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Schrodinger Operators

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Text and
Monographs
in Physics

W. Beiglb6ck
J. L. Birman
R. Geroch
E. H. Lieb
T. Rt.:egc
w. Thirring
"-"-

"-

Scric' Eclilon



H.L. Cycon R.G. Froese W. Kirsch
B. Simon

SchrOdinger Operators
with Application to Quantum Mechanics
and Global Geometry

With 2 Figures

Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo


Dr. Hans L. Cycon

Professor Dr. Werner Kirsch

Technische UniversiUit Berlin
Fachbereich 3- Mathematik
Stra6e des 17. Juni 135, D-1000 Berlin 12

Institut fur Mathematik
UniversiUit Bochum
D-4630 Bochum, Fed. Rep. of Germany


Dr. Richard G. Froese

Professor Dr. Barry Simon

Department of Mathematics
University of British Columbia
Vancouver. B.C .. Canada V6T 1W5

California Institute of Technology
Depanment of Mathematics 253-37
Pasadena, CA 91125, USA

Editors
Wolf Beiglbock

Elliott H. Lieb

lnstitut fur Angewandte Mathematik
Universitat Heidelberg
Im Neuenheimer Feld 29~
D-6900 Heidelberg I
Fed. Rep. of Germany

Depanment of Physics
Joseph Henry Laboratories
Princeton University
Princeton, NJ 08540. USA

Joseph L. Birman


Tullio Regge

Department of Physics. The City College
of the City University of New York
NewYork. NY 100~1. USA

Istituto di Fisica Teorica
Universita di Torino. C. so M. d·Azeglio, 46
1-10125 Torino. Italy

Robert Geroch
University of Chicago
Enrico Fermi Institute
5640 Ellis Ave.
Chicago. IL 60637. USA

Walter Thirring
Institut fur Theoretische Physik
der Universitat Wien. Boltzmanngasse 5
A-1090 Wien, Austria

ISBN 3-540-16758-7 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-16758-7 Springer-Verlag New York Berlin Heidelberg
Library of Congrc~s Cataloging-in-Publication Data. Schrooingcr operators. with application to quantum
mechanics and glohal geometry. (Texts and monographs in physics) Chapters 1-11 arc revised notes taken
from a ~ummcr course gi\·cn in 1982 in Thurnau. West Germany by Barry Simon. ··Springer Study Edition ...
Bibliography: p. Includes index. 1. Schrodingcr operator. 2. Quantum theory. 3. Global differential geometry. I. Cycon. H.L. (llans Ludwig). 1942-. II. Simon. Barry. Ill. Series. QCI74.17.S6S37 1987 515.7"246
~h-13953


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2153'3 1SO-S·B210


Preface

In the summer of 1982, I gave a course of lectures in a castle in the small town
of Thurnau outside of Bayreuth, West 'Germany, whose university hosted the
lecture series. The Summer School was supported by the Volkswagen foundation and organized by Professor C. Simader, assisted by Dr. H. Leinfelder. I
am grateful to these institutions and individuals for making the school, and
thus this monograph, possible.
About 40 students took part in a grueling schedule involving about 45 hours ,

of lectures spread over eight days! My goal was to survey the theory of
Schr()dinger operators emphasizing recent results. While I would emphasize
that one was not supposed to know all of Volumes 1 - 4 of Reed and Simon (as
some of the students feared!), a strong grounding in basic functional analysis
and some previous exposure to Schr()dinger operators was useful to the
students, and will be useful to the reader of this monograph.
Loosely speaking, Chaps. 1 - 11 of this monograph represent "notes" of
those lectures taken by three of the "students" who were there. While the general organization does follow mine, I would emphasize that what follows is far
from a transcription of my lectures. Even with 45 hours, many details had to be
skipped, and quite often Cycon, Froese and Kirsch have had to flesh out some
rather dry bones. Moreover, they have occasionally rearranged my arguments,
replaced them with better ones and even corrected some mistakes!
Some results such as Lieb's theorem (Theorem 3.17) that were relevant to
the material of the lectures but appeared during the preparation of the monograph have been included.
Chapter 11 of the lectures concerns some beautiful ideas of Witten reducing
the Morse inequalities to the calculation of the asymptotics of eigenvalues of
cleverly chosen Schr()dinger operators (on manifolds) in the semiclassical limit.
When I understood the supersymmetric proof of the Gauss-Bonnet-Chern
theorem (essentially due to Patodi) in the summer of 1984, and, in particular,
using Schr()dinger operator ideas found a transparent approach to its analytic
part, it seemed natural to combine it with Chap. 11, and so I wrote a twelfth
chapter. Since I was aware that Chaps. 11 and 12 would likely be of interest to a
wider class of readers with less of an analytic background, I have included in ,
Chap. 12 some elementary material (mainly on Sobolev estimates) that have
been freely used in earlier chapters.
I

