the whole space R+ × Rn−1, i.e. weintegrate over r from 0 to ∞ and integrate over the whole tangent planeat x0. These integrals are standard Gaussian integrals and we obtain theleading asymptotics as λ→∞

F (λ) ∼ −λ−(n+1)/2(2π)(n−1)/2 exp[λS(x0)]

×[∂rS(x0)]−1[− det ∂2ξS(x0)]−1/2J(x0)f(x0) (3.60)

where J(x0) is the Jacobian of change of variables.The general form of the asymptotic expansion is given by the following

theorem.

Laplace Method 41

Theorem 38 Let f, S ∈ C∞(Ω) and let S have a maximum only at a non-degenerate boundary maximum point x0 ∈ ∂Ω. Then as λ→∞, λ ∈ Sε,

F (λ) ∼ λ−(n+1)/2 exp[λS(x0)]∞∑k=0

akλ−k (3.61)

3.3.3 Integral Operators with Singular Kernels

Let Ω be a bounded domain in Rn including the origin, 0 ∈ Ω. Let S be areal valued non-positive function on Ω of class C2 that has maximum equalto zero, S(0) = 0, only at a nondegenerate maximum critical point x0 = 0.Let Kλ : C∞(Ω)→ C∞(Ω) be a linear integral operator defined by

(Kλf)(x) =

(λ

2π

)n/2 ∫Ω

exp[λS(x− y)]f(y) dy . (3.62)

Let M be a compact subset of Ω. Then

limλ→∞

(Kλf)(x) = [− det ∂2xS(0)]−1/2f(x) (3.63)

uniformly for x ∈M .Formally(

λ

2π

)n/2exp[λS(x− y)] −→ [− det ∂2

xS(0)]−1/2 δ(x− y) (3.64)

42 Laplace Method

Chapter 4

Stationary Phase Method

4.1 Stationary Phase Method in One Dimen-

sion

4.1.1 Fourier Integrals

Let M = [a, b] be a closed bounded interval, S : M → R be a real valuednonconstant function, f : M → C be a complex valued nonzero functionand λ be a large positive parameter. Consider the integrals of the form

F (λ) =

∫ b

a

f(x) exp[iλS(x)] dx . (4.1)

The function S is called phase function and such integrals are calledFourier Integrals. We will study the asymptotics of such integrals.

As λ→∞ the integral F (λ) is small due to rapid oscillations of exp(iλS).

Lemma 4 (Riemann-Lebesgue) Let f be an integrable function on thereal line, i.e. f ∈ L1(R). Then∫

R

f(x)eiλx dx = o(1), (λ→∞) . (4.2)

Definition 51 1. A point x0 is called the regular point of the Fourierintegral F (λ) if the functions f and S are smooth in a neighborhood ofx0 and S ′(x0) 6= 0.

43

44 Stationary Phase Method

2. A point x0 is called the critical point of the integral F (λ) if it is nota regular point.

3. A critical point x0 is called isolated critical point if there is aneighborhood of x0 that does not contain any other critical points.

4. An interior isolated critical point is called stationary point.

5. The integral over a neighborhood of an isolated critical point that doesnot contain other critical points will be called the contribution ofthe critical point to the integral.

Clearly the main contribution comes from the critical points since closeto these points the oscillations slow down. As always, we will assume thatfunctions S and f are smooth, i.e. of class C∞(M). Otherwise, the sin-gularities of the functions S and f and their derivatives would contributesignificantly to F (λ).

4.1.2 Localization Principle

Lemma 5 Let S ∈ C∞(R) be smooth function and f ∈ C∞0 (R) be a smoothfunction of compact support. Then as λ→∞∫

R

f(x) exp[iλS(x)] dx = O(λ−∞) (4.3)

Remarks.

1. Since the function f has compact support, the integral is, in fact, overa finite interval.

2. This is the main technical lemma for deriving the (power) asympoticsof the Fourier integrals. It means that such integrals can be neglectedin a power asymptotic expansion.

3. The Fourier integrals are in general much more subtle object than theLaplace integrals. Instead of exponentially decreasing integrand onehas a rapidly oscillating one. This requires much finer estimates andalso much stronger conditions on the phase function S and the inte-grand f .

