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Graduate Studies
in Mathematics
Editorial Board David Cox (Chair) Steven G. Krants Rafe Mazzeo Martin Scharlemann
2000 Mathematics subject classification. 81-01, 81Qxx, 46-01, 34Bxx, 47B
Abstract. This book provides a self-contained introduction to mathematical methods in quan- tum mechanics (spectral theory) with applications to Schr¨odinger operators. The first part cov- ers mathematical foundations of quantum mechanics from self-adjointness, the spectral theorem, quantum dynamics (including Stone’s and the RAGE theorem) to perturbation theory for self- adjoint operators. The second part starts with a detailed study of the free Schr¨odinger operator respectively position, momentum and angular momentum operators. Then we develop Weyl–Titchmarsh the- ory for Sturm–Liouville operators and apply it to spherically symmetric problems, in particular to the hydrogen atom. Next we investigate self-adjointness of atomic Schr¨odinger operators and their essential spectrum, in particular the HVZ theorem. Finally we have a look at scattering theory and prove asymptotic completeness in the short range case. For additional information and updates on this book, visit: http://www.ams.org/bookpages/gsm-99/
Typeset by LATEXand Makeindex. Version: February 17, 2009.
Library of Congress Cataloging-in-Publication Data Teschl, Gerald, 1970– Mathematical methods in quantum mechanics : with applications to Schr¨odinger operators / Gerald Teschl. p. cm. — (Graduate Studies in Mathematics ; v. 99) Includes bibliographical references and index. ISBN 978-0-8218-4660-5 (alk. paper)
Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgement of the source is given. Republication, systematic copying, or multiple reproduction of any material in this pub- lication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permissions should be addressed to the Assistant to the Publisher, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permission@ams.org.
©c 2009 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted too the United States Government.
Preface xi
xii Preface
Part 2 starts with the free Schr¨odinger equation and computes the free resolvent and time evolution. In addition, I discuss position, momen- tum, and angular momentum operators via algebraic methods. This is usually found in any physics textbook on quantum mechanics, with the only difference that I include some technical details which are typically not found there. Then there is an introduction to one-dimensional mod- els (Sturm–Liouville operators) including generalized eigenfunction expan- sions (Weyl–Titchmarsh theory) and subordinacy theory from Gilbert and Pearson. These results are applied to compute the spectrum of the hy- drogen atom, where again I try to provide some mathematical details not found in physics textbooks. Further topics are nondegeneracy of the ground state, spectra of atoms (the HVZ theorem), and scattering theory (the Enß method).
I assume some previous experience with Hilbert spaces and bounded linear operators which should be covered in any basic course on functional analysis. However, while this assumption is reasonable for mathematics students, it might not always be for physics students. For this reason there is a preliminary chapter reviewing all necessary results (including proofs). In addition, there is an appendix (again with proofs) providing all necessary results from measure theory.
The present book is highly influenced by the four volumes of Reed and Simon [ 40 ]–[ 43 ] (see also [ 14 ]) and by the book by Weidmann [ 60 ] (an extended version of which has recently appeared in two volumes [ 62 ], [ 63 ], however, only in German). Other books with a similar scope are for example [ 14 ], [ 15 ], [ 21 ], [ 23 ], [ 39 ], [ 48 ], and [ 55 ]. For those who want to know more about the physical aspects, I can recommend the classical book by Thirring [ 58 ] and the visual guides by Thaller [ 56 ], [ 57 ]. Further information can be found in the bibliographical notes at the end.
There is some intentional overlap between Chapter 0, Chapter 1, and Chapter 2. Hence, provided you have the necessary background, you can start reading in Chapter 1 or even Chapter 2. Chapters 2 and 3 are key
Preface xiii
chapters and you should study them in detail (except for Section 2.6 which can be skipped on first reading). Chapter 4 should give you an idea of how the spectral theorem is used. You should have a look at (e.g.) the first section and you can come back to the remaining ones as needed. Chapter 5 contains two key results from quantum dynamics: Stone’s theorem and the RAGE theorem. In particular the RAGE theorem shows the connections between long time behavior and spectral types. Finally, Chapter 6 is again of central importance and should be studied in detail.
The chapters in the second part are mostly independent of each other except for Chapter 7, which is a prerequisite for all others except for Chap- ter 9.
If you are interested in one-dimensional models (Sturm–Liouville equa- tions), Chapter 9 is all you need.
If you are interested in atoms, read Chapter 7, Chapter 10, and Chap- ter 11. In particular, you can skip the separation of variables (Sections 10. and 10.4, which require Chapter 9) method for computing the eigenvalues of the hydrogen atom, if you are happy with the fact that there are countably many which accumulate at the bottom of the continuous spectrum.