I

1


I

1

1

I

1

I

1

I

1

I

1

1

Los Angeles, Fall 1986

Barry Simon




Contents

t. Self-Adjointness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Basic Perturbation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
TheClassesSv andKv . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kato's Inequality and All That . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Leinfelder-Simader Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
3
8
11

2. LP-Properties of Eigenfunctions, and All That . . . . . . . . . . . . . . . . . . .
2.1 Semigrou p Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Estimates on Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. 3 Local Estimates on Gradients ............. ! • • • • • • • • • • • • • • •
2.4 Eigenfunctions and Spectrum (Sch'nol's Theorem) . . . . . . . . . . .
2.5 The Allegretto-Piepenbrink Theorem . . . . . . . . . . . . . . . . . . . . . .
2.6 Integral Kernels for exp(- tH) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13
13
17
19
20
22
24


3. Geometric Methods for Bound States . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Partitions of Unity and the IMS Localization Formula . . . . . . . .
3.2 M ultiparticle SchrOdinger Operators . . . . . . . . . . . . . . . . . . . . . . .
3. 3 The HVZ-Theorem ....................... , . . . . . . . . . . . . . .
3.4 More on the Essential Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 A Theorem of Klaus: Widely Separated Bumps . . . . . . . . . . . . . .
3.6 Applications to Atomic Physics: A Warm-Up . . . . . . . . . . . . . . .
3. 7 The Ruskai-Sigal Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Lieb's Improvement of the Ruskai-Sigal Theorem . . . . . . . . . . . .
3.9 N-Body Systems with Finitely Many Bound States . . . . . . . . . . . .
Appendix: The Stone-Weierstrass Gavotte . . . . . . . . . . . . . . . . . . . . . .

27
27
29
32
36
39
41
43
SO

1.1
1.2
1.3
1.4

'2
58


4. Local Commutator Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Putnam's Theorem and the Mourre Estimate . . . . . . . . . . . . . . . .
4.2 Control of Imbedded Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Absence of Singular Continuous Spectrum . . . . . . . . . . . . . . . . . .
4.4 Exponential Bounds and Nonexistence of Positive Eigenvalues .
4.5 The Mourre Estimate for N-Body SchrOdinger Operators . . . . . .

60
60
65
67
74
82

5. Phase Space Analysis of Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Some Notions of Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Perry's Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89
89
92


VIII

Contents

5.3 Enss' Version of Cook's Method . . . . . . . . . . . . . . . . . . . . . . . .
5.4 RAGE Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.5 Asymptotics of Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Asymptotic Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. 7 Asymptotic Completeness in the Three-Body Case . . . . . . . . . .

95
97
101
105
106

6. Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

114
116
119
120
125
129
130

6.1
6.2
6.3
6.4
6.5
6.6

Gauge Invariance and the Essential Spectrum . . . . . . . . . . . . . .
A SchrOdinger Operator with Dense Point Spectrum . . . . . . . .
Supersymmetry (in O-S pace Dimensions) . . . . . . . . . . . . . . . . . .

The Aharonov-Casher Result on Zero Energy Eigenstates . . .
A Theorem of Iwatsuka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An Introduction to Other Phenomena in Magnetic Fields . . . .

7. Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 3
133

7.1 The Two-Body Stark Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 A Theorem Needed for the Mourre Theory of the
One-Dimensional Electric Field . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Propagators for Time-Dependent Electric Fields . . . . . . . . . . .
7.4 Howland's Formalism and Floquet Operators . . . . . . . . . . . . .
7.5 Potentials and Time-Dependent Problems . . . . . . . . . . . . . . . .

135
138
144
147

8. Complex Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

150

8.1
8.2
8.3
8.4
8.5

8.6
8. 7

Review of "Ordinary" Complex Scaling . . . . . . . . . . . . . . . . . .
Translation Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Higher Order Mourre Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
Computational Aspects of Complex Scaling . . . . . . . . . . . . . . .
Com.plex Scaling and the DC-Stark Effect . . . . . . . . . . . . . . . . .
Complex Scaling and the AC-Stark Effect . . . . . . . . . . . . . . . . .
Extensions and Generalizations . . . . . . . . . . . . . . . . . . . . . . . . .

150
155
156
157
158
160
162

9. Random Jacobi Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

164

9.1 Basic Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 The Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 The Lyaponov Exponent and the Ishii-Pastur-Kotani
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4 Subharmonicity of the Lyaponov Exponent and the
Thouless Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5 Point Spectrum for the Anderson Model . . . . . . . . . . . . . . . . . .


165
171

10. Almost Periodic Jacobi Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1 Almost Periodic Sequences and Some General Results . . . . . .
10.2 The Almost Mathieu Equation and the Occurrence of
Singular Continuous Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . .

176
182
186,
197
197
199


Contents

IX

Pure Point Spectrum and the Maryland Model 0 0 0 •• 0 ••• 0 0 •
Cantor Sets and Recurrent Absolutely Continuous Spectrum

203
212

11. Witten's Proof of the Morse Inequalities 0 • 0 0 0 0 • 0 0 0 •••• 0 0 •• 0 0 • 0
11.1 The Quasiclassical Eigenvalue Limit . 0 0 0 0 •• 0 0 0 • 0 0 0 0 0 0 • 0 0
11.2 The Morse Inequalities . 0 • 0 • 0 0 0 0 0 0 0 0 • 0 0 0 0 • 0 0 • 0 •• 0 •• 0 • 0 0

11.3 Hodge Theory 0 0 • 0 0 • 0 • 0 • 0 0 ••• 0 0 0 • 0 0 • 0 0 0 • 0 0 •• 0 0 • 0 0 • 0 0 0
11.4 Witten's Deformed Laplacian .. 0 • 0 0 •• 0 0 0 0 • 0 •••• 0 0 •• 0 0 • 0
110 5 Proof of Theorem 11.4 ••••• 0 • 0 0 • 0 • 0 • 0 0 0 • 0 0 0 0 ••• 0 0 0 0 0 0 •

217
217
223
226
231
234

10.3
10.4

12. Patodi's Proof of the Gauss-Bonnet-Chern Theorem and
Superproofs of Index Theorems . 0 0
0 ••• 0 • 0 0 0 ••• 0 0 0 0 ••• 0 0 0 0 0
12.1 A Very Rapid Course in Riemannian Geometry 0 0 0 0 0. 0 0 0 0 0
1202 The Berezin-Patodi Formula .. 0. 0 0 0 0 0 0 •• 0. 0 0 0 0 0 0. 0 0 0 0 0.
12.3 The Gauss-Bonnet-Chern Theorem: Statement and Strategy
of the Proof 0 0 0 0 0 0 0 •• 0 0 0 0 ••• 0 •• 0 ••••• 0 0 0 0 0 0 0 • 0 • 0 0 • 0 • 0
12.4 Bochner Laplacian and the WeitzenbOck Formula .... 0 0 • 0 •
12 05 Elliptic Regularity 0 0 0 0 • 0 0 •• 0 0 0 •• 0 ••• 0 0 0 0 0 0 0 0 0 0 0 0 0 • 0 0 • 0
12.6 A Canonical Order Calculus 0 0 0 0 0 0 0 0 • 0 • 0 0 • 0 0 0 • 0 0 0 • 0 0 0 0 0
12.7 Cutting and Pasting 0 0 0 0 0 0 0 0 • 0 0 0 0 0 • 0 0 0 0 0 0 0 0 ••• 0 0 •• 0 0 0 0
12.8 Completion of the Proof of the Gauss-Bonnet-Chern
Theorem 0 0 0 • 0 0 • 0 0 0 0 0 0 0 • 0 •• 0 0 0 • 0 0 • 0 0 0 0 0 0 0 0 0 • 0 0 0 • 0 0 0 0 0
12.9 Mehler's Formula 0 0 • 0 • 0 • 0 • 0 0 0 • 0 0 • 0 • 0 0 0 0 0 0 0 0 •• 0 0 0 • 0 0 0 0
12.10 Introduction to the Index Theorem for Dirac Operators 0 0 0 0


284
286
297

Bibliography 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 • 0 • 0 0 0 0 • 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 •••• 0 • 0

301

List of Symbols 0 0 0 0 0 0 0 0 0 0 0 0 • 0 0 0 • 0 0 0 0 0 0 0 0 • 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 • 0 •• 0

315

Subject Index 0 0 0 • 0 0 • 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 • 0 0 • 0 ••• 0 • 0

317

o ••

238
238
247
252
258
265
273
283



t. Self-Adjointness


Self-adjointness of Schrodinger operators has been a fundamental mathematical
problem since the beginning of quantum mechanics. It is equivalent to the unique
solvability of the time-dependent Schrodinger equation, and it plays a basic role
in the foundations of quantum mechanics, since only self-adjoint operators can
ben understood as quantum mechanical observables (in the sense of von Neumann
[361] ).
It is an extensive subject with a large literature (see e.g. [293, 107, 196]) and
the references given there), and it has been considerably overworked. There are
only a few open problems, the most famous being Jorgens' conjecture (see [293,
p. 339; 71, 317]).
We will not go into an exhaustive overview, but rather pick out some subjects
which seen to us to be worth emphasizing. We will begin with a short review of
the basic perturbation theorems and then discuss two typical classes of perturbations. Then we will discuss Kato's inequality. Finally, using an idea of Kato,
we give some details of the proof of the theorem of Leinfelder and Simader on
singular magnetic fields.

1.1 Basic Perturbation Theorems
First, we give some definitions (see [293, p. 162] for a more detailed discussion).
We denote by A and B, densely-defined linear operators in a Hilbert space H,
and by D(A) and Q(A), the operator domain and form domain of A respectively.
Definition 1.1. Let A be self-adjoint. Then B is said to be A-bounded if and only
if
(i) D(A) £ D(B)
(ii)

there are constants a, b > 0 such that
IIBcpll S aiiAcpll

+ bllcpll


for

cpeD(A) .