Stationary Phase Method 45

Theorem 39 Let the Fourier integral F (λ) have finite number of isolatedcritical points. Then as λ → ∞ the integral F (λ) is equal to the sum of thecontributions of all critical points up to O(λ−∞).

Thus, the problem reduces to computing the asymptotics of the contribu-tions of critical points. In a neighborhood of a critical point we can replacethe functions S and f by more simple functions and then compute somestandard integrals.

4.1.3 Boundary Points

If the phase function does not have any stationary points, then by integrationby parts one can easily obtain the asymptotic expansion.

Theorem 40 Let S ′(x) 6= 0 ∀x ∈M . Then as λ→∞

F (λ) ∼∞∑k=0

(iλ)−k−1

(1

−S ′(x)

∂

∂x

)k (f(x)

S ′(x)

)eiλS(x)

∣∣∣∣∣b

a

. (4.4)

The leading asymptotics is

F (λ) = (iλ)−1f(b) exp[iλS(b)]− f(a) exp[iλS(a)]+O(λ−2) (4.5)

The same technique, i.e. integration by parts, applies to the integralsover an unbounded interval, say,

F (λ) =

∫ ∞0

f(x) exp[iλS(x)] dx . (4.6)

with some additional conditions that guarantee the converges at ∞ as wellas to the integrals of the form

F (x) =

∫ ∞x

f(t) exp[iS(t)] dt (4.7)

as x→∞.


Examples

1. Let α > 0. Then as λ→∞

(a) ∫ ∞0

eiλx

(1 + x)αdx = iλ−1 +O(λ−2) . (4.8)

(b) ∫ ∞0

sin(λx)

(1 + x)αdx = λ−1 +O(λ−2) . (4.9)

(c) ∫ ∞0

cos(λx)

(1 + x)αdx = αλ−2 +O(λ−3) . (4.10)

2. Let α > 0. Then as x→∞

F (x) =

∫ ∞x

t−α eit dt ∼ ieixx−α∞∑k=0

Γ(k + α)

Γ(α)(ix)−k . (4.11)

In particular, the Frenel integral has the asymptotic expansion as x→∞

Φ(x) =

∫ ∞x

t−α eit dt ∼ i

2√πeix

2

x−1/2

∞∑k=0

(−1)kΓ

(k +

1

2

)x−k .

(4.12)

4.1.4 Standard Integrals

Consider the integral

Φ(λ) =

∫ a

0

f(x)xβ−1eiλxα

(4.13)

Lemma 6 (Erdeyi) Let α ≥ 1, β > 0 and f is a smooth function on aclosed bounded interval [0, a], f ∈ C∞([0, a]), that vanish at x = a with allits derivatives. Then as λ→∞

Φ(λ) =

∫ a

0

f(x)xβ−1eiλxα ∼

∞∑k=0

akλ−(β+k)/α , (4.14)


where

ak =f (k)(0)

k!

1

αΓ

(β + k

α

)exp

[iπβ + k

α

](4.15)

This lemma plays he same role in the stationary phase method as Watsonlemma in the Laplace method.

4.1.5 Stationary Point

Theorem 41 Let M = [a, b] be a closed bounded interval, S ∈ C∞(M)be a smooth real valued nonconstant function, f ∈ C∞0 (M) be a complexvalued function with compact support in M . Let S have a single isolatednondegenerate critical point x0 in M , i.e. S ′(x0) = 0 and S ′′(x0) 6= 0. Thenas λ→∞ there is asymptotic expansion of the Fourier integral

F (λ) =

∫ b

a

f(x) exp[iλS(x)] dx

∼ exp[iλS(x0) + [sgnS ′′(x0)] i

π

4

]λ−1/2

∞∑k=0

λ−k . (4.16)

The coefficients ak are determined in terms of the derivatives of the functionsS and f at x0.

The leading asymptotics as λ→∞ is

F (λ) =

√2π

|S ′′(x0)|λ−1/2 exp

[iλS(x0) + [sgnS ′′(x0)] i

π

4

] (f(x0) +O(λ−1)

).

