Mathematical Quantum Mechanicsstock/uebung/mqm2011/Notes.pdf · Chapter 0 Preliminaries Following is a collection of basic notions from functional analysis regarding operators on

Mathematical Quantum Mechanics

January 25, 2012

2

These short notes are a guideline to the topics covered in class. Motivations, moreexamples and derivations were discussed in class.

Contents

0 Preliminaries 9

1 Self-adjoint operators 131.1 Friedrichs extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Applications in quantum mechanics . . . . . . . . . . . . . . . . . . . . 141.3 Perturbations of self-adjoint operators . . . . . . . . . . . . . . . . . . . 161.4 Spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.5 Spectral decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Quantum time evolution 212.1 Stone’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 RAGE theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 Excursion:

Trace class and Hilbert-Schmidt operators . . . . . . . . . . . . . . . . . 25

3 Scattering theory 293.1 Wave and scattering operators . . . . . . . . . . . . . . . . . . . . . . . 293.2 Cook’s criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3 Asymptotic completeness . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4 Existence and asymptotic completeness for short-range perturbations . . . 323.5 Outlook:

Stationary scattering theory . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Methods for bound states 414.1 Weyl’s theorem and HVZ . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3 Riesz projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 Analytic perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . 484.5 Semiclassical approximation for sums of eigenvalues . . . . . . . . . . . 514.6 Lieb-Thirring inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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4 CONTENTS

5 Many-particle methods 595.1 Density functional theories . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Thomas-Fermi theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

CONTENTS 5

Most of the material is taken from the following textbooks where more informationcan be found:

6 CONTENTS

Bibliography

[1] M. S. Birman, M. Z. Solomjak, Spectral theory of sel-adjoint operators in Hilbert space,Reidel 1987.

[2] H.L. Cycon,, R. G. Froese, W. Kirsch, B. Simon, Schrodinger operators, Springer 2008.

[3] E. H. Lieb, R. Seiringer, The stability of matter in quantum mechanics, Cambridge 2010.

[4] M. Reed and B. Simon, Methods of modern mathematical physics I: Functional analysis,rev. and enl. ed., Academic 1981.

[5] M. Reed and B. Simon, Methods of modern mathematical physics II: Fourier analysis, Self-adjointness, Academic 1975.

[6] M. Reed and B. Simon, Methods of modern mathematical physics III: Scattering theory,Academic 1979.

[7] M. Reed and B. Simon, Methods of modern mathematical physics VI: Analysis of operators,Academic 1978.

[8] G. Teschl, Mathematical Methods in Quantum Mechanics, AMS 2009.

[9] J. Weidmann, Linear operators in Hilbert spaces, Springer 1980.

7

8 BIBLIOGRAPHY

Chapter 0

Preliminaries

Following is a collection of basic notions from functional analysis regarding operators on a Hilbertspace, i.e., linear mappings A : D(A) → H defined on some subspace D(A) ⊂ H. Subsequently,H stands for a complex Hilbert space, (·, ·) for its inner product and ‖ · ‖ the induced norm.

Definition 0.1. The graph of an operator A : D(A)→ H is the subspace

gr (A) := (ψ,Aψ) |ψ ∈ D(A) ⊂ H× H .

The operator A : D(A)→ H is called an extension of A if gr (A) ⊂ gr (A).

1. The operator A is called closed if its graph gr (A) is a closed subset of H× H (with respectto the product topology).

2. The operator A is called closable if there exists a closed extension.

Closability of A is equivalent to the property (0, ψ) ∈ gr (A)⇒ ψ = 0, i.e., the closure of thegraph gr (A) is the graph of some operator A.

Definition 0.2. The resolvent set of a closed operator A : D(A)→ H is the set

%(A) := z ∈ C |A− z : D(A)→ H is a bijection .

For z ∈ %(A) the operator (A− z)−1 is called the resolvent. The spectrum is

σ(A) := C\%(A).

1. If A − z is a bijection of D(A) to H, the inverse (A − z)−1 is a closed operator defined onall of H. Hence (A− z)−1 is bounded by the closed graph theorem.

2.

9

10 CHAPTER 0. PRELIMINARIES

The following is a convenient characterization of the spectrum in terms of so-called Weyl se-quences.

Lemma 0.1. A value λ ∈ σ(A) if there is a sequence ψn ∈ D(A) such that ‖ψn‖ = 1 andlimn→∞ ‖(A− z)ψn‖ = 0. If λ is a boundary point of %(A), then the converse is also true. (Sucha sequence is called a Weyl sequence.)

Definition 0.3. A (densely defined) operator A : D(A) → H is called symmetric if for all ϕ,ψ ∈D(A):

(Aψ , ϕ) = (ψ,Aϕ) .

Definition 0.4. For a densely defined operator A : D(A) → H, the adjoint is the unique operatorA∗ : D(A∗)→ H with

D(A∗) := ψ ∈ H | ∃b ∈ R : ∀ϕ ∈ D(A) : |(ψ,Aϕ)| ≤ b‖ϕ‖

and the property that for all ψ ∈ D(A∗) and ϕ ∈ D(A):

(A∗ψ , ϕ) = (ψ,Aϕ) .

It is called self-adjoint ifA = A∗ (which is equilent to (i)A being symmetric and (ii)D(A∗) =D(A)).

1. By the Riesz theorem any bounded linear functional Φ ∈ H∗ on a Hilbert space can beidentified with some ϕ ∈ H such that Φ(ψ) = (φ, ψ) for all ψ ∈ H. As a consequence, forany densely defined operator A : D(A) → H and all ψ ∈ D(A∗) there is ψ ∈ H such that(ψ , ϕ) = (ψ,Aϕ) for all ϕ ∈ D(A).

2. The adjoint A∗ is always closed.

3. If A : D(A) → H is a densely defined symmetric operator, then A∗ is a closed extension ofA.

4. The spectrum of a self-adjoint operator A is real, σ(A) ⊂ R. Moreover, one has the estimate

‖(A− z)−1‖ ≤ dist (z, σ(A))−1

for any z ∈ %(A). Recall that A : H → H is bounded if there is b ∈ R such that for allψ ∈ H:

‖Aψ‖ ≤ b‖ψ‖ .

The smallest such constant b is called the operator norm ‖A‖ := supψ∈H\0‖Aψ‖‖ψ‖ .

Not all symmetric operators are self-adjoint.

11

Example 0.1. Consider A = −i ddx on D(A) = ψ ∈ C1[0, 1] |ψ(0) = ψ(1) which is a densesubspace of L2[0, 1]. Then partial integration shows that for all ϕ,ψ ∈ D(A) and

(ψ,Aϕ) = (Aψ,ϕ) ,

i.e., A is symmetric. However, the partial integration formula for absolutely continuous functionsAC[0, 1] = ψ ∈ L2[0, 1] | ∃ϕ ∈ L2[0, 1] : ψ(x) = ψ(0) +

∫ x0 ϕ(t)dt shows that AC[0, 1] ⊂

D(A∗). Hence A is not self-adjoint.

The basic criterion for self-adjointness is the following.

Theorem 0.2. Let A : D(A) → H be a densely defined symmetric operator. Then the followingare equivalent:

1. A is self-adjoint.

2. ran (A− z) = H for some z ∈ C.

3. A is closed and ker(A− z) = 0 for some z ∈ C.

Proof. (The proof was discussed in the tutorium.)

It is usually rather easy to check, whether an operator is symmetric. In contrast, the constructionof self-adjoint operators as well as to check whether a given operator is self-adjoint is usuallytedious. The next chapter discusses a simple criterion.

12 CHAPTER 0. PRELIMINARIES

Chapter 1

Self-adjoint operators

1.1 Friedrichs extension

The aim of this section is to discuss a convenient way to construct semibounded self-adjoint oper-ators, i.e., self-adjoint operators A : D(A)→ H such that for some α ∈ R and all ψ ∈ D(A):

(ψ,Aψ) ≥ −α‖ψ‖2 .

In physical applications, energy operators usually have this property.In essence, the main message is that there is a one-to-one correspondence between semi-

bounded self-adjoint operators and closed semibounded quadratic forms.

Definition 1.1. A quadratic form is a map

a : Q(a)×Q(a)→ C,(ϕ,ψ) 7→ a(ϕ,ψ),

whereQ(a) is a dense linear subset of a Hilbert space H with the property that a linear in its secondentry and conjugate linear in its first entry.

If a(ϕ,ψ) = a(ψ,ϕ) for all ϕ,ψ ∈ Q(a), then a is called symmetric. If furthermore a(ψ,ψ) ≥0 for all ψ ∈ Q(a), a is called non-negative, and if a(ψ,ψ) ≥ −α‖ψ‖2 for some α ∈ R and allψ ∈ Q(a), a is called semibounded.

Any semibounded quadratic form a (with a ≥ −α) gives rise to an inner product given by(·, ·)a,α := a(·, ·) + (α+ 1)(·, ·) and hence also to a norm on Q(a):

‖ψ‖2a,α := a(ψ,ψ) + (α+ 1)‖ψ‖2 .

Definition 1.2. A semibounded quadratic from a (with a ≥ −α) is called

1. closed, if Q(a) is complete with respect to the norm ‖ · ‖a,α.

13

14 CHAPTER 1. SELF-ADJOINT OPERATORS

2. closable, if for every sequence (ψn) ⊂ Q(a) which is Cauchy with respect to the norm‖ · ‖a,α one has the implication: ‖ψn‖ → 0⇒ ‖ψn‖a,α → 0.

If a semibounded quadratic form a is closable, the completion Ha,α := Q(a)‖·‖a,α of the form

domain Q(a) is hence a complete linear subspace of H. As a consequence, a can be naturallyextended to a closed semibounded quadratic form.

One important way of constructing semibounded, closable quadratic forms is via symmetricsemibounded operators.

Theorem 1.1. Let A : D(A) → H be a symmetric, semibounded operator. Then the quadraticform a : D(A)×D(A)→ C, (ϕ,ψ) 7→ (ϕ, Aψ) is semibounded and closable.

Proof. (The non-trivial part of this theorem is the content of Exercise 1 on Sheet 3. Proof also themore trivial parts!)

The following theorem shows that any semibounded, closed quadratic form can be uniquelyassociated with a self-adjoint operator.

Theorem 1.2. Let a : Q(a) × Q(a) → C be a semibounded, closed quadratic form. Then thereexists a unique self-adjoint operator A : D(A) → H with D(A) ⊂ Q(a) such that for all ψ ∈D(A), ϕ ∈ Q(a):

(ϕ, Aψ) = a(ϕ,ψ) .

Explicitly, one has

D(A) = ψ ∈ Q(a) | ∃ψ ∈ H ∀ϕ ∈ Q(a) : (ϕ, ψ) = a(ϕ,ψ)

Aψ = ψ .

Proof. (The proof of this theorem rests on the Riesz theorem and is non-trivial. Try at least tounderstand the bounded case in which Q(a) = H.)

The two theorems taken together immediately imply:

Corollary 1.3 (Friedrichs extension). Let A : D(A) → H be a symmetric, semibounded operator.

Then there exists a unique self-adjoint extension A of A with D(A) ⊂ Q(a)‖·‖a,α (where a is the

form corresponding to A).

1.2 Applications in quantum mechanics

Following are important physical applications in which one may show the existence of a self-adjointextension via the above construction.

1.2. APPLICATIONS IN QUANTUM MECHANICS 15

Example 1.1 (Free particle). The energy of the free quantum particle in d dimension is describedby the Laplacian, H0 = −∆. For D(H0) = C∞0 (Rd) this operator is densely defined on L2(Rd)and symmetric. Since for any ϕ,ψ ∈ C∞0 (Rd):

(ϕ, (−∆)ψ) =

∫∇ϕ(x)∇ψ(x) =: h0(ϕ,ψ) ,

it follows (ψ, (−∆)ψ) ≥ 0. It corresponds to the quadratic form h0 defined onQ(h0) := H1(Rd).The latter is closed and non-negative. The domain of the unique self-adjoint operator H0 turns outto be D(H0) = H2(Rd).

Example 1.2 (Hydrogen atom). In suitable units, the energy of an atom with charge Z > 0 isdescribed by the operator

H = −∆− Z

|x|.

To check that this operator may be densely defined on D(H) := C∞0 (R3), it remains to bound thepotential energy for any ψ ∈ C∞0 (R3):∫

Z

|x||ψ(x)|2dx ≤ Z ‖ψ‖

√∫|ψ(x)|2|x|2

dx ≤ 2Z ‖ψ‖

√∫|∇ψ(x)|2dx <∞ .

The second inequality is Hardys inequality, Lemma 1.4 below. Hence the operator H is denselydefined on L2(R3). Using partial integration it is straightforward to check that H is symmetric.Hardys inequality may also be employed to show that H bounded from below:

(ψ,Hψ) =

∫|∇ψ(x)|2dx−

∫Z

|x||ψ(x)|2dx

≥∫ (

1

4|x|2− Z

|x|

)|ψ(x)|2dx ≥ −Z2‖ψ‖2 .

Also with ground state transformation

Lemma 1.4 (Hardys inequality). For any ψ ∈ H1(Rd):∫|∇ψ(x)|2dx ≥ (d− 2)

4

∫|ψ(x)|2

|x|2dx.

Proof. (See Exercise 3 on Sheet 3.)

Example 1.3 (Atom with N electrons). For a system of N ∈ N interacting charged particles in anatomic potential with central charge Z > 0 the energy is given by

HN,Z :=

N∑ν=1

(−∆ν −

Z

|xν |

)+

∑1≤ν<µ≤N

1

|xν − xµ|.

The operator is well defined on C∞0 (R3N ) and hence densely defined on L2(R3N ). (Check this!) Itis symmetric and bounded from below.


Another example is the δ-potential in one dimension discussed in Exercise 2 on Sheet 3.

Lemma 1.5 (Hardy-Littlewood-Sobolev inequality). Let p, r > 1 and 0 < λ < d with 1/p +1/r + λ/d = 2. Then there is some finite constant C(d, p, r, λ) such that for all f ∈ Lp(Rd) andg ∈ Lr(Rd): ∫ ∫

f(x)g(y)

|x− y|λdxdy ≤ C(d, p, r, λ) ‖f‖p ‖g‖r .

(For a proof, see Lieb/Loss, Analysis, AMS 2001.)

