Quantum Mechanics for Hamiltonians Defined as Quadratic Forms 9781400868834

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Quantum Mechanics for Hamiltonians Defined as Quadratic Forms

Quantum Mechanics for Hamiltonians Defined as Quadratic Forms by Barry Simon

Princeton Series in Physics

Princeton University Press Princeton, New Jersey *1971

Copyright © 1971, by Princeton University Press All rights reserved L.C. Card: 73-146648 I.S.B.N.: 0-691-08090-9

Printed in the United States of America by Princeton University Press

To my Parents

INTRODUCTION

It is our purpose in this monograph to present a complete, rigorous mathematical treatment of two body quantum mechanics for a wider class of potentials than is normally treated in the literature. At the same time, we will review the theory of the "usual" Kato classes, although no at­ tempt has been made to make this review exhaustive or complete. The scope of what we present is best delineated by stating the limits of this work: we take for granted the standard Hilbert space formalism, and our main goal is to prove forward dispersion relations from first principles. For example we do not assume the Lippman-Schwinger equation but prove it within the framework of time-dependent scattering theory. Mathematical physics of the type we discuss here seems to me to be of interest to the wider physics community for a variety of reasons: (a) It can often clarify concepts considerably. On the pedagogic level, we might mention the relation between boundary conditions and self-adjointness; this connection should help students understand one of those points most often found puzzling to the novice, the seemingly ad hoc introduction of boundary conditions. On the professional level, we might mention the Jauch-Cook work [24, 61] on scattering: the more mathematical approach clekred the cobwebs considerably, eliminating the ρ if I nonsense of adiabatic switching, i.e., the replacement of V with Ve fc I 1 and the taking of the Iim S(e). (One can't help picturing a pair of experiε j. ο menters in the lab, one shouting to the other: "I have the beam ready, turn on the target but do it very slowly!") (b) It can uncover some interesting subtleties. The standard paradigm which comes to mind is the Wigner-von Neumann example [93] of a potential with a positive energy bound state.

viii

INTRODUCTION

(c) It can often serve as a guide to other fields, mostly notably high energy physics. The classic example is Regge's famous work on analytic continuation in angular momentum. (It must be mentioned that Regge's work came as an outgrowth of his study of the Mandelstam conjecture for Yukawian potentials, a problem suggested by high energy physics.) (d) Finally, on a certain philosophical level, it is of use to know exactly which "reasonableness" arguments are really true and which re­ quire additional conditions. There is a standard and highly developed lore, most often associated with the work and name of T. Kato, which studies the quantum mechanics of four closely related classes of potentials: L 2 +L°°, L 2 +(L 00 ) , L 2 and L2 Π L 1 ,

ε

[L 2 +(L 00 ) g = iV|(Ve)V = V le +V 2 ; V lee L 2 , ||V 2e || eo (4) Follows from Proposition II.4. (4) -> (1) Immediate.! Thus, at least for monotone potentials, if we have at worst V barely not in

(and, e.g.

Theorem II.5 leads one to feel that for some purposes a slightly larger class than

is more natural. However, when V is not non-nasty,

this class doesn't seem to have a simple direct characterization. Moreover, when we study scattering theory, we will see R arising quite naturally. We thus avoid introducing additional classes of potentials and study the classes

R and

only.

After defining the Hamiltonian by a forms method in II.2, and alternately by Neumann (and summed Neumann) series in II.3, we investigate various general properties of these Hamiltonians. In this chapter, we treat mainly those features not directly related to spectra; in Chapter III, we turn to the bound state spectrum and in Chapter IV to the continuous spectrum and the related problem of scattering.

38

II. THE HAMILTONIAN

II.2. Definition as a Quadratic Form In this section we will define the Hamiltonian

for

by a method closely related to the "Friedrichs extension." The idea of extending semi-bounded operators is due to von Neumann [92]; the existence of a natural extension was proved by Stone [T26] and Friedrichs [37]; various formal simplifications may be found in the work of Freudenthal [36], Calkin [11], Eberlein [28], Lions [T19] and Nelson [T22], We use explicitly the idea of a scale of spaces. Faris has used this method to define

for

Simultaneous to the preparation of this

work, Nelson has discussed this latter case in more detail [T23], Kato, in his book [T17], has some discussion of the use of Freidrichs extensions in defining quantum mechanical Hamiltonians. DEFINITION. Let

endowed with the Hilbert space struc-

ture given by

We will denote

. It is clear that

By the Riesz lemma, we can associate continuous conjugate linear functions on tion and regard

for

and the space

We suppress this associa-

as a separate space with norm

the original space

Then

of

Let

defines a conjugate lin-

ear function which is || ||+1 continuous since (II. 1) Thus we can regard (by (II. 1)). In particular,

and with this association and this inclusion is distinct from

the natural map which we have suppressed. If

we have:

39

II.2. DEFINITION AS A QUADRATIC FORM

This explains the notation To summarize, we have introduced a pair of abstract Hilbert spaces along with K, so that

and so that

Moreover, there is a "pairing" between ting the functional in

and

which we represent by

(given by evaluaso that for

We can thus drop the p i n w i t h o u t creating confusion. We then have:

Since

whenever both sides make sense, we can define when

by formally extending this relation.

40

II. THE HAMILTONIAN

We remark in passing that in the same way that , we can extend this

where

extended to

to: 8

has the norm:

These spaces are nothing but the familiar Sobolev spaces (see e.g. [T29], pg. 55). We can picture them explicitly in p-space as those locally L functions for which

r\

From this realization, we see

that m > n implies

in particular,

|| ||-dense in H and

is

dense in

The crucial fact we will use about the scale

is the

following: LEMMA II.6. Let

be a continuous map from

such that

is a bijection for some real \ and so that for all

Then there is a unique self-adjoint operator A (on K)

so that Proof.

and

for all

We need only prove existence for the case

general, setting

and

and

show A is self-adjoint we consider the operator

Then: g One also defines

for, in

we recover the full theorem.

We define D(A) by

tor is Hermitean for let

and

and let

To . This operais onto),

II.2. DEFINITION AS A QUADRATIC FORM

As a result

41

is an everywhere defined Hermitean operator and

thus, by the Hellinger-Toeplitz theorem (Theorem A.2), it is bounded and self-adjoint. Therefore, A, as the inverse of

is self-adjoint.

Thus an A obeying the theorem exists. Let A be another self-adjoint operator with

and

tension of A for Note.

Then A is an ex-

implies

Thus

For a brief history of results related to this theorem, see Cannon

[16], pg. 57; Cannon calls this result the "KLMN theorem" for Kato, Lions, Lax, Milgram, Nelson. This lemma allows us to prove the analogue of the Kato-Rellich theorem for forms: 9 THEOREM II.7. Let V define for some

symmetric form on

with

and some b,

(H.2) Then, there exists a unique self-adjoint operator H with

and (H.3)

Moreover

and (H.4)

Proof.

Define

To show V is

continuous, we must work on extending (II.2). (II.2) implies

A similar result can be found in many places, e.g. Kato [T17] pp. 338-343 or Nelson [l32], App. ^^ This formula is intended in the sense of expectation v a l u e s ; m a y not make sense as a bona fide vector in

42

II. THE HAMILTONIAN

for some C. Let Then h is an Hermitean bilinear form with Thus, h is an inner product and so a Cauchy-Schwartz inequality holds: Thus:

This implies V is continuous. We next show that

is bijective as a map of

For (by (II.2)): (II.5) Thus

implies

implies

is injective. Also, by (II.5), Ran closed. If it is not all of

so i is

we can find