(4.17)To prove this theorem one does a change of variables in a sufficiently

small neghborhood of x0 and applies the Erdelyi lemma.

4.1.6 Principal Values of Integrals

Let f be a smooth function and consider the integral∫ b

a

f(x)

xdx (4.18)


This integral diverges, in general, at x = 0. One can regularize it bycutting out a symmetric neighborhood of the singular point

I(ε) =

∫ −εa

f(x)

xdx+

∫ b

ε

f(x)

xdx . (4.19)

Definition 52 If the limit of I(ε) as ε → 0+ exists, then it is called theprincipal value of the integral I

P∫ b

a

f(x)

xdx = lim

ε→0+

(∫ −εa

f(x)

xdx+

∫ b

ε

f(x)

xdx

). (4.20)

In this section we consider the asymptotics of the integrals of the form

F (λ) = P∫R

e±iλS(x) f(x)dx

x(4.21)

as λ→∞.

Lemma 7 Let f ∈ C∞0 (R) be a smooth function of compact support. Thenas λ→∞

P∫R

e±iλx f(x)dx

x= ±iπf(0) +O(λ−∞) . (4.22)

Theorem 42 Let f ∈ C∞0 (R) be a smooth function of compact support,S ∈ C∞(R) be a real valued smooth function and S ′(0) 6= 0. Then as λ→∞

F (λ) = P∫R

e±iλS(x) f(x)dx

x= [sgnS ′(0)] iπf(0) exp[iλS(0)] +O(λ−∞) .

(4.23)

Theorem 43 Let f ∈ C∞0 (R) be a smooth function of compact support,S ∈ C∞(R) be a real valued smooth function. Let x = 0 be the only statinarypoint of the function S on supp f , and let it be nondegenerate, i.e. S ′(0) = 0and S ′′(0) 6= 0. Then as λ→∞ there is asymptotic expansion

F (λ) = P∫R

e±iλS(x) f(x)dx

x

∼ exp[iλS(0)]λ−1/2

∞∑k=0

akλ−k . (4.24)


The leading asymptotics has the form

F (λ) = exp[iλS(0) + [sgnS ′′(0)] i

π

4

]√ 2π

|S ′′(0)|

×λ−1/2

[− S

′′′(0)

6S ′′(0)f(0) + f ′(0) +O(λ−1)

]. (4.25)

4.2 Stationary Phase Method in Many Di-

mensions

Let Ω be a domain in Rn and f ∈ C∞0 (Ω) be a smooth function of compactsupport, S ∈ C∞(Ω) be a real valued smooth function. In this section westudy the asympotics as λ→∞ of the multi-dimensional Fourier integrals

F (λ) =

∫Ω

f(x) exp[iλS(x)] dx . (4.26)

4.2.1 Nondegenerate Stationary Point

Localization Principle

Lemma 8 Let Ω be a domain in Rn and f ∈ C∞0 (Ω) be a smooth functionof compact support, S ∈ C∞(Ω) be a real valued smooth function withoutstationary points in supp f , i.e. ∂xS(x) 6= 0 for x ∈ supp f . Then as λ→∞

F (λ) = O(λ−∞) (4.27)

This lemma is proved by integration by parts.

Definition 53 1. The set S(Rn) of all smooth functions on Rn that de-crease at |x| → ∞ together with all derivatives faster than any powerof |x| is called the Schwartz space.

2. For any integrable function f ∈ L1(Rn) the Fourier transform isdefined by

F(f)(ξ) = (2π)−n/2∫Rn

exp(i x · ξ) f(x) dx (4.28)


Proposition 9 1. Fourier transform is a one-to-one onto map (bijection)F : S(Rn)→ S(Rn), i.e. if f ∈ S(Rn), then F(f) ∈ S(Rn).