If you are interested in scattering theory, read Chapter 7, the first two sections of Chapter 10, and Chapter 12. Chapter 5 is one of the key prereq- uisites in this case.
The AMS is hosting a web page for this book at
http://www.ams.org/bookpages/gsm-99/
where updates, corrections, and other material may be found, including a link to material on my own web site:
http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/
I would like to thank Volker Enß for making his lecture notes [ 18 ] avail- able to me. Many colleagues and students have made useful suggestions and pointed out mistakes in earlier drafts of this book, in particular: Kerstin Ammann, J¨org Arnberger, Chris Davis, Fritz Gesztesy, Maria Hoffmann- Ostenhof, Zhenyou Huang, Helge Kr¨uger, Katrin Grunert, Wang Lanning, Daniel Lenz, Christine Pfeuffer, Roland M¨ows, Arnold L. Neidhardt, Harald
4 0. A first look at Banach and Hilbert spaces
Example. Euclidean space Rn^ together with d(x, y) = (
∑n k=1(xk^ −^ yk)
is a metric space and so is Cn^ together with d(x, y) = (
∑n k=1 |xk^ −yk|
The set Br(x) = {y ∈ X|d(x, y) < r} (0.2)
is called an open ball around x with radius r > 0. A point x of some set U is called an interior point of U if U contains some ball around x. If x is an interior point of U , then U is also called a neighborhood of x. A point x is called a limit point of U if (Br(x){x}) ∩ U 6 = ∅ for every ball around x. Note that a limit point x need not lie in U , but U must contain points arbitrarily close to x.
Example. Consider R with the usual metric and let U = (− 1 , 1). Then every point x ∈ U is an interior point of U. The points ±1 are limit points of U.
A set consisting only of interior points is called open. The family of open sets O satisfies the properties
(i) ∅, X ∈ O, (ii) O 1 , O 2 ∈ O implies O 1 ∩ O 2 ∈ O, (iii) {Oα} ⊆ O implies
α Oα^ ∈ O.
That is, O is closed under finite intersections and arbitrary unions.
In general, a space X together with a family of sets O, the open sets, satisfying (i)–(iii) is called a topological space. The notions of interior point, limit point, and neighborhood carry over to topological spaces if we replace open ball by open set.
There are usually different choices for the topology. Two usually not very interesting examples are the trivial topology O = {∅, X} and the discrete topology O = P(X) (the powerset of X). Given two topologies O 1 and O 2 on X, O 1 is called weaker (or coarser) than O 2 if and only if O 1 ⊆ O 2.
Example. Note that different metrics can give rise to the same topology. For example, we can equip Rn^ (or Cn) with the Euclidean distance d(x, y) as before or we could also use
d˜(x, y) =
∑^ n
k=
|xk − yk|. (0.3)
Then
1 √ n
∑^ n
k=
|xk| ≤
∑n
k=
|xk|^2 ≤
∑^ n
k=
|xk| (0.4)
0.1. Warm up: Metric and topological spaces 5
shows Br/√n(x) ⊆ B˜r(x) ⊆ Br(x), where B, B˜ are balls computed using d,
d˜, respectively.
Example. We can always replace a metric d by the bounded metric
d˜(x, y) = d(x, y) 1 + d(x, y)
without changing the topology.
Every subspace Y of a topological space X becomes a topological space of its own if we call O ⊆ Y open if there is some open set O˜ ⊆ X such that O = O˜ ∩ Y (induced topology).
Example. The set (0, 1] ⊆ R is not open in the topology of X = R, but it is open in the induced topology when considered as a subset of Y = [− 1 , 1].
A family of open sets B ⊆ O is called a base for the topology if for each x and each neighborhood U (x), there is some set O ∈ B with x ∈ O ⊆ U (x). Since an open set O is a neighborhood of every one of its points, it can be written as O =
O⊇ O˜∈B O˜ and we have
Lemma 0.1. If B ⊆ O is a base for the topology, then every open set can be written as a union of elements from B.
If there exists a countable base, then X is called second countable.
Example. By construction the open balls B 1 /n(x) are a base for the topol- ogy in a metric space. In the case of Rn^ (or Cn) it even suffices to take balls with rational center and hence Rn^ (and Cn) is second countable.
A topological space is called a Hausdorff space if for two different points there are always two disjoint neighborhoods.
Example. Any metric space is a Hausdorff space: Given two different points x and y, the balls Bd/ 2 (x) and Bd/ 2 (y), where d = d(x, y) > 0, are disjoint neighborhoods (a semi-metric space will not be Hausdorff).
The complement of an open set is called a closed set. It follows from de Morgan’s rules that the family of closed sets C satisfies
(i) ∅, X ∈ C, (ii) C 1 , C 2 ∈ C implies C 1 ∪ C 2 ∈ C, (iii) {Cα} ⊆ C implies
α Cα^ ∈ C.