( 1.1)

The infimum of all such a is called the A-bound (or relative-bound) of B.
There is an analogous notion for quadratic forms:
Definition 1.2. Let A be self-adjoint and bounded from below. Then a symmetric
operator B is said to be A-form bounded if and only if


2

I. Self-Adjointncss

(i) Q(A) c Q(B)
(ii) there are constants a, b > 0 such that
l(cp,Bcp)l < a(cp,Acp) + b(cp,cp)

for

cpeQ(A) .

The infimum of all such a is called the A-jorn1-bound (relative form-bound)
of B.
Note that the operators in the above definitions do not need to be self-adjoint
or symmetric [ 196, p. 190, p. 319]. We require it here because later propositions
will be easier to state or prove for the self-adjoint case.

A subspace in His called a core for A if it is dense in D(A) in the graph norm.
It is called a form core if it is dense in Q(A) in the form norm.
There is an elementary criterion for relative boundedness.
Proposition 1.3. (i) Assume A to be self-adjoint and D(A) c D(B). Then B is
A-bounded if and only if B(A + i)- 1 is bounded. The A-bound of B is equal to
lim II B(A

+ ii·)- 1 II .

lt·l-x

(ii) (form version). Assume A to be self-adjoint, bounded from below and
Q(A) c Q(B). Then B is A-form-bounded if and only if (A + i)- 1' 2 B(A + i)- 1' 2 is
bounded. The A-form-bound of B is equal to
lim II(A + ii')- 1 '2 B(A + ii')- 112 11

•

lt'l-:x:

The assertion (i) can easily be seen by replacing cp by (A + iy)- 1 t/1 in ( 1.1) and
observing that IIB(A + ii·)- 1 11 <[a+ (b/lyl)]. (ii) follows analogously. Note that
there is an extension of this notion which we use occasionally: We say that B is
A-compact if and only if B(A + i)- 1 is compact. Here i can be replaced by any
point of the resolvent set.
Now we will state the basic perturbation theorem which was proven by Kato
over 30 years ago, and which works for most perturbations of practical interest.
Theorem 1.4 (Kato-Rellich). Suppose that A is self-adjoint, B is symmetric and
A-bounded with A-bound a < 1. Then A + B [which is defined on D(A)] is self
adjoint, and any core for A is also a core for A + B.

We give a sketch of the proof. Note that self-adjointness of A is equivalent
to Ran(A + iJI) = H for some Jl > 0 [292, Theorem VIII.3]. Then, as above, we
conclude from ( 1.1) that
IIB(A

+ ip)- 1 II <

a

+b

.

Jl

Thus, for JIIarge enough C := B(A + iJI)- 1 has norm less than 1, and this implies
that Ran( 1 + C) = H. This, together with the equation


1.2 The Classes S" and K"

( 1 + C)(A

+ iJI)cp

=(A

+ B + iJI)cp

3


cp e D(A)

and the self-adjointness of A, implies that Ran(A + B + iJI) = H. The second part
of the theorem is a simple consequence of ( 1.1 ).
There are various improvements due to Kato [196] and Wust [371] for the
case a = 1, but in fact all the perturbations one usually deals with in the theory
of Schrodinger operators have relative bound 0.
There is also a form version of Theorem 1.4 (due to Kato, Lax, Lions,
Milgram and Nelson):
Theorem 1.5 (KLMN). Suppose that A is self-adjoint and bounded from below
and that B is symmetric and A-bounded with form-bound a< 1. Then
the sum of the quadratic forms of A and B is a closed symmetric form on
Q(A) which is bounded from below.
(ii) There exists a unique self-adjoint operator associated with this form which
we call the form sum of A and B.
(iii) Any form core for A is also a form core for A + B.
(i)

+

For a proof, see [293, Theorem X.l7]. We will denote the form sum by A B
when we want to emphasize the form character of the sum. otherwise we will
write A +B.
Note that in spite of the parallelism between operators and forms, there is a
fundamental asymmetry. There are symmetric operators which are closed but
not self-adjoint. But a closed form which is bounded from below is automatically
the form of a unique, self-adjoint operator [ 196, Theorem VI.2.1 ]. The form
analog of essential self-adjointness, however, does exist: a suitable set being a
form core. If one defines something to be a closed quadratic form, it is automatic

that the associated operator is self-adjoint-one knows nothing, however, about
the operator domain or the form domain. It is therefore a nontrivial fact that a
convenient set (e.g. C~) is a form core.