2. The inverse Fourier transform is

F−1(f)(x) = (2π)−n/2∫Rn

exp(−i x · ξ) f(ξ) dξ (4.29)

Theorem 44 Let Ω be a finite domain in Rn, f ∈ C∞0 (Ω) be a smooth

function with compact support in Ω and S ∈ C∞(Ω) be a real valued smoothfunction. Let S have a single stationary point x0 in Ω and let it be non-degenerate. Then as λ→∞ there is asymptotic expansion

F (λ) ∼ λ−n/2 exp[iλS(x0)]∞∑k=0

akλ−k . (4.30)

The coefficients ak are determined in terms of derivatives of the functions fand S at x0.

The leading asymptotics is

F (λ) =

(2π

λ

)n/2exp

[iλS(x0) + [sgn ∂2

xS(x0)] iπ

4

]×∣∣det ∂2

xS(x0)∣∣−1/2 [

f(x0) +O(λ−1)]. (4.31)

Recall that sgnA = ν+(A)− ν−(A) denotes the signature of a real symmet-ric nondegenerate matrix A, where ν±(A) are the number of positive andnegative eigenvalues of A.

4.2.2 Integral Operators with Singular Kernels

Let Ω be a bounded domain in Rn including the origin, 0 ∈ Ω. Let f ∈ C∞0 (Ω)be a smooth function with compact support and S ∈ C∞(Rn) be a real valuednon-positive smooth function. Let S have a single stationary point x = 0,and let it to a nondegenerate, i.e. let S(0) = S ′(0) = 0 and ∂2

xS(0) 6= 0. LetKλ : C∞0 (Ω)→ C∞(Ω) be a linear integral operator defined by

(Kλf)(x) =

(λ

2π

)n/2 ∫Ω

exp[iλS(x− y)]f(y) dy . (4.32)


Then as λ→∞

(Kλf)(x) = exp[[sgn ∂2

xS(0)] iπ

4

] ∣∣det ∂2xS(0)

∣∣−1/2 [f(x) +O(λ−1)

](4.33)

uniformly for x ∈ Ω. On the other hand, if x 6∈ Ω, then as λ→∞

(Kλf)(x) = O(λ−∞) . (4.34)

Chapter 5

Saddle Point Method

5.1 Saddle Point Method for Laplace Inte-

grals

Let γ be a contour in the complex plane and the functions f and S areholomorphic in a neighborhood of this contour. In this section we will studythe asymptotics as λ→∞ of the Laplace integrals

F (λ) =

∫γ

f(z) exp[λS(z)] dz . (5.1)

5.1.1 Heuristic Ideas of the Saddle Point Method

The idea of the saddle point method (or method of steepest descent)is to deform the contour in such a way that the main contribution to theintegral comes from a neighborhood of a single point. This is possible sincethe functions f and S are holormorphic.

First of all, let us find a bound for |F (λ)|. For a contour γ = γ0 of finitelength l(γ0) we have, obviously,

|F (λ)| ≤ l(γ0) maxz∈γ0

f(z) exp[λ ReS(z)] . (5.2)

Now, let Γ be the set of all contours obtained by smooth deformations of thecontour γ0 keeping the endpoints fixed. Then such an estimate is valid forany contour γ ∈ Γ, hence,

|F (λ)| ≤ infγ∈Γ

l(γ) max

z∈γf(z) exp[λ ReS(z)]

. (5.3)

53

54 Saddle Point Method

Since we are interested in the limit λ→∞, we expect that the length of thecontour does not affect the accuracy of the estimate. Also, intuitively it isclear that the behavior of the function S is much more important that thatof the function f (since S is in the exponent and its variations are scaledsignificantly by the large parameter λ). Thus we expect an estimate of theform

|F (λ)| ≤ C(γ, f) infγ∈Γ

maxz∈γ

exp[λReS(z)]

, (5.4)

where C(γ, f) is a constant that depends on the contour γ and the function fbut does not depend on λ. So, we are looking for a point on a given contourγ where the maximum of ReS(z) is attained. Then, we look for a contour γ∗

where the minimum of this maximum is attained, i.e. we assume that thereexists a contour γ∗ where

minγ∈Γ

maxz∈γ

ReS(z) (5.5)

is attained. Such a contour will be called a minimax contour.Let z0 ∈ γ∗ be the only point on the contour γ∗ where the maximum of

ReS(z) is attained. Then, we have an estimate

|F (λ)| ≤ C(γ∗, f) exp[λ ReS(z0)] . (5.6)

By deforming the contour of integration to γ∗ we obtain

F (λ) =

∫γ∗

f(z) exp[λS(z)] dz . (5.7)

The asymptotics of this integral can be computed by Laplace method.