That is, closed sets are closed under finite unions and arbitrary intersections.
The smallest closed set containing a given set U is called the closure
U =
C∈C,U ⊆C
0.1. Warm up: Metric and topological spaces 7
dense, choose y ∈ Y. Then there is some sequence xnk with d(xnk , y) < 1 /k. Hence (nk, k) ∈ J and d(ynk ,k, y) ≤ d(ynk ,k, xnk ) + d(xnk , y) ≤ 2 /k → 0.
A function between metric spaces X and Y is called continuous at a point x ∈ X if for every ε > 0 we can find a δ > 0 such that
dY (f (x), f (y)) ≤ ε if dX (x, y) < δ. (0.9)
If f is continuous at every point, it is called continuous.
Lemma 0.5. Let X, Y be metric spaces and f : X → Y. The following are equivalent:
(i) f is continuous at x (i.e, (0.9) holds). (ii) f (xn) → f (x) whenever xn → x. (iii) For every neighborhood V of f (x), f −^1 (V ) is a neighborhood of x.
Proof. (i) ⇒ (ii) is obvious. (ii) ⇒ (iii): If (iii) does not hold, there is a neighborhood V of f (x) such that Bδ(x) 6 ⊆ f −^1 (V ) for every δ. Hence we can choose a sequence xn ∈ B 1 /n(x) such that f (xn) 6 ∈ f −^1 (V ). Thus xn → x but f (xn) 6 → f (x). (iii) ⇒ (i): Choose V = Bε(f (x)) and observe that by (iii), Bδ(x) ⊆ f −^1 (V ) for some δ.
The last item implies that f is continuous if and only if the inverse image of every open (closed) set is again open (closed).
Note: In a topological space, (iii) is used as the definition for continuity. However, in general (ii) and (iii) will no longer be equivalent unless one uses generalized sequences, so-called nets, where the index set N is replaced by arbitrary directed sets.
The support of a function f : X → Cn^ is the closure of all points x for which f (x) does not vanish; that is,
supp(f ) = {x ∈ X|f (x) 6 = 0}. (0.10)
If X and Y are metric spaces, then X × Y together with
d((x 1 , y 1 ), (x 2 , y 2 )) = dX (x 1 , x 2 ) + dY (y 1 , y 2 ) (0.11)
is a metric space. A sequence (xn, yn) converges to (x, y) if and only if xn → x and yn → y. In particular, the projections onto the first (x, y) 7 → x, respectively, onto the second (x, y) 7 → y, coordinate are continuous.
In particular, by the inverse triangle inequality (0.1),
|d(xn, yn) − d(x, y)| ≤ d(xn, x) + d(yn, y), (0.12)
we see that d : X × X → R is continuous.
8 0. A first look at Banach and Hilbert spaces
Example. If we consider R × R, we do not get the Euclidean distance of R^2 unless we modify (0.11) as follows:
d˜((x 1 , y 1 ), (x 2 , y 2 )) =
dX (x 1 , x 2 )^2 + dY (y 1 , y 2 )^2. (0.13)
As noted in our previous example, the topology (and thus also conver- gence/continuity) is independent of this choice.
If X and Y are just topological spaces, the product topology is defined by calling O ⊆ X × Y open if for every point (x, y) ∈ O there are open neighborhoods U of x and V of y such that U × V ⊆ O. In the case of metric spaces this clearly agrees with the topology defined via the product metric (0.11).
A cover of a set Y ⊆ X is a family of sets {Uα} such that Y ⊆
α Uα. A cover is called open if all Uα are open. Any subset of {Uα} which still covers Y is called a subcover.
Lemma 0.6 (Lindel¨of). If X is second countable, then every open cover has a countable subcover.
Proof. Let {Uα} be an open cover for Y and let B be a countable base. Since every Uα can be written as a union of elements from B, the set of all B ∈ B which satisfy B ⊆ Uα for some α form a countable open cover for Y. Moreover, for every Bn in this set we can find an αn such that Bn ⊆ Uαn. By construction {Uαn } is a countable subcover.
A subset K ⊂ X is called compact if every open cover has a finite subcover.
Lemma 0.7. A topological space is compact if and only if it has the finite intersection property: The intersection of a family of closed sets is empty if and only if the intersection of some finite subfamily is empty.
Proof. By taking complements, to every family of open sets there is a cor- responding family of closed sets and vice versa. Moreover, the open sets are a cover if and only if the corresponding closed sets have empty intersec- tion.
A subset K ⊂ X is called sequentially compact if every sequence has a convergent subsequence.
Lemma 0.8. Let X be a topological space.
(i) The continuous image of a compact set is compact. (ii) Every closed subset of a compact set is compact. (iii) If X is Hausdorff, any compact set is closed.