1.2 The Classes S, and K,
In this book, we will study the sum -A + V in virtually all cases. But occasionally we will also study (- iV + a) 2 + Vas operators or forms in the Hilbert space
L 2 (~"). Here V is a real-valued function on ~" describing the electrostatic
potential, and a is a vector-valued function which describes the magnetic potential. We denote by H 0 the self-adjoint representation of -A in L 2 (~"). In
reasonable cases, one can think of V as a perturbation of H0 • Physically, this
ts motivated by the uncertainty principle which allows the kinetic energy to
control some singularities of V if they are not too severe. This phenomenon
has no classical analog. This is also practical since the Laplacian has an explicit


4

I. Self-Adjointness

eigenfunction expansion and integral kernel, and one knows everything about
operator cores, etc.
There are two classes of perturbations we will discuss here. The class Sv,
which is an (almost maximal) class of operator perturbations of H0 and the class
K v which is the form analog of Sv. Sv was introduced originally by Stummel [352],
and has been discussed by several authors (see e.g. [308] ).

Definition 1.6. Let V be a real-valued, measurable function on
V e Sv if and only if
a) lim[sup
al,O


X

c) sup
X

X

lx- yl 4 -viV(y)l 2 dvy] = 0 if v > 4

J

ln(lx - yl)- 1 IV(y)l 2 dvy] = 0

We say that

lx-yl Sa

b) lim [sup
al,O

J

~v.

if v = 4

lx-)'1 Sa

J


IV(y)l 2 dvy < oo if v S 3 .

l.x-yl S 1

For the reader who is disturbed by the lack of symmetry in the above definition,
we remark that for v s 3,
sup
X

J

IX-)'1 s

IV(y)l 2 dvy <

00

1

is equivalent to

We define a Sv·norm on Sv by
II VIIs.:= sup

J

K(x, y; v) I V(y)l 2 dvy ,

.x lx-yl S 1


where K is the kernel in the above definition of Sv. We now state (and prove) a
theorem which shows how these quantities arise naturally. We denote, by II· lip.,,
the operator norm for operators from LP(~v) to L 4 (~v), and by II· liP the norm in
LP(~v).

Theorem 1.7. V e Sv if and only if
lim II (Ho

E-x.

+ E)- 2 1 Vl 2 II oo. oo

= 0 .

(1.2)

Proof As with all functions of H 0 , (H0 + E)- 2 is a convolution operator with an
explicit kernel Q(x - y, E) [293, Theorem IX.29]. It has the following properties
(see [308, Theorem 3.1, Chap. 6] ).


1.2 The Classes S" and K"

5

1. Q(x - y, E) > 0 ,
O(Ix - Yl 4 -")

2. Q(x- y,E)
3.


={

OC(Inlx- yl- 1 )

sup elx-yiQ(x - y, E)--+ 0
lx-yl>6

as

if v > 4
if v = 4 as
if v ~ 3
E--+ oo,

lx- yl--+0,

for any

~

>0 .

Using the elementary fact that
sup
x

J

lx-yiSI


I V(y)l 2 dy < oo

for any V e S", it is not hard to see that V e S" if and only if
supx Q(x - y, E) IV(y)l 2 d" y--+ 0 as E--+ oo. This gives the result, since
Q( · - y, E) I V(y)l 2 is a positive integral kernel and II A II oo, X> = II A Ill :x:· holds for
any A with positive integral kernel. D

J

The above result has an L 2 consequence by a standard "duality and interpolation" argument:
Corollary 1.8. If V e S", then
II(H0

+ E)- 1 Vll 2 , 2 --+ 0

as

E--+ oo .

(1.3)

Proof Let V e S". Then it is enough to show that
(1.4)

since ( 1.3) follows then by Theorem 1. 7. Assume for a moment that Vis bounded,
and consider the function

F(z) is an operator-valued function which is L 1 and L -bounded and analytic in
(L


the interior of the strip {z e Cl Re z e [0, 1] }. Thus, by the Stein interpolation
theorem [293, Theorem IX.21] and, using that (by duality)

II (Ho + E)- 2 1 Vl 2 II oc. oo

=

I I VI 2 (Ho + E)- 2 II 1.1

,

we get

Since

( 1.4) follows for bounded V"s. and by an approximation argument, also for all
VeS". D


6

I. Self-Adjointness

Remark. Note that Corollary 1.8 implies that if V e Sv, then it is H0 -bounded with
H0 -bound 0 by Proposition 1.3 (Proposition 1.3 has to be slightly modified for
the semibounded case we are considering here).
One might think that since Sv is telling us something about L 1 -bounds and
L x is "stronger" than L 2 , there would be no way going from L 2 -bounds to Sv.
So the following theorem is interesting.


Theorem 1.9. Suppose there are a, b > 0 and a ~ with 0 <
0 < E; < 1 and all cpeD(H0 )
IIVcpll~ < E;IIHocpll~

+ aexp(bE;-

6

~

< 1 such that, for all

)IIcpll~ .

Then VeSv.
Proof We just have to pick the right cp"s. Fix y e ~v, t e ~+, and consider the
integral kernel

cp(x) := Jexp(- tH0 )(x, y) .
Then, noting that II cp 11 2 = 1 and (by scaling)

IIH0 cpll 2 = ct- 2 for suitable c > 0
we have
(1.5)

Now, take E;:= (1 + llntl)-", where y:= 2/(1 + ~), and multiply (1.5) by
t exp(- tE) forE> 0. Then the R.H.S. of (1.5) is integrable in t and its integral
goes to zero as E--+ ~-Now if we use the identity
(H0


+

7..