1. Boundary Point. Let z0 be an endpoint of γ∗, say, the initial point.Suppose that S ′(z0) 6= 0. Then one can replace the integral F (λ) by anintegral over a small arc with the initial point z0. Finally, integratingby parts gives the leading asymptotics

F (λ) =1

−S ′(z0)exp[λS(z0)]λ−1

[f(z0) +O(λ−1)

]. (5.8)

2. Interior Point. Let z0 be an interior point of the contour γ∗. Fromthe minimax property of the contour γ∗ it follows that the point z0 is

Saddle Point Method 55

the saddle point of the function Re S(z). Let z = x + iy. Since thesaddle point is a stationary point, then

∂

∂xRe S(z0) =

∂

∂yRe S(z0) = 0 . (5.9)

Then from Cauchy-Riemann conditions it follows that S ′(z0) = 0.

Definition 54 1. A point z0 ∈ C is called a saddle point of the complexvalued function S : C→ C if S ′(z0) = 0.

2. A saddle point z0 is said to be of order n if

S ′(z0) = · · · = S(n)(z0) = 0, S(n+1)(z0) 6= 0 . (5.10)

3. A first order saddle point is called simple, i.e. for a simple saddlepoint S ′′(z0) 6= 0.

4. The number Re S(z0) is called the height of the saddle point.

To compute the asymptotics at an interior saddle point, we replace thecontour γ∗ by a small arc γ0

∗ containing the point z0. Then we expand thefunction S in the Taylor series in the neighborhood of z0 and neglect theterms of third order and higher, i.e. we replace S by

S(z) = S(z0) +1

2S ′′(z0)(z − z0)2 +O((z − z0)3) . (5.11)

Finally, by changing the variables and evaluating the integral by Laplacemethod we obtain the asymptotics as λ→∞

F (λ) =

√2π

−S ′′(z0)λ−1/2 exp[λS(z0)]

[f(z0) +O(λ−1)

]. (5.12)

The saddle point method consists of two parts: the toplogical part andthe analytical part.

The topological part consists of the deformation of the contour to theminimax contour γ∗ that is most suitable for asymptotical estimates. Theanalytical part contains then the evaluation of the asymptotics over the con-tour γ∗.


The analytical part is rather straightforward. Here one can apply thesame methods and as in the Laplace method; in many cases one can evenuse the same formulas.

The topological part is usually much more complicated since it is a globalproblem. It could happen, for example, that a contour γ∗ where the minimaxminγ∈Γ maxz∈γ ReS(z) is attained does not exist at all! Next, strictly speak-ing we need to look for a contour where minγ∈Γ maxz∈γ f(z) exp[ReS(z)] isattained what makes the problem even more complicated.

Thus, if one can find the minimax contour, then one can compute theasymptotics of F (λ) as λ→∞. Unfortunately, there is no simple algorithmthat would always enable one to find the minimax contour. Nevertheless,under certain conditions one can prove that such a contour exists and, infact, find one. We will discuss this point later.

5.1.2 Level Curves of Harmonic Functions

Lemma 9 Let S : C → C be holomorphic at z0 and S ′(z0) 6= 0. Then in asmall neighborhood of the point z0 the arcs of the level curves

ReS(z) = ReS(z0), ImS(z) = ImS(z0), (5.13)

are analytic curves. These curves are orthogonal at z0.