1.3 Perturbations of self-adjoint operators

Our aim in this section is twofold. First, we investigate the stability of self-adjointness under ’small’symmetric pertubations. Second, we also collect a first (robust) property regrading the behavior ofspectrum under such perturbations.

’Smallness’ of the perturbation is captured by

Definition 1.3. Let A : D(A)→ H and B : D(B)→ H be densely defined. Suppose:

1. D(A) ⊂ D(B) ⊂ H

2. there are a, b ∈ R such that for all ψ ∈ D(A):

‖Bψ‖ ≤ a‖Aψ‖+ b‖ψ‖ ,

with ‖ ·‖ the norm on H. ThenB is calledA-bounded. The infimum of such a is called the relativebound of B with respect to A.

Theorem 1.6 (Kato-Rellich). LetA : D(A)→ H be self-adjoint andB : D(B)→ H be symmetricand A-bounded with relative bound a < 1. Then A+B : D(A)→ H is self-adjoint.

Proof. We will employ the fundamental criterion, Theorem 0.2, and show that there exists someλ ∈ R such that ran (A+B + iλ) = H.

For any λ > 0 and any ψ ∈ H one has ϕ := (A+ iλ)−1ψ ∈ D(A). From the identity

‖(A+ iλ)ϕ‖2 = ‖Aϕ‖2 + λ2‖ϕ‖2

valid for all ϕ ∈ D(A) and the bound ‖(A+ iλ)−1‖ ≤ λ−1 we conclude:

‖ψ‖2 = ‖A(A+ iλ)−1ψ‖2 + λ2‖ϕ‖2 ≥ ‖A(A+ iλ)−1ψ‖2

and hence ‖A(A+ iλ)−1‖ ≤ 1.

1.3. PERTURBATIONS OF SELF-ADJOINT OPERATORS 17

Therefore

‖B(A+ iλ)−1ψ‖ ≤ a‖A(A+ iλ)−1ψ‖+ b‖(A+ iλ)−1ψ‖

≤(a+

b

λ

)‖ψ‖ .

HenceC := B(A+iλ)−1 is a bounded operator on H with operator norm ‖C‖ = ‖B(A+iλ)−1‖ ≤(a+ b

λ

)< 1 if λ is large enough. This implies −1 ∈ %(C) and hence ran (1 + C) = H. Since

ran (A+ iλ) = H one concludes using the fact that for any ϕ ∈ D(A):

(1 + C)(A+ iλ)ϕ = (A+B + iλ)ϕ

that ran (A+B + iλ) = H.

One may estimate how the spectrum may expand under an A-bounded perturbation. In fact,the previous proof implies:

Theorem 1.7. LetA : D(A)→ H be self-adjoint and semibounded, i.e., A ≥ −α for some α ∈ R.Let B : D(B)→ H be symmetric and A-bounded with relative bound a < 1. Then

A+B ≥ −(α+ max

b

1− a, a|α|+ b

).

Proof. (Homework!)

Example 1.4. Consider the Laplacian −∆ on D(−∆) = H2(R3). Then for any real-valued

V ∈ L2(R3) + L∞(R3) := V = V1 + V2 |V1 ∈ L2(R3), V2 ∈ L∞(R3) ,

the operator −∆ + V is self-adjoint on D(−∆). The proof relies on Kato-Rellich’s theorem. Let

D(V ) := ψ ∈ L2(R3) |V ψ ∈ L2(R3).

According to Lemma 1.8 below, any ψ ∈ H2(R3) is essentially bounded and for any a > 0 thereexists b > 0 such that:

‖V ψ‖2 ≤ ‖V1ψ‖2 + ‖V2ψ‖2 ≤ ‖V1‖2‖ψ‖∞ + ‖V2‖∞‖ψ‖2≤ a ‖V1‖2‖∆ψ‖2 + b ‖V1‖2‖ψ‖2 + ‖V2‖∞‖ψ‖2 ,

This proves thatD(−∆) ⊂ D(V ). Moreover, choosing a small enough, the multiplication operatorV is seen to be relative −∆-bounded with arbitrarily small relative bound.

Lemma 1.8. For any a > 0 there is b ∈ R such that for any ψ ∈ H2(R3):

‖ψ‖∞ ≤ a ‖∆ψ‖2 + b‖ψ‖2 .

Proof. (Uses Fourier analysis).


1.4 Spectral theorem

The aim of this section is to discuss the spectral theorem and familiarize us with issues related toit. We will not prove this theorem in this course.

Definition 1.4. A projection-valued measure is a map from the Borel sets to the bounded operatorson a Hilbert space

P : B(R)→ Lin(H) , I 7→ P (I)

whose range are the orthogonal projections, i.e. P (I)∗ = P (I) and P (I)2 = P (I), and which hasthe properties:

1. P (R) = 1,

2. If I =⋃n In with In ∩ Im = ∅ for n 6= m, then for all ψ ∈ H:∑

n

P (In)ψ = P (I)ψ .

Every projection-valued measure gives rise to:

1. complex-valued Borel measures for ϕ,ψ ∈ H defined by

µϕ,ψ(I) := (ϕ, P (I)ψ) .

In case ϕ = ψ the measure µψ := µψ,ψ is non-negative and one has µψ(R) = ‖ψ‖2. (In fact.the measure µϕ,ψ may be recovered from µϕ±ψ and µϕ±iψ by the polarization identity.)

2. normal operators associated with measurable functions f ∈ B(R). (Recall thatA : D(A)→H is normal if and only if D(A) = D(A∗) and ‖Aψ‖ = ‖A∗ψ‖ for all ψ ∈ D(A).) Moreprecisely, let

Df := ψ ∈ H |∫|f(x)|2µψ(dx) <∞ (1.1)

and set for ϕ ∈ H and ψ ∈ Df :

(ϕ, P (f)ψ) :=

∫f(x)µϕ,ψ(dx) . (1.2)

The map P : B(R)→ Lin(H) has the following properties:

(a) P (1) = 1

(b) P (f) = P (f)∗

(c) P (fg) = P (f)P (g)

(d) If fn → f and supn ‖fn‖∞ <∞, then P (fn)ψ → P (f)ψ for all ψ ∈ H.

1.5. SPECTRAL DECOMPOSITION 19

In particular, one has for all ϕ,ψ ∈ H:

(P (g)ϕ, P (f)ψ) =

∫g(x) f(x)µϕ,ψ(dx) .

There is a one-to-one correspondence between projection-valued measures and self-adjointoperators.

Theorem 1.9 (Spectral theorem). To every self-adjoint operatorA : D(A)→ H there correspondsa unique projection valued measure PA such that for all ϕ ∈ H and ψ ∈ D(A):

(ϕ,Aψ) =

∫xµϕ,ψ(dx)

where µϕ,ψ := (ϕ, PAψ) stands for the spectral measure associated with ϕ,ψ.

As a consequence, given a self-adjoint operator A : D(A) → H, we may define functionsof that operator using the functional calculus described above. Namely, for f Borel measurable,(1.2) defines on (1.1) the normal operator PA(f) =: f(A). In the next chapter, we will particularlyfocus on properties of the family of operators U(t) := e−itA, t ∈ R, modelling quantum evolution.

1.5 Spectral decomposition

It is a consequence of the Radon-Nikodym theorem that every (complex-valued) Borel measure µon R can be uniquely decomposed into µ = µac + µsc + µpp, where

1. µac is absolutely continuous with respect to Lebesgue measure, i.e. µac(dx) = h(x)dx forsome h ∈ L1(R).

2. µpp is a discrete measure, i.e. µpp =∑

nwnδan for some (an) ⊂ R and (wn) ⊂ C.

3. µsc is singular continuous with respect to Lebesgue measure.

This so-called Lebesgue decomposition when applied to the spectral measures associated witha self-adjoint operator, gives rise to a decomposition of Hilbert space.

Definition 1.5. The spectral subspaces of a self-adjoint operator A : D(A)→ H are:

Hac(A) :=ψ ∈ H |µψ = µacψ

Hsc(A) :=

ψ ∈ H |µψ = µscψ

Hpp(A) :=

ψ ∈ H |µψ = µppψ


1. One can show that Hac(A), Hsc(A) and Hpp(A) are closed, orthogonal subspaces of H with

H = Hac(A)⊕ Hsc(A)⊕ Hpp(A) .

2. The orthogonal projections P# corresponding to H#(A) commute with the spectral projec-tions associated with A, i.e., for all I ∈ B(R) and # = ac, sc, pp:

P# PA(I) = PA(I)P# .

As a consequence, the restrictions of A to H#(A) are well-defined self-adjoint operators(through the spectral representation).

3. For future reference, it is useful to note that for any ϕ ∈ H and ψ ∈ H#(A), the spectralmeasure µϕ,ψ inherits the property #, since for any I ∈ B(R)

µϕ,ψ(I) = (ϕ ,PA(I)P#ψ) = (P#ϕ ,PA(I)P#ψ) = µ#ϕ,ψ(I) .

Definition 1.6. The spectral components of a self-adjoint operator A : D(A) → H are definedthrough the restrictions of A to the spectral subspaces.

1. σac(A) := σ(A|Hac(A)

)is the absolutely continuous spectrum,

2. σsc(A) := σ(A|Hsc(A)

)is the singular continuous spectrum,

3. σpp(A) := σ(A|Hpp(A)

)is the pure point spectrum.

1. One has σ(A) = σac(A) ∪ σsc(A) ∪ σpp(A).

2. The set of eigenvalues

σp(A) := a ∈ R | ∃ψ ∈ H\0 : Aψ = aψ

is a subset of σpp(A). In general, one has σpp(A) = σp(A).

Chapter 2

Quantum time evolution

2.1 Stone’s theorem

The following theorem in particular guarantees that U(t) = e−itA is unitary and ψ(t) = U(t)ψsolves the Schrodinger equation, i.e., the Cauchy problem

id

dtψ(t) = Aψ(t) , ψ(0) = ψ

for all ψ ∈ D(A).

Theorem 2.1. Let A : D(A)→ H be self-adjoint and set U(t) := e−itA with t ∈ R. Then:

1. U(t), t ∈ R, is a strongly continuous unitary group, i.e.

(a) U(t) is unitary for all t ∈ R.

(b) U(t+ s) = U(t)U(s) for all t, s ∈ R.

(c) For all t ∈ R and all ψ ∈ H: lims→t U(s)ψ = U(t)ψ.

2. the limit limt→0 t−1(U(t)ψ−ψ) exists if and only ifψ ∈ D(A) and one has limt→0 t

−1(U(t)ψ−ψ) = −iAψ for all ψ ∈ D(A).

3. U(t)D(A) = D(A) and AU(t) = U(t)A for all t ∈ R.

Proof. (Based on the spectral theorem)

The converse also holds.

Theorem 2.2 (Stone). For every strongly continuous unitary group U(t), t ∈ R, there exists aunique self-adjoint operator A : D(A) 7→ H such that U(t) = e−itA for all t ∈ R.

(We will not give a proof.)

21

22 CHAPTER 2. QUANTUM TIME EVOLUTION

2.2 RAGE theorem

The transition probability into a state ϕ when time-evolving a state ψ for time t is given by|(ϕ, e−itAψ)|2. By the spectral theorem, it is expressed in terms of the Fourier transform ofthe spectral measure:

µϕ,ψ(t) :=

∫e−itxµϕ,ψ(dx) = (ϕ, e−itAψ) .

It is folk knowledge that the ’smoothness’ properties of µϕ,ψ determine the decay properties ofits Fourier transform. For example:

1. if µϕ,ψ =∑N

n=1wnδan , then µϕ,ψ(t) =∑N

n=1wne−itan is a periodic function of t. Physi-

cally, it hence models a time evolution which shows recurrence.

2. if µϕ,ψ(dx) = hϕ,ψ(x)dx, then limt→∞ µϕ,ψ(t) = 0 by the Riemann-Lebesgue lemma.

The following theorem concerning Cesaro averages of Fourier transforms generalizes theabove observations.

Theorem 2.3 (Wiener). Let µ be a finite complex Borel measure on R. Then

limT→∞

1

T

∫ T

0|µ(t)|2 dt =

∑E∈suppµpp

|µ(E)|2 . (2.1)

Proof. The assertion follows by Fubini’s theorem:

1

T

∫ T

0|µ(t)|2 dt =

∫ ∫ (1

T

∫ T

0ei(E−E

′)t dt

)µ(dE′)µ(dE) .

Since the integrand is uniformly bounded by one and converges pointwise to 10(E − E′), thedominated convergence theorem implies

limT→∞

1

T

∫ T

0|µ(t)|2 dt =

∫ ∫10(E − E′)µ(dE′)µ(dE)

=

∫µ(E)µ(dE) =

∑E∈suppµpp

|µ(E)|2 .

In order to formulate an application of Wiener’s theorem in the context of quantum dynamics,we recall the following definitions and facts:

1. K ∈ Lin(H) is finite-rank iff rankK = dim ran K <∞.

2.2. RAGE THEOREM 23

2. The set C(H) of compact operators is the norm closure of the set of finite-rank operators, i.e.for any K ∈ C(H) there is a sequence (Kn) of finite-rank operators such that limn→∞ ‖K−Kn‖ = 0.

3. If K ∈ C(H) and (ϕ,ψn)→ (ϕ,ψ) for any ϕ ∈ H, then ‖Kψn −Kψ‖ → 0.

Definition 2.1. The operator K : D(K) → H is called relatively compact with respect to A :D(A)→ H if D(A) ⊂ D(K) and the operator K(A− z)−1 is compact for all z ∈ %(A).

Corollary 2.4. Let A : D(A) → H be a self-adjoint and K : D(K) → H be a relatively compactoperator. Then

limT→∞

1

T

∫ T

0

∥∥K e−itAPcψ∥∥2

dt = 0

for all initial states ψ ∈ D(A), for which the spectral measure of H is continuous, i.e. Pc denotesthe orthogonal projection onto Hc(A) := Hac(A)⊕ Hsc(A).

Proof. In caseK is a rank-one operator, i.e.,K = |ρ)(φ|, the claim follows from Wiener’s theorem(Theorem 2.3) applied to the spectral measure µφ,ψ. As a consequence, it also holds for finite-rankoperators K.