E)- 2

=

Jte-'80 e-rE dt
0

we get (1.2), and therefore VeSv by Theorem 1.7.

D

The second class of potentials we are considering here is K V' which is the
form analog of Sv. This type of potentials was first introduced by Kato [ 193].
See also Schechter [308] for related classes. K v was studied in some detail by
Aize1unan and Simon [1], and Simon [334].
Definition 1.10. Let V be a real-valued measurable function on ~v. We say that
V E Kv if and only if
a) lim[sup
2~0

X

J

IX-}'1 ~2


lx- yl 2 -viV(y)ldvy] = 0,

if v > 2


1.2 The Classes Sw and K w

J

b) lim [sup
al,O

x

= 0,

if v

=2

lx-yiSa

x

J

c) sup

lnl(x - y)l- 1 1 V(y)l dvy]


7

IV(y)ldvy < oo,

v= 1 .

if

lx-}'1 S I

We also define a Kv-norm by

II vII" := sup
w

x

J

K (x, y; v) Iv (y) Id vy

lx-)'1 S I

where K is the kernel in the above definition of Kv. Then virtually everything
goes through as before.
Theorem 1.11 [7]. VeKv if and only if
lim

II (Ho + E) -I I VI II x. :o


= 0 .

E-x

The proof is the same as in Theorem 1. 7.
Theorem 1.12 [7]. Suppose there are a, b > 0 and a
for all 0 < e < 1 and all cp e Q(H0 )
(cp, IVI cp) S e(cp, H0 cp)

+ aexp(be- 6 ) llcplli

~

with 0 <

~

< 1 such that,

.

Then VeKv.
The proof is again like that in Theorem 1.9 above (see also [7, Theorem 4.9]).
Remarks. ( 1) Both of the classes Sv and K v have some nice properties:
a) If J1 S v, then K~J c Kv and S~J c Sv. By these inclusions we mean the
following. Suppose We K~J (resp. S~J), and there is a linear surjective map T: ~v--+
~~J and V(x) := W(T(x)). Then V e Kv (resp. Sv). The canonical example to think
of here is an N -body system with v = N p, where a point x e ~v is thought of as
an N -tuple of 11-dimensional vectors x = (x 1 , ••• , xN) and Tx := xi - x1 for

some i,je{l, ... , N}, i =Fj.
b) There are some LP-estimates which tell you when a potential is in Kv
(resp. Sv ), i.e.
L~nif

C

Sv

if

{p>~

for

v>4

p=2

for

v<4

~

for

v>2

p=2


for

v<2

and
L~nif

C

Kv

if

{p

>


8

I. Self-Adjointness

where

J

L~nir := {VIs up

IV(y)IP dy < oo} .


lx-yiSl

x

The proof is a straightforward application of Holder's inequality (see [7, Proposition 4.3] ).
(2) If V e K", then Vis H 0 -form bounded with relative bound 0. This follows
again analogously from Proposition 1.3(ii), Theorem 1.11 and a corollary analogous to Corollary 1.8.
The classes K" and Sv, however, are not the ''maximal" classes with respect
to the perturbation theorems, that is, one just misses the "borderline cases." This
can be seen in the following:
Example. (a) Let v > 3 and

Then V e K" if and only if~ > 1, but Vis H0 -form bounded with bound 0 if and
only if <5 > 0.
(b) Let v > 5 and V as in (a). Then V e Sv if and only if~> 1/2 but it is
H0 -bounded with bound 0 if and only if~ > 0. (a) is a consequence of[7, Theorem
4.11] and general perturbation properties (see [293, Chap. X.2] ). (b) has a similar
proof.
Remark. The above example shows that it is false that S" is contained in K v·

1.3 Kato's Inequality and All That
We will now sketch a set of ideas· which go back to Kato [193], and which were
subsequently studied by Simon [322, 327] (see also Hess, Schrader and Uhlenbrock [163]).
Let us first consider a vector potential a (magnetic potential), and a scalar V
(electric potential) satisfying
a e Lroc(~")"
VeLloc(~"),

V> 0 ,


( 1.6)

Then the formal expression
r := ( - iV - a) 2

+

V

is associated with a quadratic form

and

hmax

(called the maximal form) defined by


1.3 Kato's Inequality and All That

hmax(cp, t/1):=

L"

((oj- iaj)cp, (oj- iaj)t/1)

9

+

j=l

"'E

for cp,
Q(hmax); (oj := ofoxj). Note that hmax is a closed, positive form (since it
is the sum of (v + 1) positive closed forms), and therefore there exists a selfadjoint, positive operator H associated with hm••' with
Q(H) = Q(hma.)

and

(H cp, t/1) = hma•(cp, t/1)

for

cp, t/1 e D(H) [196] .