Let ϕ(z) = S(z) − S(z0). Since S ′(z0) 6= 0 the function w = ϕ(z) is is aone-to-one holomorphic, in fact, conformal, mapping of a neghborhood ofthe point z = z0 onto a neighborhood of the point w = 0. The inversefunction z = ϕ−1(w) is holomorphic in a neighborhood of the origin w = 0.Let w = u + iv and ψ(u, v) ≡ ϕ−1(w). The arc of the level curve ReS(z) =ReS(z0) is mapped onto an open interval on the imaginary axis. It is definedby z = ψ(0, v) and is analytic. The same is true for the level curve ImS(z) =ImS(z0). It is defined by z = ψ(u, 0) and is analytic as well. The tangentvectors to the level curves at z0 are determined by

∂ψ(u, 0)

∂u

∣∣∣∣u=0

= (∂wϕ−1)(0) ,

∂ψ(0, v)

∂v

∣∣∣∣v=0

= i (∂wϕ−1)(0) (5.14)

and are obviously orthogonal. One could also conlude this from the fact thatthe map is conformal and, therefore, preserves the angles.


Lemma 10 Let z0 be a simple saddle point of the function S. Then in asmall neighborhood of the point z0 the level curve ReS(z) = ReS(z0) consistsof two analytic curves that intersect orthogonally at the point z0 and separatethe neighborhood of z0 in four sectors. The signs of the function Re [S(z)−S(z0)] in adjacent sectors are different.

In a neighborhood of a simple saddle point the there is a one-to-one holo-morphic function z = ψ(w) such that ψ(0) = z0, ψ′(0) 6= 0, and S(ψ(w)) =S(z0) + w2. In the complex plane of w the level curve ReS(z) = ReS(z0)takes the form Rew2 = 0. Its solution consists of two orthogonal linesw± = (1 ± i)t with −ε < t < ε that intersect at the point w = 0. The levelcurves z± = ψ(w±) have listed properties.

More generally :

Lemma 11 Let z0 be a saddle point of the function S of order n. Thenin a small neighborhood of the point z0 the level curve ReS(z) = ReS(z0)consists of (n+ 1) analytic curves that intersect at the point z0 and separatethe neighborhood of z0 in 2(n+ 1) sectors with angles π(n+ 1) at the vertex.The signs of the function Re [S(z)− S(z0)] in adjacent sectors are different.

Definition 55 Let S be a complex valued function and γ be a simple curvewith the initial point z0. The curve γ is called curve of steepest descentof the function ReS if ImS(z) = const and ReS(z) < ReS(z0) for z ∈ γ,z 6= z0. If ImS(z) = const and ReS(z) > ReS(z0) for z ∈ γ, z 6= z0, thenthe curve γ is called curve of steepest ascent of the function ReS.

Lemma 12 1. If z0 is not a saddle point, then there is exactly one curveof steepest descent.

2. If z0 is a simple saddle point, then there are 2 curves of steepest descent.

3. If z0 is a saddle point of order n, then there are (n+1) curves of steepestdescent.

4. In a neighborhood of a saddle point z0 in each sector in which Re [S(z)−S(z0)] > 0 there is exatly one curve of steepest descent.

This is proved by a change of variables in a neighborhood of z0.


Remarks. Let S : D → C be a nonconstant holomorphic function in adomain D. Let z = x+iy and S(z) = u(x, y)+iv(x, y). Then both u : D → R

and v : D → R are harmonic functions in D. Harmonic functions do not havemaximum or minimum points in the interior of D. They are attained onlyat the boundary of the domain D. All critical points of harmonic functions,i.e. the points where ∂u = ∂v = 0 are saddle points. These are exactly thepoints where S ′(z) = 0. That is why such points are called saddle pointsof the function S. In the simplest case S = z2 the surface u = x2 − y2 ishyperbolic paraboloid (saddle).

Definition 56 Two contours γ1 and γ2 are called equivalent if∫γ1

f(z) exp[λS(z)] dz =

∫γ2

f(z) exp[λS(z)] dz . (5.15)

Lemma 13 Let S and f be holormorphic functions on a finite contour γ.Let the points where maxz∈γ ReS(z) is attained are neither saddle points northe endpoints of the contour γ. Then there is a contour γ′ equivalent to thecontour γ and such that

maxz∈γ′

ReS(z) < maxz∈γ

ReS(z) . (5.16)

Theorem 45 Let F (λ) be a Laplace integral (5.1) If there exists a contourγ∗ such that: i) it is equivalent to the contour γ and ii) the integral F (λ)attains the minimax minγ∈Γ maxz∈γ ReS(z) on it. Then among the pointswhere maxz∈γ∗ ReS(z) is attained there are either endpoints of the contouror saddle points zj such that in aneighborhood of zj the contour γ∗ goesthrough two different sectors where ReS(z) < ReS(zj).