Any compact operatorK may in turn be approximated by finite-rank operators. More precisely,for every ε > 0 there exists Kε of finite rank such that ‖K −Kε‖ < ε and hence

‖K ψ(t)‖ ≤ ‖Kε ψ(t)‖+ ‖(K −Kε)ψ(t)‖ ≤ ‖Kε ψ(t)‖+ ε ‖ψ‖ , (2.2)

where ψ(t) := e−itAPcψ. The Cesaro average of the first term goes to zero in the long-time limit.Since ε was arbitrary, this completes the proof of the assertion in the compact case.

In case K is relatively compact and ψ ∈ D(A) we set ϕ := (A− i)Pcψ and write

Ke−itAPcψ = K(A− i)−1e−itAϕ .

Since K(A− i)−1 is compact, the claim follows from the previous step.

Theorem 2.5 (RAGE). Let A be a self-adjoint operator and Kn : H → H be a sequence ofrelatively compact operators which converge strongly to the identity. Then

Hc(A) =

ψ ∈ H

∣∣ limn→∞

limT→∞

1

T

∫ T

0

∥∥Kne−itAψ

∥∥2dt = 0

Hpp(A) =

ψ ∈ H

∣∣ limn→∞

supt∈R

∥∥(1−Kn)e−itAψ∥∥ = 0

Proof. If ψ ∈ Hc(A) the previous corollary implies limT→∞

1T

∫ T0 ‖Knψ(t)‖2 dt = 0 for every n.

If ψ ∈ Hpp(A) we claimlimn→∞

supt∈R‖(1−Kn)ψ(t)‖ = 0 . (2.3)


For a proof we expand ψ into eigenfunctions (ξn) of H , and split the sum into the first N termsand a remainder with norm less than ε,

ψ =N∑k=1

(ξk, ψ) ξk + φN , ‖φN‖ < ε . (2.4)

Since limn→∞ ‖(1−Kn)ξk‖ = 0 for every k, the first term contributes zero to the limit in thecharacterization of Hpp(A) . This completes its proof since ε may be chosen arbitrarily small.

Since every ψ may be uniquely decomposed into a component ψc ∈ Hc(A) and an orthogonalone ψpp ∈ Hpp(A), it remains to show that (i) the limit in the first identity does not tend to zero forψpp and, (ii) the limit in the second identity does not tend to zero for ψc.

The proof of the first assertion relies on

‖Knψpp(t)‖ ≥ |‖ψ‖ − ‖(1−Kn)ψpp(t)‖| > 0 (2.5)

for all t ≥ 0 and sufficiently large n.The second assertion follows by contradiction. Suppose the characterization of Hpp(A) also

applies to ψc 6= 0, then Corollary 2.4 implies

0 =

(limn→0

limT→∞

1

T

∫ T

0‖(1−Kn)ψc(t)‖2 dt

) 12

≥ ‖ψc‖ −(

limn→0

limT→∞

1

T

∫ T

0‖Knψ

c(t)‖2 dt) 1

2

= ‖ψc‖ (2.6)

which contradicts ψc 6= 0.

To interprete the RAGE theorem, we use that for a Schrodinger operator H = −∆ + V onD(−∆) = H2(Rd) where V is a relatively bounded perturbation, the multiplication operators 1BRcorresponding to balls BR := x ∈ Rd | |x| ≤ R are relatively compact with respect to H (seeLemma 2.6 below). Hence:

1. The states ψ ∈ Hc(H) are scattering states since upon time average they eventually leaveany ball:

Hc(H) =

ψ ∈ L2(Rd)

∣∣∣ limR→∞

limT→∞

1

T

∫ T

0

(∫BR

∣∣(e−itHψ) (x)∣∣2 dx)1/2

dt = 0

.

2. The states ψ ∈ Hpp(H) are bound states since they are forever confined to a suitably largeball:

Hc(H) =

ψ ∈ L2(Rd)

∣∣∣ limR→∞

supt∈R

∫Rd\BR

∣∣(e−itHψ) (x)∣∣2 dx = 0

.

2.3. EXCURSION: TRACE CLASS AND HILBERT-SCHMIDT OPERATORS 25

For the interpretation of the RAGE theorem the following lemma was useful.

Lemma 2.6. Let B ∈ B(Rd) be a bounded Borel set. Then:

1. 1B(−∆− z)−1 is compact for all z ∈ C\[0,∞).

2. if H = −∆ + V is self-adjoint on D(−∆) = H2(R) and V is relatively bounded withrespect to V , then 1B(H − z)−1 is compact for any z ∈ %(H).

Proof. We will give the proof only in case d = 3 and proof that 1B(−∆ − z)−1 for Im z > 0 isa Hilbert-Schmidt operator. (The case Im z < 0 is treated similarly.) First note that by means ofFourier transformation one shows that (−∆− z)−1 : L2(R3)→ L2(R3) is an integral operator

(−∆− z)−1ψ(x) =

∫G(x− y, z)ψ(y)dy , G(ξ, z) :=

ei√z |ξ|

4π |ξ|.

(Note that since Im z > 0 also Im√z > 0 and hence one has exponential decay in the integral.) As

a consequence, 1B(−∆− z)−1 is also an integral operator with kernel given by 1B(x)G(x− y, z)which is sqaure integrable:∫

R3

∫R3

| 1B(x)G(x− y, z)|2dxdy = |B|∫R3

|G(ξ, z)|2dξ < ∞ .

By Lemma 2.7 below the operator is Hilbert-Schmidt, and hence compact.

The second claim is left as a homework.

2.3 Excursion:Trace class and Hilbert-Schmidt operators

Definition 2.2. For a bounded, non-negative operator A ≥ 0 on a Hilbert space H one defines the(possibly divergent) trace as

tr |A| :=∑n

(ϕn, Aϕn) ,

where (ϕn) is any orthonormal basis in H.

The trace is well defined since it is independent of the chosen orthonormal basis.

In case A is compact and hence A∗A ≥ 0 is compact, the spectral theorem for self-adjointcompact operators asserts that there is an orthonormal basis (ϕn) of H and corresponding non-negative eigenvalues (an) such that

|A| :=√A∗A =

∑n

an|ϕn)(ϕn| .


We may hence consider the following (possibly divergent) series:

‖A‖p := (tr |A|p)1/p =

(∑n

apn

)1/p

, p ∈ [1,∞) .

Their finiteness defines subspaces of the space of compact operators C(H).

Definition 2.3. For any p ∈ [1,∞) the set

Jp(H) := A ∈ C(H) | tr |A|p <∞ ,

is called the Schatten-p class and ‖ · ‖p is the Schatten-p-norm on Jp(H).

1. It is not hard to see that ‖ · ‖p is indeed a norm on Jp(H). Moreover, Jp(H) is complete withrespect to this norm, i.e., it is a Banach space.

2. By definition J1(H) ⊂ J2(H) ⊂ · · · ⊂ C(H).

There are two important special cases which often appear in quantum mechanics:

1. p = 1 in which case J1(H) is called trace class, and

2. p = 2 in which case J2(H) is called the set of Hilbert-Schmidt operators. (The space J2(H)is in fact a Hilbert space.)

Since Hilbert-Schmidt operators are characterized by the finiteness of

tr A∗A =∑n

‖Aϕn‖2 <∞

for some orthonormal basis (ϕn), it is relatively easy (in comparison to the other Schatten classes)to check whether a given operator is Hilbert-Schmidt operator. A special class of Hilbert-Schmidtoperators are integral operators with square-integrable kernel.

Lemma 2.7. Let Λ ⊂ Rd be an open set and a ∈ L2(Λ × Λ). Then the operator A : L2(Λ) →L2(Λ) with

(Aψ)(x) :=

∫Λa(x, y)ψ(y) dy

is Hilbert-Schmidt with

‖A‖22 =

∫Λ

∫Λ|a(x, y)|2dx dy .

2.3. EXCURSION: TRACE CLASS AND HILBERT-SCHMIDT OPERATORS 27

Proof. Consider any orthonormal basis (ϕn) of L2(Λ). Then their product is an orthonormal basisof L2(Λ × Λ) and set αn,m :=

∫Λ

∫Λ ϕn(x)ϕm(y) a(x, y)dxdy the coefficients of a with respect

to this basis. For N ∈ N the operator

ANψ :=

N∑n,m=1

αn,m |ϕn)(ϕm|

is a finite-rank operator which approximates A, namely ‖A − AN‖ → 0 as N → ∞. Hence A iscompact and one has

∞∑n=1

‖Aϕn‖2 =

∞∑n,m=1

|αn,m|2 =

∫Λ

∫Λ|a(x, y)|2dx dy .

Here the last equality is the Plancherel identity on L2(Λ× Λ).


Chapter 3

Scattering theory

3.1 Wave and scattering operators

Scattering theory revolves about the asymptotic comparison of two time-evolutions generated byself-adjoint operators H : D(H) → H and H0 : D(H) → H. One thinks of the latter as the freetime evolution and asks whether

1. a state ψ looks in the remote past like a free state ψ− in the sense that

e−itHψ ≈ e−itH0ψ− , as t→ −∞.

2. a state ψ looks in the remote future like a free state ψ+ in the sense that

e−itHψ ≈ e−itH0ψ+ , as t→∞.

Therefore we should should have ψ = limt→±∞ eitHe−itH0ψ±. These limits relate the outgo-

ing/incoming asymptotic states ψ± to states ψ, which have an outgoing/incoming asymptotic state,i.e., under the time evolution generated by H they evolve asymptotically free in the remote futureor past.

Definition 3.1. For a pair of self-adjoint operators H and H0, the wave operators (or: Mølleroperators) are

Ω±ψ := limt→±∞

eitHe−itH0ψ

with domain D(Ω±) :=ψ ∈ H | limt→±∞ e

itHe−itH0ψ exists

.

Following are a few first properties:

1. D(Ω±) is closed and e−itH0D(Ω±) ⊂ D(Ω±).

2. Ω± : D(Ω±)→ H are isometries.

29

30 CHAPTER 3. SCATTERING THEORY

3. For all ψ ∈ D(Ω±) and t ∈ R one has the intertwining relation

e−itHΩ±ψ = Ω±e−itH0ψ .

Proof. The first assertion is an exercise. The second follows from

‖Ω±ψ‖ = limt→±∞

‖eitHe−itH0ψ‖ = ‖ψ‖

for all ψ ∈ D(Ω±). The intertwing relation is checked by showing that for all ψ ∈ D(Ω±) ands ∈ R:

Ω±ψ = limt→±∞

eitHe−itH0ψ = limt→±∞

eisHei(t−s)He−i(t−s)H0e−isH0ψ = eisHΩ±e−isH0ψ

where the last equality relied on the fact that e−isH0ψ ∈ D(Ω±).

The above considerations describe a physical scattering situation only if H0 possess enoughfree scattering states which are in the domain of the wave operators. This motivates:

Definition 3.2. We will say that Ω± exist if Hac(H0) ⊂ D(Ω±).

In case H0 has purely absolutely continuous spectrum

H = Hac(H0) , (3.1)

the existence of Ω± requires D(Ω±) = H. In the following, we will restrict most of the discussionto the case (3.1). In a more general situation, one would restrict Ω± to the closed subspace Hac(H0).

The following lemma shows that the wave operators Ω± : H → H which as limits of unitaryoperators are isometric, i.e., Ω∗±Ω± = 1, they are generally not unitary.

Lemma 3.1. Assume that Ω± : H → H exist. Then Ω±Ω∗± is an orthogonal projection onto the(closed) subspace ran Ω±.

Proof. The operator P± := Ω±Ω∗± is self-adjoint P ∗± = P± and one has

P 2± = Ω±Ω∗±Ω±Ω∗± = Ω±Ω∗± = P± ,

since Ω± is an isometry, i.e., Ω∗±Ω±Ω = 1.

The domain D(Ω±) are the sets of outgoing/incoming asymptotic states ψ±. The range of thewave operators ran Ω± are those states which have an outgoing/incoming asymptotic state. Thescattering operator (or scattering matrix) is defined as the one relating the incoming states to theoutgoing states.

3.2. COOK’S CRITERION 31

Definition 3.3. Assume that Ω± exist. Then

S := Ω∗+Ω−

defines the scattering operator on D(S) = Hac(H0).

In a physical scattering situation, the probability should be conserved and hence S should beunitary. This is guaranteed in case ran Ω− = ran Ω+.

Lemma 3.2. Assume that Ω± : H→ H exist and ran Ω− = ran Ω+. Then S : H→ H is unitary.

Proof. Denoting by P = Ω±Ω∗± the projection onto ran Ω±, one has:

S∗S = Ω∗−Ω+Ω∗+Ω− = Ω∗−PΩ− = Ω∗−Ω− = 1 ,

SS∗ = Ω∗+Ω−Ω∗−Ω+ = Ω∗+PΩ+ = Ω∗+Ω+ = 1 ,

which shows that S : H→ H is unitary.

Among further properties of the scattering operator S : Hac(H0) → H are the energy conser-vation in a scattering situation and the fact that S inherits the symmetries of H0 and H:

1. e−itH0Sψ = Se−itH0ψ for any ψ ∈ Hac(H0) and any t ∈ R.

2. If U : H→ U is a unitary operator which leaves invariant the dynamics, i.e.

[e−itH0 , U ] = [e−itH , U ] = 0

for all t ∈ R, then USψ = SUψ for any ψ ∈ Hac(H0).

Proof. (This is left as an exercise.)

3.2 Cook’s criterion

The aim in this section is to present one criterion due to Cook for the existence of the wave opera-tors. We will supply a proof a Cook estimate in case of short-range perturbations of the Laplacianbelow.

Theorem 3.3 (Cook’s criterion). Let D ⊂ D(H0) suppose there is some T ≥ 0 such that for allψ ∈ D:

1. e−itH0ψ ∈ D(H) for all |t| ≥ T , and

2.∫∞T ‖(H −H0)e∓itH0ψ‖ dt <∞.

Then D ⊂ D(Ω±). Moreover, for all ψ ∈ D one has the estimate:

‖(Ω± − 1)ψ‖ ≤∫ ∞T‖(H −H0)e∓itH0ψ‖dt .