Note also that ( 1.6) are the weakest possible conditions for defining a (closable
positive) quadratic form associated with r on C~(IR"). The closure of this form
[which is the restriction of hma• to Csays that these two forms coincide. Thus, the self-adjoint operator associated
with the formal expression r is, in a sense, unique.

Theorem 1.13 [329, 195]. c:(R") is a form core for H.
We give only a sketch of the proof (see [329]).

Step 1.

e- rH : L 2 ( IR") --+ L 00 ( IR" ), t E IR +

(1.7)

•

We only need to show that

le-' 8 cpl < e-rHolcpl, cpeL 2 (1R")

(1.8)

(which is the semigroup version of Kato's inequality, sometimes also called
Kato-Simon inequality or diamagnetic inequality; see [327]), since (1.7) follows
from ( 1.8) by using Young's inequality and the fact that exp(- tH 0 ) is a convolution with an L 2 -integral kernel.
We know that H is a form sum of v + 1 operators. Therefore, we can use a
generalized version of Trotter's product formula (shown by Kato and Masuda
[198]) and get

exp(

-tH)

= s ;_~m[expGDf

)expGDi)·· .expGD;)exp(-~ v)J,
(1.9)

where
Dj := oj - iaj, j E { 1' ... ' v} .

Now, let
Xj

)..1(x):= Ja(x 1 ,
0

.....

x1 _ 1 ,y.. x1+ 1 ,

.....

x")dy.


10

I. Self-Adjointness

Then [329]

- iDi = euJ(- ioi )e -uJ .
(Note that, in a "physicist's language", this means that in one dimension, magnetic vector potentials can always be removed by a gauge transformation.)
Therefore

expGD/) = exp(ii.)iexpGO/ )exp( -ii.i) , so that
lexp(tD/ )cpl < exp(tcf) Icpl,

cp e L 2 (1R") .

( 1.1 0)

Now (1.8) follows from (1.10), (1.9) and lexp( -t·V/n)l S 1.

Step 2. L 1 (1R") n Q(H) is a form core for H.
This follows from ( 1. 7) and the fact that Ran [ exp(- t H)] is a form core for
H by the s¢pectral theorem.

Step 3. L~omp(IR") n Q(H) is a form core for H [where L~mp(IR") := { cp e L 2 (1R")I
cp e L x ( IR" ), supp cp is compact}].
This follows by a usual cut-ofT approximation argument, i.e. choose '7 e
C~ (IR") with '1 = 1 near 0, then consider, for any cp e L Xl n Q(H)
cp,.(x) := '1 (:) cp(x)

then
~

cp, (n

~

(n e 1\1)

oo) in the form sense. Now the proof will be finished by

Step 4. C0 (IR") is a form core for H.
This follows by a standard mollifier argument, i.e. choosej e CJj(x)d" x = 1; set ir. := c-"j(x/e), then for cp e L':omp n Q(H) cpt :=it • cp e C0 and

Note that in the last two steps, it is crucial that the approximated function
is in L x.
The next theorem is also a well-known result [ 193].
Theorem 1.14. Let V > 0, V e Lroc(IR") and a = 0. Then H := H 0 + V is essentially self-adjoint on C0 (IR"), i.e. C~ (IR") is an operator core for H, and its closure
is the form sum.
The proof is exactly the same as in Theorem 1.13 (replacing form cores by
operator cores and form domains by operator domains) with one additional step.
Once one notices that L x·(IR") n D(H) is an operator core for H one uses the
formula
(1.11)


1.4 The Leinfelder-Simader Theorem

11

for cpeL 1 (1Rv) n D(H) and '7EC0'(1Rv). The right-hand side of(l.ll) makes sense
since we know from Theorem 1.13 that cp e Q(H) and therefore Vcp e L 2(1Rv)v.
Equation (1.11) can then be used to show the analogous steps of Step 3 and Step
4 in Theorem 1.13. D

1.4 The Leinfelder-Simader Theorem
Our last theorem in this chapter is a result due to Leinfelder and Simader [229].
It finishes the problem of self-adjointness of Schrodinger operators with singular
potentials and V > 0 by giving a definitive result.

Theorem 1.15 (Leinfelder, Simader [229]). Let V > 0, VeLfoc(IRv) and ae
Li!,c(!Rv)v and V ·a e Lfoc(!Rv). Then H [the operator associated with the maximal
form of ( - iV - a) 2 + V] is essentially self-adjoint on C0 (!Rv ).
Though not explicitly mentioned in [229], the key lemma in the proof of