5.1.3 Analytic Part of Saddle Point Method

In this section we always assume that γ is a simple smooth (or piece-wisesmooth) curve in the complex plane, which may be finite or infinite. Thefunctions f and S are assumed to be holomorphic on γ. Also, we assumethat the integral F (λ) converges absolutely.

First of all, we have

Lemma 14 If maxz∈γ ReS(z) ≥ C, then

F (λ) = O(eCλ), (λ ≥ 1) . (5.17)


Theorem 46 Let z0 be the initial endpoint of the curve γ. Let f and S areanalytic at z0, ReS(z0) > ReS(z) ∀z ∈ γ, and S ′(z0) 6= 0. Then, as λ→∞there is asympotic expansion

F (λ) ∼ exp[λS(z0)]∞∑k=0

akλ−k−1 (5.18)

where

ak = −(− 1

S ′(z)

∂

∂z

)k [f(z)

S ′(z)

]∣∣∣∣∣z=z0

(5.19)

The proof is by integration by parts.

Theorem 47 Let z0 be an interior point of the curve γ. Let f and S areanalytic at z0, ReS(z0) > ReS(z) ∀z ∈ γ. Let z0 be a simple saddle point ofS such that in a neighborhood of z0 the contour γ goes through two differentsectors where ReS(z) < ReS(z0). Then, as λ → ∞ there is asympoticexpansion

F (λ) ∼ exp[λS(z0)]∞∑k=0

akλ−k−1/2 . (5.20)

The branch of the square root is choosen so that arg√−S ′′(z0 is equal to the

angle between the positive direction of the tangent to the curve γ at the pointz0 and the positive direction of the real axis.

In a neighborhood of z0 there is a mapping z = ψ(w) such that

S(ψ(w)) = S(z0)− w2

2. (5.21)

After this change of variables and deforming the contour to the steepestdescent contour the integral becomes

F (λ) = exp[λS(z0)]

∫ ε

ε

e−λw2/2f(ψ(w))ψ′(w) dw +O(λ−∞) . (5.22)

Since both f and ψ are holomorphic there is a Taylor expansion

f(ψ(w)ψ′(w) =∞∑k=0

ckwk . (5.23)


Then the coefficients ak are easily computed in terms of ck

ak = 2k+1/2Γ

(k +

1

2

)c2k . (5.24)

Theorem 48 Let γ be a finite contour and f and S be holomorphic in aneighborhood of γ. Let maxz∈γ ReS(z) be attained at the points zj and thesepoints are either the endpoints of the curve or saddle points such that ina neighborhood of a saddle points the contour γ goes through two differentsectors where ReS(z) < ReS(zj). Then as λ → ∞ the integral F (λ) isasymptotically equal to the sum of contributions of the points zj.

Remark. The integral over a small arc containing a saddle point is calledthe contribution of the saddle point to the integral.

Proposition 10 Let f and S be holomorphic functions on γ and ImS(z) =const on γ. If γ has finite number of saddle points, then as |lambda| → ∞,| arg λ| ≤ π/2 − ε < π/2, the asymptotic expansion of the integral F (λ) isequal to the sum of contributions of the saddles points and the endpoints ofthe contour.

5.1.4 Examples

1. ∫ ∞−∞

eiλx(1 + x2)−λ dx ∼√π(1− c)e−λcλ−1/2(2c)−λ (λ→∞)

(5.25)where c =

√2− 1. There are two saddle points z1,2 = i(−1±

√2). The

asymptotics is determined by the contribution from the point z1.

2. ∫ i+∞

i−∞e−z

2

(1 + z)−n dz ∼√π

2i−n e(n−1)/2

(n2

)−n/2, (n→∞) .

(5.26)Here n > 0 is a positive integer.