Proof. We will give the poof for + (the case − is similar). Let ψ(t) := eitHe−itH0ψ with ψ ∈ D.Since e−itH0ψ ∈ D(H)∩D(H0) for |t| ≥ T , the function t 7→ ψ(t) is (strongly) differentiable for|t| > T and one has

d

dtψ(t) = i eitH(H −H0)e−itH0ψ ,

and hence T ≤ s < t:

ψ(t)− ψ(s) =

∫ t

seiτH(H −H0)e−iτH0ψ dτ . (3.2)

The triangle inequality implies:

‖ψ(t)− ψ(s)‖ ≤∫ t

s‖eiτH(H −H0)e−iτH0ψ‖dτ .

By assumption, the right side goes to zero as |s| → ∞ such that ψ(t) is a Cauchy sequence in H.Therefore, the limits limt→±∞ ψ(t) exist, i.e. ψ ∈ D(Ω±).

3.3 Asymptotic completeness

In the standard case in which H = Hac(H0) and if Ω± exist, one hasD(Ω±) = H. The intertwiningrelation then shows that the operator e−itH restricted to ran Ω± is unitarily equivalent to e−itH0 .This implies that the asymtotically free states are a subset of the scattering states:

ran Ω± ⊂ Hac(H) . (3.3)

It is physically reasonable to expect also the converse, i.e., any scattering state evolves asymptoti-cally free. This is the requirement of asymptotic completeness.

Definition 3.4. We say that Ω± are asymptotically complete if ran Ω± = Hc(H).

1. In the standard situation and if Ω± exists, then asymptotic completeness implies ran Ω± =Hac(H) and in particular Hsc(H) = 0.

2. Asymptotic completeness in particular guarantees the unitarity of the scattering operator.

3.4 Existence and asymptotic completeness for short-range perturba-tions

The main aim of this section is to establish existence and asymptotic completeness in the standardsituation in which H = H0 + V , where:

1. H0 = −∆ on D(−∆) = H2(Rd), and

3.4. EXISTENCE AND ASYMPTOTIC COMPLETENESS FOR SHORT-RANGE PERTURBATIONS33

2. V is a relatively bounded perturbation of −∆ with bound a < 1 which is short-range in thefollowing sense.

Definition 3.5. A relatively bounded perturbation V of the Laplacian −∆ : H2(Rd)→ L2(Rd) iscalled short-range if ∫ ∞

0‖V (−∆ + 1)−1 1Rd\Br ‖ dr <∞ .

Here Br := x ∈ Rd | |x| < r.

We will collect some properties of short range perturbations below.

The main result of this section is Enss’ famous result.

Theorem 3.4 (Enss). Let V be relatively bounded perturbation of −∆ with bound a < 1 which isshort-range. Then:

1. the wave operators Ω± exist and are asymptotically complete.

2. σsc(H) = ∅.

For a proof of this theorem we proceed in five steps:

Step 0 (Properties of short-range perturbations). Before we start with the main part of theproof of Theorem 3.4, it is useful to collect a few properties of short range perturbations. Thefollowing lemma is proven as a homework. As is shown there, it provides a convenient characteri-zation of short range perturbations.

Lemma 3.5. Let V be relatively bounded perturbation of −∆ with bound a < 1. Then V is shortrange if and only if ∫ ∞

0‖V (1− χR) (H0 + 1)−1‖ dr <∞

for some χR ∈ C∞0 ((0,∞), [0, 1]) with χR(x) = 1 for |x| ≤ R/2 and χR(x) = 0 for |x| ≥ R.

One consequence is that any potential with |V (x)| ≤ C(1 + |x|)−1−ε with some ε > 0, is ashort-range perturbation.

The following properties of short-range potentials will be used in the proof of Theorem 3.4.

Lemma 3.6. Let V be relatively bounded perturbation of H0 = −∆ with bound a < 1 which isshort-range and H := −∆ + V . Then:

1. (H − z)−1 − (H0 − z)−1 is compact for any z ∈ %(H) ∩ %(H0)

2. f(H)− f(H0) is compact for any f ∈ C∞(R).


Proof. By the Stone-Weierstrass theorem (cf. Tutorium), it suffices to prove the first assertion.The resolvent identity yields

(H0 − z)−1 − (H − z)−1 = (H − z)−1V (H0 − z)−1

= (H − z)−1V χR(H0 − z)−1 + (H − z)−1V (1− χR)(H0 − z)−1 .

The first term is a compact operator for every R > 0 by Lemma 2.6. Lemma 3.5 and the resolventidentity shows that for all z ∈ %(H0):∫ ∞

0‖V (1− χR) (H0 − z)−1‖ dr <∞ .

This shows that for some subsequence ‖(H − z)−1V (1− χR)(H0 − z)−1‖ → 0 as R → 0. Thisimplies the first assertion.

Step 1 (Perry’s estimate for the free evolution). It is intuitive to expect that the incomingand outgoing states, i.e., the scattering states of H0 = −∆, are characterized by a decreasingrespectively increasing second moment. Since

d

dt

∫|x|2 |e−itH0ψ(x)|2dt = 4 (ψ(t), Dψ(t)) ,

where D is the dilation operator acting as

Dψ(x) := x · ∇ψ(x)− id2ψ(x) ,

we expect that the subspaces corresponding to the spectral projections

P± := PD((0,±∞))

are associated with the outgoing respectively incoming states. In this context, recall from theexcercise sheet that:

1. D is the self-adjoint generator of the dilation group.

2. D is diagonalized, i.e.Me−isD = e−is(·)M , s ∈ R ,

using the Mellin transformationM : L2(Rd)→ L2(R× Sd−1) which is given by

Mψ(λ, ω) := (2π)−1/2

∫ ∞0

r−iλψ(r, ω) rd/2−1dr .


3. σ(D) = σac(D) = R and σsc(D) = σpp(D) = ∅.

4. The Fourier transform (·) anticommutes with D, i.e., for all ψ ∈ S(Rd)

Dψ = −Dψ .

Perry’s estimate is a propagation estimate for the free time evolution. Roughly speaking, itstates that for t > 0 outgoing states with energy bounded from below by v2 leave the ball B2vt

with radius 2vt. Analogous statements apply for incoming states.

Theorem 3.7 (Perry’ estimate). Suppose f ∈ C∞0 (R) with supp f ⊂ [v20, v

21] for some v0, v1 > 0

and pick v ∈ (0, v0). Then for any N ∈ N and R ∈ R there exists C > 0 such that for ±t > 0:

‖ 1B2v|t| e−itH0 f(H0)PD((±R,±∞))‖ ≤ C

(1 + |t|)N.

Proof. We will give the proof only for + (the − case is similar). Abbreviating

Kt,x(p) :=ei(tp

2−p·x)

(2π)d/2f(p2) , P±R := PD((±R,±∞))

we write for ψ ∈ S(Rd) using the Fourier transform:

ψ(t, x) := e−itH0 f(H0)P+R ψ(x)

= (Kt,x, P+R ψ) = (Kt,x, P

−R ψ) .

Hence ‖ψ(t)‖ ≤ ‖P−RKt,x‖‖ψ‖. It hence suffices to proof that for for t > 0 and |x| < 2vt:

‖P−RKt,x‖ ≤C

(1 + t)N.

Invoking the Mellin transformation und using its unitarity, we write

‖P−RKt,x‖2 =

∫ −R−∞

∫Sd−1

|MKt,x(λ, ω)|2 dλdω

withMKt,x(λ, ω) =

1

(2π)d/2+1

∫ ∞0

eiαt(r) f(r2) rd/2−1dr (3.4)

and αt,λ(r) := tr2 + rω · x− λ log r. Since λ < 0, t > 0 and |x| < 2vt:

α′t,λ(r) = 2tr − ω · x− λ

r≥ 2tr − ω · x ≥ 2t(r − v) > 0

for all r ∈ [v0, v1]. We hence find some C > 0 such that for all r ∈ r|f(r2) 6= 0 and t > 0,λ < 0:

|α′t,λ(r)| ≥ C(1 + t+ |λ|) .The integral in can hence be estimated with the method of stationary phase:


1. Write eiαt,λ(r) =(−i α′t,λ(r)−1 d

dr

)Neiαt(r),

2. Integrate by parts N times:

(2π)d/2+1 |MKt,x(λ, ω)| =∣∣∣∣∫ ∞

0eiαt,λ(r) f(r) rd/2−1dr

∣∣∣∣=

∣∣∣∣∣∫ ∞

0eiαt,λ(r)

[d

dr

(α′t,λ(r)−1 d

dr

)N−1 (α′t,λ(r)−1f(r2) rd/2−1

)]dr

∣∣∣∣∣This yields:

|MKt,x(λ, ω)| ≤ C

(1 + |t|+ |λ|)N,

which completes the proof.

As a first consequence, we note

Corollary 3.8. Let f ∈ C∞0 ((0,∞)). Then for all ψ ∈ L2(Rd):

limt→∓∞

∥∥P±f(H0) e−itH0ψ∥∥ = 0 .

Proof. We estimate∥∥P±f(H0) e−itH0ψ∥∥

≤∥∥∥P±f(H0) e−itH0 1B2v|t|

∥∥∥ ‖ψ‖+∥∥P±f(H0) e−itH0

∥∥∥∥∥(1− 1B2v|t|)ψ∥∥∥

≤∥∥∥1B2v|t| e

itH0f(H0)∗P±

∥∥∥ ‖ψ‖+ ‖f‖∞∥∥∥(1− 1B2v|t|)ψ

∥∥∥ .The first term converges to zero as t → ∓∞ by Perry’s estimate. The second term convergens tozero in this limit by the dominated convergence theorem.

Step 2 (Cook’s estimate). We will now prove the validity of a Cook estimate for short-rangepotentials on the set

D := ψ ∈ L2(Rd) | ∃ϕ ∈ L2(Rd), f ∈ C∞0 (R) , supp f ⊂ [v20, v

21] for some v0, v1 > 0 , R ∈ R

s.t. ψ = f(H0)PD((±R,±∞))ϕ ⊂ H2(Rd) ,

which is dense in L2(Rd). By Theorem 3.3 this implies the existence of the wave operators.

Corollary 3.9. In the situation of Theorem 3.4 , we have∫ ∞0‖V e∓itH0ψ‖ dt <∞

for all ψ ∈ D. [Hence the wave operators Ω± : L2(Rd)→ L2(Rd) exist.]


Proof. We estimate for all ψ ∈ D:

‖V e∓itH0ψ‖ ≤ ‖V (H0 + 1)−1‖ ‖ 1B2vt e∓itH0(H0 + 1)ψ‖

+ ‖V (H0 + 1)−1 1Rd\B2vt‖ ‖(H0 + 1)ψ‖ . (3.5)

The first term is bounded using Perry’s theorem (Theorem 3.7):

‖ 1B2vt e∓itH0(H0 + 1)ψ‖ ≤ C

(1 + |t|)2‖ϕ‖ (3.6)

where ψ = f(H0)PD((±R,±∞))ϕ. Hence, this term is integrable. The second term is integrableby the short-range assumption.

Corollary 3.10. Let f ∈ C∞0 ((0,∞)). Then (Ω± − 1)f(H0)P± is compact.

Proof. We will give the proof only for + (the − case is similar). We first write

f(x) =: (x+ i)−1g(x)

with g ∈ C∞0 ((0,∞)). The intertwing property yields:

(Ω± − 1)f(H0) = (H + i)−1(Ω± − 1)g(H0) +((H + i)−1 − (H0 + i)−1

)g(H0) .

From Lemma 3.6 we infer that the second term is a compact operator. To show that the first termalso gives rise to a compact operator we pick (ψn) with (ϕ,ψn) → 0 for all ϕ ∈ L2(Rd). Thenusing (3.2):

‖(H + i)−1(Ω+ − 1)g(H0)P+ψn‖ ≤∫ ∞

0‖(H + i)−1V e−itH0g(H0)P+ψn‖ dt

The integrand converges to zero as n→∞, since

(H + i)−1V (H0 + i)−1 = (H + i)−1 − (H0 + i)−1

is compact by Lemma 3.6 and (H0 + i) e−itH0g(H0)P+ is bounded. Estimating the integrand asin (3.5) and (3.6), one proves that the integrand is an integrable function which is independent ofn.

Step 3 (Absence of sc spectrum). In the standard situation discussed in Theorem 3.4 the exis-tence of the wave operators Ω± : L2(Rd)→ L2(Rd) implies (3.3). Hence, Hsc(H) ⊂ (ran Ω±)⊥.

Let P := Psc PH((x0, x1)) stand for the projection onto the singular continuous part of thespectrum in the interval (x0, x1) ⊂ R\0. Choosing f(x) = 1 for x ∈ [x0, x1] we conclude that

P = Pf(H) = Pf(H0) + P (f(H)− f(H0)) .


Since f(H) − f(H0) is compact by Lemma 3.6, the second term is compact. In case supp f ⊂(−∞, 0) we conclude that P is compact, since then f(H0) = 0. Otherwise, if supp f ⊂ (0,∞)we use PΩ± = 0 and write

Pf(H0) = P (1− Ω+)f(H0)P+ + P (1− Ω−)f(H0)P− .

Both operators in the right side are compact by Corollary 3.10. Hence P is compact so that PHis finite-dimensional. This implies that σsc(H) ∩ (x0, x1) is a finite set. But a singular measurecannot be supported on a finite set. We therefore conclude that σsc(H) = ∅.

Using essentially the same argument as above, we also conclude that σac(H) ⊂ [0,∞).

Step 4 (Asymptotic completeness). Thanks to (3.3) and the fact that Hsc(H) = 0, a proofof asymptotic completeness requires to establish that

Hac(H) ⊂ ran Ω±.

Following are essential inputs to the proof:

1. By the Riemann-Lebesgue lemma, for any ψ ∈ Hac(H) the sequence ψ(t) := e−itHψconverges weakly to zero as t→ ±∞.

2. It suffices to prove that ψ ∈ ran Ω± for all functions of the form ψ = f(H)ψ with f ∈C∞0 ((0,∞)).

We now decompose

ψ(t) = f(H0)P+ψ(t) + f(H0)P−ψ(t) + (f(H)− f(H0))ψ(t)

= Ω+f(H0)P+ψ(t) + Ω−f(H0)P−ψ(t)

+ (1− Ω+)f(H0)P+ψ(t) + (1− Ω−)f(H0)P−ψ(t) + (f(H)− f(H0))ψ(t) .