Leinfelder and Simader is

Lemma 1.16 (Kato's Version [197]). Let cpeL~mp(IRv), aeLi!,c(IRvy. If Vcpe
L 2(1Rv)v and - Jcp + 2ia · Vcp e L 2(1Rv), then Jcp e L 2(1Rv) and Vcp e L 4 (1Rv)v.
Proof(of Lemma 1.16) [227]. By a scaling argument, it is clear that without loss
one can choose supp cp to be contained in the unit ball B1 • One needs, as a basic
step, the following inequality which goes back to Gagliardo [ 127] and Nirenberg
[264]
(1.12)
for any p e (I, oo ), cp e L:-c,mp and a suitable constant d(p) depending on p. (Note
II V cp II := Ill V cp Ill). Equation ( 1.12) can be shown by using
II cjcp IIi= = lim J{[(cjcp) 2 + t;Jp-l cjcp} cjcp
t-O

partial integration and controlling all second derivatives by the formula
IID 2cpiiP < d'(p) IIL1cpiiP (see [350, p. 59]). If we choose I concern ourselves with p e [p 0 , p 1 ], then d := max d(p) can be chosen independently of p.
From (1.12) we get, forE;> 0 and pe [p 0 ,p 1 ]
II V cp ll2p < !dE;- 1 II cp llx + !dE; II Jcp II p

•

'

Thus, with g := - Jcp + 2ia · V cp
11Vcpll2p ~ !dt;- 1 llcpll1 + !dt;llgllp + !dt;ll2ia· Vcpllp .
Now, since supp cp is in the unit ball, if we choose p

~

p1

~

2, we can always


12

I. Self-Adjointness

estimate llgiiP by llgll 2, and by using ae L 4 (1R\It' and Holder's inequality, we get

IIP'cpll2p < c(e) + !ceiiP'cpll,
for some c, c(e) > 0, where 1/r
I

I

rn := ( 4 + 2n+l

)-t

'

n EN

= 1/p -

1/4. Now take

.

Then rn < 4 and rn ? 4 and

2~n =~en + ~) = rn~l

.

If we choose e > 0 suitably, we get inductively that IV cp Ie L'" and II V cp II, S D +
"
1/211 V cp II, , for some constant D and all 'n· Here we used the fact that rn S 2pn
"
and that supp cp is contained in the unit ball. This implies II Vcp II," S 2D, (n eN)
and therefore IIP'cpll 4 < oo, and this proves Lemma 1.16. D
Having this result, the proof of Theorem 1.1 S is as elementary as the above
theorems.

Proof (of Theorem 1.15). The only problem in following the proof of Theorem
1.13 is Step 4, since the mollifier it does not commute with (V - ia). All other
steps work as in Theorems 1.13 and 1.14, i.e. we can prove as above that
L~mp n D(H) is an operator core for H. So, for cp e L:,mp n D(H)
H cp = - A cp

+ 2ia · (V cp) + (- iV · a + a 2 + V) cp .

( 1.13)

By the assumptions of Theorem 1.15 and Lemma 1.16, each individual term in
( 1.13) is in L 2 (1Rv) and V cp e L 4 (1Rv)v. This suffices to show that the "mollified"

sequence cpt :=it • cp converges to cp in the operator norm as e ~ 0. D


2. LP-Properties of Eigenfunctions, and All That

In this chapter, we study properties of eigenfunctions and some consequences
for the spectrum of H.
We begin with some semigroup properties which turn out to be useful for
showing essential self-adjointness of H0 + V when the negative part of Vis in Kv
(Section 2.1 ). In Sects. 2.2 and 3, we give some estimates for eigenfunctions, which
we use in Sect. 2.4, to give a characterization of the spectrum of H.
In Sect. 2.5, we make some assertions about positive solutions, and in Sect.
2.6 we give an alternative proof of the result of Zelditch, that the time evolution
exp( -it H) has a weak integral kernel under suitable hypotheses on V.
We will only prove a few things, and refer the reader to the review article of
Simon [334] which has fairly complete references and results. Some of the results
are also contained in the Brownian motion paper of Aizenman and Simon [7].

2.1 Semigroup Properties
The first theorem states a basic "smoothing" property of the semigroup associated with H = H 0 + V where H 0 is the self-adjoint realization of (-A). We will
give a complete proof of it. The following Corollary 2.2 is an immediate consequence of the L 2 ~ L 00 -boundedness of the semigroup. It is an extension of
Theorem 1.14, i.e. it gives essential self-adjointness of H if V_ (the negative part
of V) is in Kv. In the last proposition, we give (without proof) a semigroup
criterion for V being in K v if V is negative. This illustrates the "naturalness" of
the class K v for these LP-properties.
Theorem 2.1 [7]. If V e Kv and t > 0, then exp(- tH) is a bounded operator from
LP to Lq for all 1 < p < q < oo.
Remark. Note that V e K v implies that Vis H 0 -form bounded with relative bound
0. So H := H 0 + V is well defined and self-adjoint as a form sum (see Theorem
1.5).

Proof (of Theorem 2.1). We divide it into six steps.
Step 1. exp(- tH): L J2(1Rv)
We have, for VeKv

~

L x (IRv) is bounded for small t.

I

lim II Je-sllol VI ds II x.
1'1.0

0

I

=0 .

(2.1)