3. Let a > 0. As λ→∞∫ ia+∞

ia−∞exp(−2λz2−4λ/z) dz ∼ π1/6(6λ)−1/2 exp(3λ+ i3

√3λ) . (5.27)

There are three saddle points, which are the roots of the equationz3 = 1. The asymptotics is determined by the contribution from thepoint z1 = ei 2π/3.

4. ∫ i∞

−i∞exp(−λz2 + z2 ln z) dz ∼ i

√π exp

(−1

2e2λ−1

), (λ→∞) .

(5.28)There are two saddle points z1 = 0 and z2 = eλ−1/2. The asymptoticsis equal to the contribution from the saddle point z2.

Notation

Logic

A =⇒ B A implies B

A⇐= B A is implied by B

iff if and only if

A⇐⇒ B A implies B and is implied by B

∀x ∈ X for all x in X

∃x ∈ X there exists an x in X such that

Sets and Functions (Mappings)

x ∈ X x is an element of the set X

x 6∈ X x is not in X

x ∈ X | P (x) the set of elements x of the set X obeying the propertyP (x)

A ⊂ X A is a subset of X

X \ A complement of A in X

A closure of set A

X × Y Cartesian product of X and Y

f : X → Y mapping (function) from X to Y

f(X) range of f

63

64 Notation

χA characteristic function of the set A

∅ empty set

N set of natural numbers (positive integers)

Z set of integer numbers

Q set of rational numbers

R set of real numbers

R+ set of positive real numbers

C set of complex numbers

Vector Spaces

H ⊕G direct sum of H and G

H∗ dual space

Rn vector space of n-tuples of real numbers

Cn vector space of n-tuples of complex numbers

l2 space of square summable sequences

lp space of sequences summable with p-th power

Normed Linear Spaces

||x|| norm of x

xn −→ x (strong) convergence

xnw−→ x weak convergence

Function Spaces

supp f support of f

H ⊗G tensor product of H and G

Notation 65

C0(Rn) space of continuous functions with bounded support inRn

C(Ω) space of continuous functions on Ω

Ck(Ω) space of k-times differentiable functions on Ω

C∞(Ω) space of smooth (infinitely diffrentiable) functions on Ω

D(Rn) space of test functions (Schwartz class)

L1(Ω) space of integrable functions on Ω

L2(Ω) space of square integrable functions on Ω

Lp(Ω) space of functions integrable with p-th power on Ω

Hm(Ω) Sobolev spaces

C0(V,Rn) space of continuous vector valued functions with boundedsupport in Rn

Ck(V,Ω) space of k-times differentiable vector valued functions onΩ

C∞(V,Ω) space of smooth vector valued functions on Ω

D(V,Rn) space of vector valued test functions (Schwartz class)

L1(V,Ω) space of integrable vector valued functions functions onΩ

L2(V,Ω) space of square integrable vector valued functions func-tions on Ω

Lp(V,Ω) space of vector valued functions functions integrable withp-th power on Ω

Hm(V,Ω) Sobolev spaces of vector valued functions

Linear Operators

Dα differential operator

66 Notation

L(H,G) space of bounded linear transformations from H to G

H∗ = L(H,C) space of bounded linear functionals (dual space)

Bibliography

[1] M. Reed and B. Simon, Methods of Mathematical Physics. I: FunctionalAnalysis, (New York: Academic Press, 1972)

[2] R. D. Richtmyer, Principles of Advanced Mathematical Physics, vol. I,(Berlin: Springer, 1985)

[3] L. Debnath and P. Mikusinski, Introduction to Hilbert Spaces with Ap-plications, (Boston: Academic Press, 1990)

[4] M. V. Fedoriuk, Asymptotics: Integrals and Series, (Moscow: Nauka,1987)

[5] E. T. Copson, Asymptotic Expansions, (Cambridge, UK: CambridgeUniversity Press, 1965)

[6] F. W. J. Olver, Introduction to Asymptotics and Special Functions,(New York: Academic Press, 1973)

[7] F. W. J. Olver, Asymptotics and Special Functions, (New York: Aca-demic Press, 1974)

[8] A. Erdelyi, Asymptotic Expansions, (New York: Dover Publications,1956)

[9] R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpre-tation, (London: Academic Press, 1973)

67

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