Since f(H) − f(H0) is compact by Lemma 3.6 and so is (1 − Ω±)f(H0)P± by Corollary 3.10,we hence conclude that

limt→±∞

‖ψ(t)− Ω+f(H0)P+ψ(t)− Ω−f(H0)P−ψ(t)‖ = 0 .

As a consequence, the norm of the wave packet, which stays invariant under the unitary timeevolution can be written as a sum of two components:

‖ψ‖2 = limt→±∞

‖ψ(t)‖2

= limt→±∞

(ψ(t) , Ω+f(H0)P+ψ(t) + Ω−f(H0)P−ψ(t))

= limt→±∞

(ψ , Ω+eitH0f(H0)P+ψ(t)) + lim

t→±∞(ψ , Ω−e

itH0f(H0)P−ψ(t)) .

3.5. OUTLOOK: STATIONARY SCATTERING THEORY 39

Here the last equality relies on the interwining relation. Suppose now that ψ ∈ (ran Ω±)⊥. Then

‖ψ‖2 = limt→±∞

(ψ , Ω∓eitH0f(H0)P∓ψ(t))

= limt→±∞

(P∓ f(H0)∗e−itH0Ω∗∓ψ , ψ(t)) = 0

by Corollary 3.8. This concludes the proof.

3.5 Outlook:Stationary scattering theory

The main aim in this section is to formally derive the Lippmann-Schwinger equation which relatesthe generalized eigenfunctions Ω±ψp of theH = H0+V in terms of the generalized eigenfunctionsψp of H0. The starting point here is the following observation.

Lemma 3.11 (Abel). Let f ∈ L∞((0,∞)) and suppose that limt→∞ f(t) =: f∞. Then

limε↓0

ε

∫ ∞0

e−εtf(t) dt = f∞ .

Proof. (Exercise.)

In case the wave operators Ω± : H→ H exist, then Abel’s lemma implies that

(ϕ , Ω±ψ) = limt→±∞

(ϕ , eitH e−itH0ψ) = limε↓0

ε

∫ ∞0

e−εt (ϕ , e±itH e∓itH0ψ) dt

for all ϕ,ψ ∈ H.Suppose we could now take ψp(x) = eip·x, i.e., a generalized eigenfunction of H0 = −∆ with

momentum p ∈ Rd. Then we should have:

Ω±ψp = limε↓0

ε

∫ ∞0

e−εt e±itHe∓itp2ψpdt = lim

ε↓0

±iεH − p2 ± iε

ψp

= ψp − limε↓0

1

H0 − p2 ± iεV

±iεH − p2 ± iε

ψp = ψp −1

H0 − p2 ± iεV Ω±ψp .

In other words, the generalized eigenfunctions ϕ± := Ω±ψp of H with energy p2 should satisfythe Lippmann-Schwinger equation

Ω±ψp = ψp − limε↓0

1

H0 − p2 ± iεV Ω±ψp .

In case d = 3 and V is a multiplication operator, this equations takes the following explicit formof an integral equation:

ϕ±(x) = eip·x −∫R3

e∓i|p||x−y|

4π|x− y|V (y)ϕ±(y) dy .

The right side represents the unscattered wave plus a superpositions of spherical waves (centeredat y).


Chapter 4

Methods for bound states

In this chapter, we will discuss methods for addressing the bound states of self-adjoint operators. Ina previous chapter, the notion of bound states was identified with the eigenstates corresponding tothe pure point spectrum of the operator. We have also remarked that there are Schrodinger operatorswith dense pure point spectrum. However, in most examples such as

H = −∆− V in L2(Rd),

with a (short range) negative potential V ≥ 0, the low lying eigenvalues in the pure point spec-trum are isolated and possibly only accumulate towards the continuous spectrum. It is thereforeconvenient to introduce another decomposition of the spectrum of a self-adjoint operator.

Definition 4.1. Let A : D(A) 7→ H be a self-adjoint operator.

1. The essential spectrum σess(A) is the set of λ ∈ R for which the spectral projectionPA((−ε, ε)) is infinite dimensional for all ε > 0.

2. The discrete spectrum is σdisc(A) := σ(A)\σess(A).

Some easy facts:

1. σc(A) ⊂ σess(A) and σdisc(A) ⊂ σpp(A).

2. If λ is an accumulation point of a sequence of eigenvalues of A, then λ ∈ σess(A). Also, ifλ is an eigenvalue of infinite multiplicity of A, then λ ∈ σess(A).

4.1 Weyl’s theorem and HVZ

The goal of this section is to characterize the essential spectrum. We first note that the essentialspectrum is stable under compact perturbations.

41

42 CHAPTER 4. METHODS FOR BOUND STATES

Lemma 4.1. Let A and B be bounded, self-adjoint operators on a Hilbert space H and A−B becompact. Then σess(A) = σess(B).

Proof. By symmetry, it is enough to show that σess(A) ⊂ σess(B). For λ ∈ σess(A) ⊂ σ(A) wepick a Weyl sequence (ψn) of orthonormal vectors in H such that limn→∞(A − λ)ψn = 0. Since(ψn) converges weakly to zero one and B − A is compact, one has limn→∞(B − A)ψn = 0 andhence limn→∞(B − λ)ψn = limn→∞(A− λ)ψn = 0.

Theorem 4.2 (Weyl characterization of the essential spectrum). Let A : D(A) → H be a self-adjoint operator, z ∈ %(A) and λ 6= z. Then λ ∈ σess(A) if and only if there is a weaklyconvergent sequence ψn 0 in H, which does not strongly go to zero, such that

limn→∞

((A− z)−1 − (λ− z)−1

)ψn = 0 .

Proof. Assume that λ ∈ σess(A) ⊂ σ(A). Then there is a sequence (ψn) with ψn ∈ PA((λ −1n , λ + 1

n))H which may be chosen to be orthonormal and hence weakly converges to zero, suchthat∥∥((A− z)−1 − (λ− z)−1

)ψn∥∥2

=

∫(λ− 1

n,λ+ 1

n)

∣∣∣∣ 1

E − z− 1

λ− z

∣∣∣∣2 µψn(dE) ≤ C

n‖ψn‖2

for some C <∞.Conversely, let λ 6∈ σess(A), i.e., dim ran PA((λ− ε, λ+ ε)) <∞ for some ε > 0. If (ψn) is

a weakly convergent null sequence, then

limn→∞

PA((λ− ε, λ+ ε))ψn = 0 .

Corollary 4.3 (Weyl’s theorem). Let A1, A2 be self-adjoint on H and z ∈ %(A1)∩%(A2) such that(A1 − z)−1 − (A2 − z)−1 compact. Then σess(A1) = σess(A2).

Proof. Pick λ ∈ σess(A1). Then there is ψn 0, which does not strongly go to zero, such thatlimn→∞

((A1 − z)−1 − (λ− z)−1

)ψn = 0. By compactness

limn→∞

((A1 − z)−1 − (A2 − z)−1

)ψn = 0 ,

and hence limn→∞((A2 − z)−1 − (λ− z)−1

)ψn = 0. Hence λ ∈ σess(A2).

As an application, consider a relatively compact potential V with respect to−∆. Then σess(−∆+V ) = σess(−∆) = [0,∞].

The proof of the next theorem, which was given independently by W. Hunziger, C. van Winterand G. Zhislin, is also based on the Weyl’s theorem. It essentially corresponds to the physicalplausible statement that the lowest energy in the essential spectrum of an atom with central chargeZ andN electrons arises by peeling off one electron, which is thought off as being send off infinity.

4.1. WEYL’S THEOREM AND HVZ 43

Theorem 4.4 (HVZ). Consider the N -particle atomic Schrodinger operator

HN,Z =N∑n=1

(−∆n − Z|xn|−1

)+

∑1≤n<m≤N

|xn − xm|−1 ,

in L2(R3N ). Then σess(HN,Z) = [Σ,∞) where Σ := inf σ(HN−1,Z).

Proof. (To be inserted)

The proof of the HVZ theorem relied on a localization technique due to Martin and I. Sigal.

Lemma 4.5 (IMS localization). Consider non-negative J1, . . . Jk ∈ C∞(Rd) with J21 + · · ·+J2

k =1. Then

−∆ =

k∑j=1

Jj(−∆)Jj −k∑j=1

|∇Jj |2 .

Proof. (Straighforward calculation.)

Let us close this section with a remark.Hamiltonians such HN,Z describe the electronic degrees of freedom of an atom. In case the

quantum mechanical nature of the nuclei are also taken into account, one ends up with operators inL2(R3N ) of the form:

H =

N∑j=1

−∆j

mj+

∑1≤j<k≤N

Vjk(xj − xk) ,

where mj > 0 is the mass of the jth particle and Vjkdescribes the interaction of particle j withparticle k. Evidently, such operators commute with translations of the center of mass:

Taψ(x1, . . . , xN ) := ψ(x1 − a, . . . , xN − a).

Therefore, the operator has no discrete spectrum. Physically however, there exits bound statessuch as the electrons in the atom. To determine them, one needs to fix the center of mass R :=(∑n

j=1mj)−1∑N

j=1mjxj and consider the operator in that coordinate system. Since the nucleiare much heavier than the electrons, their kinetic energies are usually neglected and one ends upwith an operator of the form HN,Z for the remaining degrees of freedom.


4.2 Variational principle

The main aim of this section is to discuss the various form of the variational principle for theeigenvalues of self-adjoint operators A : D(A)→ H which are bounded from below. We start withthe min-max principle by E. Fischer and R. Courant. For its formulation we denote by

E1(A) ≤ E2(A) ≤ · · · ≤ En(A) ≤ . . .

the eigenvalues of A (taking into account multiplicity by repeating the eigenvalue) below the es-sential spectrum σess(A) with the convention that EN (A) = inf σess(A) in case the number ofthese eigenvalues is N ∈ N0.

Theorem 4.6 (Min-max principle). Let A : D(A) → H be a self-adjoint operator which isbounded from below. Then

E1(A) = infϕ∈D(A), ‖ϕ‖=1

(ϕ,Aϕ)

En(A) = supψ1,...,ψn−1∈H

infϕ∈D(A), ‖ϕ‖=1

ϕ⊥spanψ1,...,ψn−1

(ϕ,Aϕ) , n ≥ 2 . (4.1)

Proof. Without loss of generality we assume that there are n− 1 eigenvalues E1, . . . , En−1 belowEn and we denote by ϕ1, . . . , ϕn−1 the corresponding orthonormal eigenfunctions.

Then for any ϕ ⊥ spanϕ1, . . . , ϕn−1 with ϕ ∈ D(A), one has ϕ = PA([En,∞))ϕ suchthat by the spectral representation

(ϕ ,Aϕ) = (ϕ , APA([En,∞))ϕ) =

∫ ∞En

λµϕ(dλ) ≥ En ‖ϕ‖2 .

Pickingϕ1, . . . , ϕn−1 for ψ1, . . . , ψn−1 in the supremum in (4.1) shows that the left side is boundedfrom above by the right side.

To establish the converse inequality, we pick for ϕn ⊥ spanϕ1, . . . , ϕn−1 either a nor-malized eigenfunction corresponding to En or, in case the latter is the bottom of the esssentialspectrum, an approximate normalized eigenfunction, i.e. ‖(A − En)ϕn‖ < ε, cf. Theorem 4.2.Then any normalized linear combination, ϕ =

∑nj=1 αjϕj obeys:

(ϕ ,Aϕ) ≤n∑j=1

|αj |2Ej + ε ≤ En + ε .

Since at least one of the ϕj’s in belongs to spanψ1, . . . , ψn−1⊥ and ε can be chosen arbitrarilysmall, this completes the proof.

The proof shows that the infima and suprema are attained as long as En(A) is an eigenvalue.The special case n = 0 is called the Rayleigh-Ritz principle. It is the most important method forapproximating the ground state.

Theorem 4.6 implies that the eigenvalues and the counting function of a self-adjoint operatoris monotone. Namely, if A ≤ B then

4.2. VARIATIONAL PRINCIPLE 45

1. En(A) ≤ En(B) for all n, and

2. for NA(E) := tr PA((−∞, E)) with E < inf σess(A) one has

NB(E) ≤ NA(E) . (4.2)

More generally, any sum of a monotone function of the eigenvalues is monotone in the operator.

To estimate the ground-state energy of free Fermions it is also of interest to approximate thesums of eigenvalues. Here the following versions of the variational principle is of help.

Corollary 4.7. Let A : D(A) → H be self-adjoint and bounded below and pick n ∈ N withEn(A) < inf σess(A). Then

infP∈Pn(A)

tr AP =n∑j=1

Ej(A)

where Pn(A) := P ∈ LinH |P orthogonal projection with dim ran P = n , ran P ⊂ D(A).

Proof. Let ϕ1, . . . ϕn stand for the orthonormal eigenvalues corresponding to the eigenvaluesE1 ≤. . . En. Since tr APA([E1, En]) =

∑nj=1Ej , it remains to prove that tr AP ≥

∑nj=1Ej for any

P ∈ Pn(A). The is done by induction on n.In case n = 1 one has Pψ = (χ, ψ)χ for some χ ∈ D(A) and hence tr AP = (χ,Aχ) ≥ E1

by the Rayleigh-Ritz principle.For the induction step, suppose that infQ∈Pn(A) tr AQ ≥

∑nj=1Ej and considerP ∈ Pn+1(A).

There is some χ ∈ ran P with ‖χ‖ = 1 such that χ ∈ spanϕ1, . . . , ϕn⊥. Let Qψ :=Pψ − (χ, ψ)χ. Then using the induction hypothesis and the min-max principle:

tr AP = tr AQ+ (χ,Aχ) ≥n∑j=1

Ej + (χ,Aχ) ≥n+1∑j=1

Ej ,

which completes the proof.

As we will see, the one-body properties of systems of many Fermions are described by one-body reduced particle density operators for Fermions, i.e., trace-class operators γ ∈ J1(H) withthe properties

1. 0 ≤ γ ≤ 1, and

2. tr γ = n.

As will be explained below, the first property is a consequence of the Pauli exclusion principle.Particular examples (which describe the reduced state of n non-interacting Fermions at zero tem-perature) are orthogonal projections P ∈ Pn(A). In this context, the above variational principlefor sums of eigenvalues has the following generalization due to E. Lieb.


Corollary 4.8. In the situation of Corollary 4.7 asssume that A ≥ −α for some α ∈ R. Then

infγ∈Dn(A)

tr γ1/2(A+ α)γ1/2 =n∑j=1

(Ej(A) + α)

where Dn(A) := γ ∈ J1(H) | 0 ≤ γ ≤ 1 , tr γ = n , ran γ1/2 ⊂ D(A).

The trace in the left side is always well-defined (possibly equal to ∞) since the operatorγ1/2(A + α)γ1/2 is non-negative. In case the trace is finite, cyclic invariance yields tr γ1/2(A +α)γ1/2 = tr Aγ + nα.

Proof of Corollary 4.8. Let ϕ1, . . . ϕn stand for the orthonormal eigenvalues corresponding to theeigenvaluesE1 ≤ . . . En. Since tr APA([E1, En]) =

∑nj=1Ej , it remains to prove that tr γ1/2(A+

α)γ1/2 ≥∑n

j=1Ej for any γ ∈ Dn(A).We use the spectral decomposition for self-adjoint γ ∈ J1(H). Let 1 ≥ γ1 ≥ γ2 ≥ · · · ≥

γn ≥ · · · > 0 denote the positive eigenvalues of γ and φ1, φ2, · · · ∈ D(A) the correspondingorthonormal eigenfunctions. For ε ∈ (0, 1) we pick Nε ∈ N such that Γε :=

∑Nεj=1 γj ≥ n − ε.

Then

tr γ1/2Aγ1/2 ≥Nε∑j=1

γj(φj , Aφj)

= γNε

Nε∑j=1

(φj , Aφj) + (γNε−1 − γNε)Nε−1∑j=1

(φj , Aφj) + · · ·+ (γ1 − γ2)(φ1, Aφ1)

≥ γNεNε∑j=1

Ej + (γNε−1 − γNε)Nε−1∑j=1

Ej + · · ·+ (γ1 − γ2)E1

=

Nε∑j=1

γj Ej ≥n−1∑j=1

Ej + (Γε − (n− 1))En =

n∑j=1

Ej − εEn

Here the second inequality is a consequence of Corollary 4.7. The last inequality follows from thefact that the minimum of

∑Nεj=1 γj Ej over all 1 ≥ γ1 ≥ · · · ≥ γNε > 0 with the property that

Γε =∑Nε

j=1 γj > n− 1 is attained when ’filling the eigenvalues from the bottom’. This completesthe proof, since ε ∈ (0, 1) can be chosen arbitrarily small.

4.3 Riesz projections

The main aim of this short section is a useful representation of eigenprojections for self-adjointoperators. For future use, we will investigate the so-called Riesz projections

P :=−1

2πi

∮C

(A− z)−1dz (4.3)

4.3. RIESZ PROJECTIONS 47

in the more general set-up of closed operators A : D(A) 7→ H.In this context, it is useful to recall that the resolvent of a closed operator A

%(A) 3 z 7→ (A− z)−1 =: RA(z)

is an analytic function with values in the bounded operators on the Hilbert space, i.e., the normlimit limz→z0(z − z0)−1(RA(z)−RA(z0)) exists.

Theorem 4.9. Let A : D(A) 7→ H be closed operator and En be an isolated point of σ(A). Thenthere is a punctured open neighborhood U ⊂ %(A) of E such that the Riesz projection defined in(4.3) exists and is independent of the closed contour C ⊂ U as long as the latter encircles E. onlyonce Moreover:

1. P is a projection, i.e., P 2 = P ,

2. [P, (A− z)−1] = 0 for all z ∈ %(A).

3. ker(A− E) ⊂ ran P .

4. If A is self-adjoint, then P is the orthogonal projection onto ker(A− E).

Proof. Since %(A) 3 z 7→ (A− z)−1 is an analytic function, the integral is well defined (as normlimit of Riemann integrals) and its independence of the contour follows from the Cauchy integraltheorem.

1. We pick two circles Cr := z ∈ C | |z − E| = r and CR := z ∈ C | |z − E| = R with0 < r < R small enough. Then

P 2 =1

(2πi)2

∮CR

∮Cr

(A− z)−1(A− z′)−1dz dz′

=1

(2πi)2

∮CR

∮Cr

(z′ − z)−1[(A− z′)−1 − (A− z)−1

]dz dz′

=1

(2πi)2

[∮CR

(A− z′)−1

∮Cr(z′ − z)−1dz dz′ −

∮Cr

(A− z)−1

∮CR(z′ − z)−1dz′ dz

]= 0− 1

2πi

∮Cr

(A− z)−1dz = P .

2. The fact that [P, (A− z)−1] = 0 for all z ∈ %(A) follows by a similar computation.

3. We pick ψ ∈ ker(A− E) and use (A− z′)−1ψ = (E − z′)−1ψ for all z′ ∈ %(A):

Pψ =−1

2πi

∮Cr

(A− z′)−1ψ dz′ =−1

2πi

∮Cr

(E − z′)−1ψ dz′ = ψ .


4. The spectral theorem implies:

(ϕ, Pψ) =−1

2πi

∮Cr

∮(E − z)−1µϕ,ψ(dE) dz

=−1

2πi

∮Cr

(E − z)−1dz µϕ,ψ(E) +−1

2πi

∮|E−E|>r

∮Cr

(E − z)−1µϕ,ψ(dE) dz

=(ϕ ,PA(E)ψ)

for every ϕ,ψ ∈ H.

Several remarks apply:

1. The set of contours C in %(A) ∪ E which encircle E only will be referred as admissiblefor E.

2. We stress that the Riesz projections are in general not orthogonal and there range generallyonly includes the eigenspace ker(A− E).

3. More generally one may proof along the same lines that integrals of the form (4.3) withcontours C ⊂ %(A) are always projections even if the encircled spectrum is more or lesscomplicated.

4.4 Analytic perturbation theory

In this section, we will present a method to determine the isolated eigenvalues and eigenprojectionsof operators of the form

H(α) := A+ αB

perturbatively in terms of the one’s of A. The main example will be operators satisfying:

1. A : D(A)→ H is a self-adjoint operator on some domain D(A) ⊂ H,

2. B is A-bounded.

This guarantees thatA+αB is a closed operator on the domainD(A) for all α ∈ C. Moreover, forevery z ∈ %(A) there is α0 > 0 such that the von Neumann series

∑∞n=0(−α)n(B(A− z)−1)n is

norm convergent for all |α| < α0 and one has

(H(α)− z)−1 = (A− z)−1∑n=0

(−α)n(B(A− z)−1)n . (4.4)

The family of operators H(α) with α ∈ Bα0(0) is hence a special case of an analytic family in thesense of Kato.

4.4. ANALYTIC PERTURBATION THEORY 49

Definition 4.2. Let G ⊂ C be a connected open set. A family of operators T (α) with is calledanalytic in the sense of Kato if and only if for all α ⊂ G:

1. the operator T (α) is closed with non-empty resolvent set,

2. there is some z0 ∈ %(T (α)) such that for some ε > 0 the function Bε(α) 3 β 7→ (T (β) −z0)−1 is analytic.

Remark 4.10. A function α 7→ T (α) on an open subset of C with values in the bounded operatorsover some Hilbert space H is analytic if the norm limits limβ→α(β − α)−1(T (β) − T (α)) exist.It can be shown that this holds if and only if α 7→ (ϕ, T (α)ψ) is analytic for all ϕ,ψ ∈ H. (Thehard part of this statement is proven with the help of Cauchy’s integral representation for analyticfunctions and the uniform boundedness principle.)

The following somewhat technical lemma clarifies the analytic properties of the resolvents ofsuch families in the above sense.

Lemma 4.11. Suppose that T (α), with α ∈ G, is an analytic family in the sense of Kato. ThenΓ := (α, z) |α ∈ G , z ∈ %(T (α)) is an open set in C2 and Γ 3 (α, z) 7→ (T (α) − z)−1 isanalytic.

Proof. Excercise.

Our first result concerns the analyticity of the Riesz projections.

Lemma 4.12. Suppose that T (α), with α ∈ G, is an analytic family in the sense of Kato. Letα ∈ G and En be an isolated eigenvalue of T (α) and

Pn(α) :=−1

2πi

∮C(T (α)− z)−1dz

be the associated Riesz projection, where C is any (fixed) admissible contour forEn(α). Then thereis δ > 0 such that:

1. Bδ(α) 3 β 7→ Pn(β) is analytic, and

2. Bδ(α) 3 β 7→ dim ran Pn(β) is constant.

Proof. We first fix the domain of analyticity of Pn(β). For every z ∈ C ⊂ %(T (α)) one has (α, z)Γ(cf. Lemma 4.11) and hence there is some δ(z) > 0 such that (β, z′) ∈ Γ for all β ∈ Bδ(z)(α)and z′ ∈ Bδ(z)(z′). Since C is compact and and ∪z∈CBδ(z)(z) is an open covering of C, there isa finite set z1, . . . zm ∈ C such that C ⊂ ∪mj=1Bδ(zj)(zj). Take δ := minδ(z1), . . . , δ(zm).Then Bδ(α) × C ⊂ Γ. Hence by Lemma 4.11, the function Bδ(α) × ∪mj=1Bδ(zj)(zj) 3 (β, z) 7→(T (β)− z)−1 is analytic. From this we conclude the analyticity of Bδ(α) 3 β 7→ Pn(β).

The proof of the constancy of β 7→ dim ran Pn(β) in a neighborhood of α is by contradiction.Suppose dim ran Pn(β) < dim ran Pn(α). Then there is ψ ∈ kerPn(β)∩ ran Pn(α) with ‖ψ‖ =1 such that ‖(Pn(β) − Pn(α))ψ‖ = ‖ψ‖ = 1. This contradicts the fact that limβ→α Pn(β) =Pn(α). (In case dimPn(β) > dimPn(α) one argues similarly.)


Finally, we came to the analyticity of the eigenvalues of analytic families. For simplicity, wewill restrict ourselves to the case of a simple eigenvalue.

Theorem 4.13. In the setting of Lemma 4.12 suppose that additionally that ran Pn(α) = spanΩnfor some normalized vector Ωn. Then there is δ > 0 and an analytic functionBδ(α) 3 β 7→ En(β)such that for all β ∈ Bδ(α):

1. En(β) is an eigenvalue of T (β).

2. Pn(β)Ωn is a non-zero eigenvector of En(β).

Proof. From the proof of Lemma 4.12 we then know that there is δ > 0 such that

1. Bδ(En) 3 β 7→ Pn(β) = −(2πi)−1∮C(T (β)−z)−1dz is analytic with dim ran Pn(β) = 1,

2. C ⊂ %(T (β)) for all β ∈ Bδ(En).

To determine En(β) we note that Pn(β)Ωn 6= 0 in a sufficiently small neighboood of α sincelimβ→α Pn(β)Ωn = Ωn 6= 0. Moreover, by an explicit calculation:

T (β)Pn(β)Ωn = T (β)Pn(β)2Ωn = Pn(β)T (β)Pn(β)Ωn .

Since dim ran Pn(β) = 1 and hence ran Pn(β) = spanPn(β)Ωn, this proves that Pn(β)Ωn isa eigenvector of T (β). We will call the corresponding eigenvalue En(β).

To prove the analyticity of En(β) we note that for z ∈ %(T (β)):

(En(β)− z)−1 =(Ωn, (T (β)− z)−1Pn(β)Ωn)

(Ωn, Pn(β)Ωn).

The analyticity in β then follows from that of the resolvent and the Riesz projection.

The above theorem may be extended in various directions:

1. It is not hard to see that in the setting of the theorem En(β) is an isolated eigenvalue ofT (β) of multiplicity one. The proof requires to study T (β) in kerPn(β). Since Pn(β) is aprojection, the spaces G(β) := ran Pn(β) and F (β) := kerPn(β) span the Hilbert spaceand G(β) ∩ F (β) = 0. (They are not necessarily orthogonal.) We already kow thatG(β) is an eigenspace. The operator T (β)− z restricted to D(T (β)) ∩ F (β) (which by theclosedness of T (β) is dense in F (β)) is invertible for all z ∈ Br(En). Namely, on D(T (β))one finds that for all z ∈∈ Br(En)

(T (β)− z)Rz(β) = Rz(β)(T (β)− z) = 1− Pn(β) ,

with Rz(β) := (2πi)−1∮Cr(z

′ − z)−1(T (β) − z′)−1dz′. Hence Rz(β) is the inverse ofT (β)− z restricted to D(T (β)) ∩ F (β).

4.5. SEMICLASSICAL APPROXIMATION FOR SUMS OF EIGENVALUES 51

2. In case the eigenvalue En(α) has multiplicity m there are in general m not necessarily dis-tinct functions E(1)

n (β), . . . E(m)n (β) which are analytic on a sufficiently small neighborhood

of α and which are eigenvalues of T (β).

Let us discuss the consequences of the previous theorem in more detail in the setting describedabove, i.e. for H(α) = A+ αB in case En is an isolated eigenvalue of the self-adjoint operator Awith multiplicity one and normalized eigenvector Ωn Then

En(α) =(Ωn, H(α)Pn(α)Ωn)

(Ωn, Pn(α)Ωn)= En + α

(Ωn, B Pn(α)Ωn)

(Ωn, Pn(α)Ωn).

The Rayleigh-Schrodinger series is obtained from the resolvent expansion (4.4). Namely,

(Ωn, B Pn(α)Ωn) =∞∑k=0

akαk

with an :=(−1)k+1

2πi

∮Cr

(Ωn, (B(A− z)−1)k+1Ωn) dz

(Ωn, Pn(α)Ωn) =∞∑k=0

bkαk

with bk :=(−1)k+1

2πi

∮Cr

(Ωn, (A− z)−1 (B(A− z)−1)kΩn) dz ,

with r > 0 sufficiently small. This yields the known expansion for the eigenvalues, e.g., to firstorder En(α) = En + α (Ωn, BΩn).

4.5 Semiclassical approximation for sums of eigenvalues

Our aim is to prove the following:

Theorem 4.14. In the above setting assume that V ∈ C2(Rd), d ≥ 3 and ‖HessV (q)‖∞ < ∞.Moreover, suppose

H = −~2∆− V

is self-adjoint operator in L2(Rd) with domain D(H) = H2(Rd). Then:

tr H− =

∫(p2 − V (q))−

dpdq

hd+O(h−d−1)

where h− := minh, 0 denotes the negative part (which we chose to be negative.)


For the proof we introduce coherent states

fp,q(x) := eipx/~ g(x− q) (4.5)

labeled by (p, q) ∈ Rd×Rd and associated with g ∈ H1(Rd), ‖g‖ = 1, g ≥ 0. We will abbreviatethe orthogonal projection onto fp,q by Πp,q. These coherent states enjoy the following properties:

1. Completeness: ∫Rd×Rd

Πpqdpdq

hd= (ψ ,ϕ) . (4.6)

The integral is understood in the weak sense, i.e.,∫Rd×Rd(ψ, Πpq ϕ) dpdq

hd= (ψ ,ϕ) for all

ψ,ϕ ∈ L2(Rd). This follows by an explicit computation.

2. Suppose H : Rd × Rd → [α, β] with α < β. Then

H :=

∫Rd×Rd

H(p, q) Πpqdpdq

hd(4.7)

defines a bounded operator with α ≤ H ≤ β. This follows from (4.6).

3. Suppose that H ∈ L1(Rd × Rd). Then the operator H is trace class and

tr H =

∫Rd×Rd

H(p, q)dpdq

hd.

For a proof it is enough to suppose that H ≥ 0. Then H ≥ 0 and for any orthonormal basisψj , j ∈ N:

tr H =1

hd

∞∑j=1

∫Rd×Rd

H(p, q)(ψj H(p, q)ψj) dpdq

=1

hd

∫Rd×Rd

H(p, q)∞∑j=1

(ψj H(p, q)ψj) dpdq

=1

hd

∫Rd×Rd

H(p, q) dpdq . (4.8)

Here the first equality is the definition of the trace for a non-negative operator. The secondequality is Fubini’s theorem and the third equality is Parseval’s identity and uses ‖fp,q‖ = 1.

Proof of Theorem 4.14. (To be inserted)

4.6. LIEB-THIRRING INEQUALITY 53

4.6 Lieb-Thirring inequality

Theorem 4.15 (Lieb-Thirring inequality). There is some finite constant Ld > 0 such that for all0 ≥ V ∈ L1+d/2(Rd) the sum of negative eigenvalues of H = −∆− V satisfies:

tr H− ≥ −Ld∫V (x)1+d/2dx . (4.9)


1. The assumption V ∈ L1+d/2(Rd) in particular guarantees that the operator H = −∆ − Vmay be defined as self-adjoint operator via the KLMN theorem. Include in Notes!

2. The integral in the right side is proportional to the phase space integral∫ (p2 − V (q)

)− dpdq.

3. Extensions of the theorem concern more general Riesz means of eigenvalues, where for anyγ ≥ 1

2 in case d = 1, any γ > 0 in case d > 2 and any γ ≥ 0 in case d ≥ 3 there is somefinite constant Lγ,d > 0 such that∑

j

|Ej(H)|γ ≤ Lγ,d∫V (x)γ+d/2dx . (4.10)

The special case γ = 0 in the above inequality is known as Cwickel-Lieb-Rozenblum in-equality. For more information, see [3].

A proof of Theorem 4.15 rests on the so-called Birman-Schwinger principle which are goingto explain next.

To motivate this principle fix an energy E > 0 and consider an eigenfunction ψ of the operatorH = −∆− V corresponding to eigenvalue −E, i.e.

(−∆ + E)ψ(x) = V (x)ψ(x) . (4.11)

Defining ϕ(x) :=√V (x)ψ(x), we may hence rewrite the eigenfunction equation as

ϕ(x) = (KEϕ)(x) :=

∫KE(x, y)ϕ(y) dy , (4.12)

where the Birman-Schwinger operator KE is an integral operator with kernel given by

KE(x, y) :=√V (x)G(x− y;−E)

√V (y)

in terms of the the integral kernel (or Green’s function) of (−∆ + E)−1:

G(ξ;−E) :=

∫eikξ

k2 + E

dk

(2π)d.

Apriori it is not clear whether the above formal calculation is sound. For example, it is not evidentwhether KE is well defined. Let us therefore address some properties of the Birman-Schwingeroperator first;


1. In case 0 ≤ V ∈ Lm(Rd for some m > d/2 the Birman-Schwinger operator KE is a non-negative integral operator on L2(Rd) which is in the Schatten class Jm(L2(Rd)). We willproof this in case m = 2 where

‖KE‖22 = tr K∗EKE =

∫|KE(x, y)|2dxdy =

∫V (x)GE(x− y;−E)V (y) dxdy

≤∫V (x)2dx

∫|G(y;−E)|2 dy =

∫V (x)2dx

∫1

(k2 + E)2

dk

(2π)3. (4.13)

2. KE is monotone decreasing in E, i.e., KE ≤ KE′ if E′ ≤ E. Therefore eigenvaluesκ1(E) ≥ κ2(E) ≥ · · · ≥ 0 are monotone decreasing in E.

Lemma 4.16 (Birman-Schwinger principle). Assume that 0 ≤ V ∈ Lm(Rd) for some m > d/2.Then −E < 0 is an eigenvalue of H = −∆− V if and only if 1 is an eigenvalue of KE .

Proof. For any ψ ∈ H1(Rd) which satisfies the eigenfunction equation (4.11), the function ϕ :=√V ψ is square integrable with ϕ 6= 0 (since otherwise −∆ψ = −Eψ which is impossible for

a function in H1(Rd). Moreover ϕ satisfies (4.12). It is hence an eigenfunction of KE witheigenvalue one. Conversely, if ϕ ∈ L2(Rd) is an eigenfunction of KE with eigenvalue one, thenψ = (−∆ + E)−1

√V ϕ satisfies (−∆ + E)ψ =

√V ϕ =

√V KEϕ = V ψ.

The Birman-Schwinger principle implies that

N(E;V ) :=

∞∑j=1

1Ej(−∆−V )≤−E =

∞∑j=1

1κj(E)≥1 ≤∞∑j=1

κj(E)m = tr KmE .

We are now ready to give a proof of the Lieb-Thirring inequality for d = 3. For a proof formore general d, see [3].

Proof of Theorem 4.15 in case d = 3. By partial integration we write

tr H− =∑j

Ej(H) = −∫ ∞

0N(E;V ) dE .

In order to apply the Birman-Schwinger principle we first estimate

N(E;V ) = N(E/2;V − E/2) ≤ N(E;WE) , WE(x) := maxV (x)− E/2, 0 .

One WE ∈ L2(R3) and hence by the Birman-Schwinger principle

N(E;WE) ≤ tr K2E ≤

∫WE(x)2dx

∫1

(k2 + E/2)2

dk

(2π)3.


Since∫

(k2 + E/2)−2dk = C E−1/2, we conclude that

∫ ∞0

N(E;V ) dE ≤ C∫ (∫ 2V (x)

0(V (x)− E/2)2 dE√

E

)dx = C

∫V (x)5/2dx .

For an application of the Lieb-Thirring inequality in the context of N -particle quantum me-chanics, we consider the kinetic energy

Tψ := (ψ ,

N∑j=1

−∆jψ)

of any N -particle wave function ψ ∈ L2(Rd;Cq)N of space and spin. We will assume

1 = (ψ, ψ) =∑

σ∈1,...,qN

∫|ψ(x1, σ1, . . . , xN , σN )|2dx1 . . . dxN =

∫|ψ(z1, . . . , zN )|2dz1 . . . dzN

(If not expliciltly states we will not assume any further symmetry properties of this function.) Theone-body (spatial) density at x ∈ Rd is

%ψ(x) =

N∑j=1

∑σ∈1,...,qN

∫|ψ(x1, σ1, . . . , xj−1, σj−1, x, σj , xj+1, σj+1, . . . xN , σN )|2dx1 . . . dxj . . . dxN

where by dxj we indicate that the integration is over all variables but xj . More generally, wemay introduce the one-particle density matrix γψ corresponding to ψ which is defined through thekernel

γψ(z, z′) :=N∑j=1

∫ψ(z1, . . . , zj−1, z, zj+1, . . . , zN )ψ(z1, . . . , zj−1, z

′, zj+1, . . . , zN ) dz1 . . . dzj . . . dzN ,

as well as the spin-reduced one-particle density matrix γψ which is defined through

γψ(x, x′) :=

q∑σ=1

γψ(x, σ, x′, σ) .


1. γψ is a bounded operator with norm ‖γψ‖ ≤ N . This bound is saturated for simple productwave functions ψ(x1, . . . , xN ) =

∏Nj=1 φ(xj).


2. For any ψ ∈ ∧Nj=1L2(Rd;Cq) =: HN , i.e., an antisymmetric function of its coordinates, the

one-particle reduced density matrix γψ satisfies

γψ ≤ 1 ,

and as a consequence ‖γψ‖ ≤ q‖γψ‖ ≤ q.

Proof. We pick ϕ ∈ L2(Rd;Cq) and define the associated annihilation operator cN,ϕ :HN → HN−1 through

(cN,ϕψ)(z1, . . . , zN−1) :=√N

∫ψ(z1, . . . , zN )ϕ(zN ) dzN .

The corresponding creation operator cϕ : HN−1 → HN is

(c†N,ϕψ)(z1, . . . , zN ) :=√N

1√N !

∑π∈SN

signπ ψ(zπ(1), . . . , zπ(N−1))ϕ(zπ(N)) ,

where the summation is over the all permutations of 1, . . . , N . We identify

c†N,ϕcN,ϕ + cN+1,ϕc†N+1,ϕ = (ϕ ,ϕ) 1 .

to be proportional to the indentity on HN . As a consequence, we have

(ϕ , γψϕ) = (ψ, c†N,ϕcN,ϕψ) = (ϕ ,ϕ)(ψ, ψ)− (ψ, cN+1,ϕc†N+1,ϕψ)

= (ϕ ,ϕ)(ψ, ψ)− (c†N+1,ϕψ, c†N+1,ϕψ) ≤ (ϕ ,ϕ)(ψ, ψ) .

This completes the proof.

3. If we take q = 1 and ψ a determinantal function

ψ(x1, . . . , xN ) = (N !)−1/2 det φj(xk)Nj,k=1

we find ϕψ(x) =∑N

j=1 |φj(xj)|2 and ‖γψ‖ = 1.

We now come to following important corollary of the Lieb-Thirring inequality:

Theorem 4.17 (Kinetic energy inequality). There is some Cd > 0 such that for any normalizedψ ∈ L2(Rd;Cq)N one has:

Tψ ≥Cd

‖γψ‖2/d

∫%ψ(x)(d+2)/ddx . (4.14)


Proof. We consider the potentialV (x) = C%ψ(x)2/d

with a constant C > 0 over which we will optimize later, and the one-body operator H = −∆−Vin L2(Rd). The corresponding N -body operator is denoted by

H :=

N∑j=1

Hj

Since H is a sum of one-particle operators which do not depend on the spin variables one has:

Tψ − C∫%ψ(x)1+2/ddx = (ψ,Hψ) = tr H γψ (4.15)

where the trace extends over L2(Rd). By Lieb’s variational principle, Corollary 4.8, the trace(which extends over L2(Rd) is lower bounded by the sum of the negative eigenvalues of H times‖γψ‖. Hence by the Lieb-Thirring inequality

(ψ,Hψ) ≥ ‖γψ‖∑j

Ej(H) ≥ −‖γψ‖LdC1+d/2

∫%ψ(x)(d+2)/ddx . (4.16)

Combing this estimate with the first equality in (4.15) we obtain:

Tψ ≥(C − ‖γψ‖LdC1+d/2

)∫%ψ(x)(d+2)/ddx .

Optimizing, i.e, choosing

C =

(d+ 2

2‖γψ‖Ld

)−2/d

completes the proof.


Chapter 5

Many-particle methods

5.1 Density functional theories

Consider the operator of a huge atom

HN =N∑j=1

(−∆j −

Z

|xj |

)+

∑1≤<j<j′≤N

1

|xj − xj′ |

in L2(R3;Cq)N (or the antisymmetric subspace). Since this is a partial differential operator inN(d+ q) variables, it is almost impossible to determine its (spectral) properties even numerically.

Therefore the idea, which can be traced to the advent of quantum mechanics, was to describesuch systems in an effect way by means of a density functional. The relation of density functionaltheory to quantum mechanics is based on a theorem due to Hohenberg and Kohn. We will presentthe version due to Levy and Lieb.

Consider an operator of the above form in which (for the sake of generality) Z|x| replaced by an

arbitrary potential v (which is relatively compact with respect to −∆). Then

(ψ, HNψ) = (ψ ,N∑j=1

−∆j +∑

1≤<j<j′≤N

1

|xj − xj′ |ψ) +

∫v(x)%ψ(x)dx .

The functional

G(%) := inf

(ψ ,

N∑j=1

−∆j +∑

1≤<j<j′≤N

1

|xj − xj′ |ψ)∣∣ % = %ψ

,

in which the infimum is taken over all densities % which coincide with a one-body reduced density

59

60 CHAPTER 5. MANY-PARTICLE METHODS

has the property:

E1(HN ) := inf σ(HN ) = inf

(ψ, HNψ) |ψ ∈N∧j=1

H2(R3;Cq) , ‖ψ‖ = 1

= inf

G(%) +

∫v(x)%dx | % ∈ L1(R3), % ≥ 0,

∫%(x) dx = N

.

The functionalEHK(%) := G(%) +

∫v(x)%(x)dx

is known as the Hohenberg-Kohn functional.Unfortunately, the first term in this functional is not explicit. In practice, one therefore approx-

imates it. We will learn about one particular approximation, namely Thomas-Fermi theory, in thenext section.

An idea for going beyond density functional theory due to Gilbert is to study insetad a universalfunctional of the one-body reduced density matrix (in quantum chemistry abbreviated by 1RDM).Introducing

G(γ) = inf(ψ,∑

1≤j<j′≤N

1

|xj − xj′ |ψ) | γψ = γ ,

one may write

E0(HN ) = inf(ψ, HNψ) = inf

tr (−∆ + v)γ + G(γ) | 0 ≤ γ ≤ 1 , tr γ = N , ∆γ ∈ J1

.

The only non-explicit term is G(γ).In the special case of Slater determinats γψ(x, y) =

∑j ej(x)ek(y) this term is explicit

(ψ ,∑

1≤j<j′≤N

1

|xj − xj′ |ψ) =

1

2

∫γψ(x, x)γψ(y, y)− |γψ(x, y)|2

|x− y|dxdy .

The Coulomb energy is hence a sum of the direct energy and a negative exchange energy. Thelatter stems from the effective repulsion of Fermions caused by the Pauli principle. The aboveformula will be the basis of the Hartree-Fock approximation, which we will discuss below.

5.2. THOMAS-FERMI THEORY 61

5.2 Thomas-Fermi theory

Thomas-Fermi theory is the oldest density functional theory. It aims to describe the energy of Nelectrons in R3 which move among M nuclei located at positions R1, . . . , RM ∈ R3 with nuclearcharges Z1, . . . , ZM ≥ 0 through the Thomas-Fermi functional

E(%) = T (%)− V (%) +D(%) + U

where:

T (%) :=3

5γ

∫%(x)5/3dx , V (%) :=

∫ ∑1≤m≤M

Zm|x−Rm|

%(x) dx

D(%) :=1

2

∫%(x)%(y)

|x− y|dxdy , U :=

∑1≤m<m′≤M

ZmZm′

|Rm −Rm′ |

model the kinetic energy, the potential energy, the interaction energy among the electrons and theelectrostatic energy among the nuclei. The physical value for γ in case of Fermions with q spindegrees of freedom is (6π/q)2/3. (This is the exact value for the kinetic energy a a gas of freeFermions.)

Let ψ ∈∧Nj=1 L

2(R3;C2) stand for the normalized ground-state of the operator

HN =N∑j=1

−∆j −N∑j=1

M∑m=1

Zm|xj −Rm|

+∑

1≤j<j′≤N

1

|xj − xj′ |+ U ,

the hope of Thomas-Fermi theory is that

%ψ ≈ %

(ψ ,HNψ) ≈ E(%) = infE(%) |∫%(x)dx = N, % ≥ 0 .

The terms T (%) and D(%) in general only approximate the kinetic energy and interaction. Bythe kinetic energy inequality the former is lower bounded by T (%ψ). The interaction term D(%ψ)misses the effects of exchange correlations.

The natural domain of the Thomas-Fermi functional is

Dλ :=

% ∈ L1(R3) | % ≥ 0 , % ∈ L5/3(R3) ,

∫%(x)dx ≤ λ

,

where physically λ ≥ 0 stands for the number of electrons. The inequality accounts for the possi-bility that some of those electrons disappear to infinity.

Let us collect some first elementary observations:


1. All terms in E(%) are finite for % ∈ Dλ.

Proof. This evident for T (%). To proof the assertion for V (%), we isolate the Coulombsingularities from its Coulom tail and abbreviate for a > 0:

v>(x) :=

M∑m=1

Zm min|x−Rm|−1 , a−1

and v<(x) :=

∑Mm=1 Zm|x−Rm|−1 − v>(x). Holder’s inequality then yields:

0 ≤ V (%) =

∫%(x)v<(x)dx+

∫%(x)v>(x)dx

≤ ‖%‖5/3‖‖v<‖5/2 + ‖%‖1 ‖v>‖∞ <∞ .

The finiteness of the interaction term may be seen with the help of the Hardy-Littlewood-Sobolev inequality, Lemma 1.5,

D(%) ≤ C ‖%‖26/5 ≤ C ‖%‖7/61 ‖%‖

5/65/3 <∞ .

2. The above estimates also show that for all % ∈ Dλ:

E(%) ≥ T (%)− V (%) ≥ 3γ

5‖%‖5/35/3 − ‖%‖5/3‖v<‖5/2 − λ ‖v>‖∞ .

Infimizing over ‖‖%‖5/3, we hence conclude that there is some Cλ <∞ such that

inf%∈Dλ

E(%) ≥ −Cλ .

3. Dλ is convex and E(%) is strictly convex on Dλ.

Proof. The functional T (%) is strictly convex and V (%) is linear. To see that D(%) is convex,we use the positivity of the Fourier transform of x 7→ |x|−1 to conclude that for %1, %2 ∈ Dλ:

D(%1, %2) :=

∫%1(x) %2(y)

|x− y|dxdy ≤

√D(%1)D(%2) ≤ 1

2(D(%1) +D(%2)) .

and hence for all α ∈ [0, 1]:

D(α%1 + (1− α)%2) = α2D(%1) + (1− α)2D(%2) + 2α(1− α)D(%1, %2)

≤ αD(%1) + (1− α)D(%2) .


4. The functional E(%) is weakly lower semicontinuous.

Proof idea. This derives from the following observations: Suppose (%n) ⊂ L1(R3)∩L5/3(R3)such that %n % weakly in L5/3(R3). Then:

• lim infn→∞ T (%n) ≥ T (%)

(since T (%) is essentially the ‖·‖5/3-norm and as such is weakly lower semicontinuous.)• limn→∞ V (%n) = V (%).• lim infn→∞D(%n) = D(%).

By standard arguments, the lower boundedness,convexity and lower semicontinuity imply theexistence of a unique minimizer of the Thomas-Fermi functional.

Theorem 5.1 (Existence and uniqueness of minimizers). There is a unique %λ ∈ Dλ such that

E(%λ) = inf%∈Dλ

E(%) =: E(λ) .

Moreover, the function λ 7→ E(λ) is convex and non-increasing.

Proof idea. Since E(%) is strictly convex, the uniqueness of the minimizer follows immediatelyfrom its existence. For the latter, let (%n) ⊂ Dλ be a minimizing sequence, i.e. limn→∞ E(%n) =E(λ). By virtue of the above bounds, we see that there is some finite constant C, such that thesequence is bounded in L5/3(R3):

lim supn→∞

‖%n‖5/3 ≤ C .

By the Banach-Alaoglu theorem, there is some %λ ∈ L5/3(R3) and a subsequence, which we willdenote again by (%n), such that %n % in L5/3(R3). In particular, for any f ∈ L5/2(R3):

limn→∞

∫f(x)%n(x) dx =

∫f(x)%λ(x)dx .

Choosing f appropriately, this implies that %λ ≥ 0 and∫%λ(x)dx ≤ λ, i.e., %λ ∈ Dλ. Since E(%)

is weakly lower semicontinuous, this implies that

E(λ) = limn→∞

E(%n) ≥ E(%λ) ≥ E(λ) .

The convexity of the function E(λ) derives from that of the functional. Let α ∈ [0, 1] andλ1, λ2 ≥ 0 and suppose that %1, %2 are the corresponding unique minimizers. Then

E(αλ1 + (1− α)λ2) ≤ E(α%1 + (1− α)%2) ≤ αE(%1) + (1− α)E(%2)

= αE(λ1) + (1− α)E(λ2) .

Since transporting part of the density to infinity does not increases the energy, the function E(λ) isnon-increasing.


In the remainder of this section, we will collect more specific properties of the Thomas-Fermifunctional and its minimizer. (Not all of them will be proven.)

Theorem 5.2 (Thomas-Fermi equation). For any λ > 0 the unique minimizer %λ of the Thomas-Fermi functional is non-zero and satisfies the Thomas-Fermi equation:

γ %(x)2/3 = max Φ%(x)− µ , 0

with some µ ≥ 0 (which depends on λ). Here

Φ%(x) :=M∑m=1

Zm|x−Rm|

−∫

%(y)

|x− y|dy

is the Thomas-Fermi potential. The latter satisfies Φ%λ ≥ 0.

Proof. We calculate the variation of E(%) about its minimizer %λ. To this end, let t ∈ [0, 1] andg ∈ L1(R3) ∩ L5/3(R3) and set:

%(x; t) := %λ(x) + t

(g(x)− %λ(x)

∫g(y)dy

/∫%λ(y)dy

).

By definition∫%(x; t)dx =

∫%λ(x)dx. Moreover, the restrictions g(x) ≥ −%λ(x)/2 and

∫g(y)dy ≤

12

∫%λ(y)dy imply that %(x; t) ≥ 0 for all t ∈ [0, 1].The function F (t) := E(%(·; t)) clearly satisfies F (0) ≥ E(λ) and its derivative at zero is

F ′(0) =

∫g(x)

[γ%λ(x)2/3 − Φ%λ(x) + µ

]dx .

whereµ := −

∫%λ(x)

[γ%λ(x)2/3 − Φ%λ(x)

]dx/∫

%λ(y)dy . (5.1)

Since %λ is a minimizer, we must have F ′(0) ≥ 0, i.e.∫g(x)

[γ%λ(x)2/3 − Φ%λ(x) + µ

]dx ≥ 0 .

Since this inequality holds for all 0 ≤ g ∈ L1(R3) ∩ L5/3(R3), we conclude that

γ%λ(x)2/3 − Φ%λ(x) + µ ≥ 0 .

From (5.1) we then conclude that γ%λ(x)2/3 − Φ%λ(x) + µ = 0, whenever %λ(x) > 0.To complete the proof of the first part, assume µ < 0. Then Thomas-Fermi equation implies

that for sufficiently large |x| the function %λ(x)2/3 does not decay to zero at infinity.


To show that the Thomas-Fermi potential of the minimizer is non-negative, we consider the set

A :=x ∈ R3 |Φ%λ(x) < 0

.

Since the Thomas-Fermi potential is continuous away fromR1, . . . , RM and one hasR1, . . . , Rm 6∈A, the set A is open. By the Thomas-Fermi equation %λ(x) = 0 for all x ∈ A and hence for allx ∈ A:

∆Φ%λ(x) = 4π%λ(x) = 0 ,

i.e. the Thomas-Fermi potential is harmonic in A. Since Φ%λ is continuous, it vanishes at theboundary of A. Since it also vanishes uniformly at ∞, one has Φ%λ(x) = 0 for all x ∈ A andhence A is empty.

In the following, we will abbreviate the total nuclear charge by

Z :=M∑m=1

Zm .

By averaging Φ%λ over a sphere of a sufficiently large radius r > 0 such that the charge distributionis well approximated by a spherical one, the non-negativity of the Thomas-Fermi potential implies:

0 ≤ Z

r− 1

r

∫|x|<r

%λ(x)dx ,

i.e., the total nuclear charge bounds the total charge from above:

Z ≥∫%λ(x)dx .

In other words, there are no negative ions in Thomas-Fermi theory. In contrast, Thomas-Fermitheory does describe positively charged and neutral atoms. This and further properties of theThomas-Fermi minimizer is the content of the following theorem.

Theorem 5.3. 1. Suppose that 0 < λ ≤ Z. Then the Thomas-Fermi minimizers satisfies:∫%λ(x)dx = λ.

2. Suppose that λ ≥ Z. Then the

(a) E(λ) = E(Z)

(b) µ = 0.


(For a proof see e.g. E. Lieb/M. Loss, Analysis, AMS 2001.)

It is instructive to discuss the Thomas-Fermi minimizer in case of a neutral atom at R = 0 andone electron N = 1. In this case the Thomas-Fermi equation reads γ%2/3 = Φ% or:

∆Φ%(x) = −4πδ(x) + 4π%(x) = −4πZδ(x) + 4πγ−3/2Φ%(x)3/2 .

The Ansatz Ψ(x) = f(r)(4π)2r

leads to the ordinary differential equation

γ3/2 r1/2f ′′(r) = f3/2(r) .

whose solution has a polynomial fall-off as r−3 for r →∞. As a consequence,

%λ(x) = γ−3/2Φ%(x)3/2

decays as |x|−6 as |x| → ∞. Moreover, the Thomas-Fermi energy of the atom is Eatom(1) :=−3.678γ−1.

More generally, in case of a neutral atom with central charge Z > 0 and N = Z electrons, thesolution is obtained from the above by scaling. One identifies Z−1/3 as relevant length scale and

Eatom(Z) = Z7/3Eatom(1) .

Our last topic is Teller’s theorem (which was proved by Lieb and Simon) on the non-existenceof moluecules in Thomas-Fermi theory.

Theorem 5.4. For each λ ≤ Z there are K numbers λj ∈ (0, Zj with∑K

j=1 λj = λ such that

E(λ) ≥K∑j=1

Eatom(λj) .

Proof idea in case λ = Z. The claim immediately follows from the following dissociation prop-erty of the Thomas-Fermi energy:

EM (Z1, . . . , ZM ;R1, . . . , RM ) ≥ EL(Z1, . . . , ZL;R1, . . . , RL)

+ EM−L(Z1, . . . , ZM−L;R1, . . . , RM−L)

for every 1 ≤ L < M , where

EM (Z1, . . . , ZM ;R1, . . . , RM ) := min

E(%) |∫%(x)dx =

M∑j=1

Zm , % ∈ D∑Mj=1 Zm

.

For its proof we study the function

f(α) := EM (αZ1, . . . , αZL, ZL+1, . . . ZM ;R1, . . . , RM )− EL(αZ1, . . . , αZL;R1, . . . , RL)

− EM−L(Z1, . . . , ZM−L;R1, . . . , RM−L)


and show that f ′(α) ≥ 0 for all α ∈ (0, 1), and hence f(1) = f(0)+∫ 1

0 f′(α)dα ≥ 0. To calculate

the derivative we use that for λ =∑

m Zm:

∂E(λ)

∂Zk=∂E(%λ)

∂Zk+

∫∂E(%λ)

∂%(x)︸︷︷︸=0

∂%λ(x)

∂Zkdx

=−∫

%λ(x)

|x−Rk|dx+

∑m 6=k

Zm|Rm −Rk|

= limx→Rk

(Φ%λ(x)− Zk

|x−Rk|

).

As a consequence:

f ′(α) =

L∑k=1

Zk

(∂EM∂Zk

− ∂EL∂Zk

)=

L∑k=1

Zk

(Φ%∑M

m=1 Zm(x)− Φ%∑L

m=1 Zm(x)).

As it turns out, the last term is non-negative due to the monotonicity of the Thomas-Fermi potentialin the nuclear charges, i.e., if Z :=

∑Mm=1 Zm ≥

∑Lm=1 Zm =: Z then

Φ%Z (x) ≥ Φ%Z(x) .

For a proof, we consider ψ(x) := Φ%Z (x)−Φ%Z(x) and the set B := x ∈ R3 |ψ(x) < 0, which

is open. For x ∈ B one has

∆ψ(x) = 4π(%Z(x)− %Z(x)) = 4πγ−3/2(Φ%Z(x)3/2 − Φ%Z (x)3/2 < 0 .

That is ψ is superharmonic on B. Since it vanishes on the boundary where its minimum is, the setB is empty.

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