Quantum Mechanics for Hamiltonians Defined as Quadratic Forms

by Barry Simon

Overview

This monograph combines a thorough introduction to the mathematical foundations of n-body Schrodinger mechanics with numerous new results.

Originally published in 1971.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

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ISBN-13:	9781400868834
Publisher:	Princeton University Press
Publication date:	03/08/2015
Series:	Princeton Series in Physics , #1846
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Quantum Mechanics for Hamiltonians Defined as Quadratic Forms

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By Barry Simon

PRINCETON UNIVERSITY PRESS

Copyright © 1971 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08090-1

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CHAPTER 1

THE ROLLNIK CONDITION

1.1 Relation to Lpspaces

In this chapter, we will consider in detail various properties of measurable functions, V(x), obeying:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.1)

Such potentials were first considered by Rollnik [98] and so we call (1.1) the Rollnik condition. We denote the set of such potentials by R and define the Rollnik norm:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.2)

In Section 1.2, we will show that R with [parallel][parallel]R is a complete normed vector space. We remark that the significance of (1.1) is that it assures us that V½[parallel] (E - Ho)-1 V ½ is a Hilbert-Schmidt operator in the limit E [up arrow] 0 (and for any other E!); this aspect of the Rollnik condition will be considered in Section 1.4.

In this section, we study the relation of the Rollnik potentials to Lp conditions. The Lp spaces, we recall, are defined by:

Definition. f [member of] Lp if and only if ∫ [absolute value of f(x)]p d3x < ∞. In this case, we set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. L∞ is defined as the set of functions bounded almost everywhere with [parallel]f[parallel]∞ = inf {L| [absolute value of f(x)] < L almost everywhere}.

The basic result relating Lp and R is:

Theorem 1.1 (Kato). If V [member of] L3/2then V [member of] R. In fact (for a constant C independent of V):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.3)

Proof. This is a direct consequence of the Sobolev inequalities which we discuss in Appendix 2 of this section.

Remark.

In Section 1.6, we will present an example of a V which is in R, but is (barely) not in L3/2.

Given a function, f, we can define f> (x) = f(x) when [absolute value of f(x)] > 1 and 0 otherwise. Also, let f< = f - f>. Then f [member of] Lp implies f> [member of] Lq for q ≤ p and f< [member of] Lr for r ≥ p Thus Theorem 1.1 has two immediate consequences:

Corollary 1.2. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; in particular

L2 + L ∞ [subset] R + L∞.

Proof. Let V = f + g; f [member of] Lp, g [member of] L∞. Then f> [member of] L3/2 and f< + g [member of] L∞.

Remark.

By putting an [member of] in the break-up f> + f< we can show L2 + (L∞)ε [subset] R + (L∞)ε where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with f1 [member of] X and [parallel]f2[parallel]∞< ε}.

Corollary 1.3. If p ≤ 3/2 ≤ q, then Lp [intersection] Lq [subset] R; in particular

L1 [intersection] L2 [subset] R [intersection] L1.

Proof. If V [member of] Lp [intersection] Lq, then V> [member of] L3/2 (since V [member of] Lq) and V< [member of] L3/2 (by V [member of] Lp) so V [member of] L3/2 [subset] R.

The crucial case of Corollary 1.3, that L1 [intersection] L2 [subset] R, has been noted by Grossman and Wu [45], There is a useful modification of their proof which allows us to relate [parallel][parallel]R, [parallel][parallel]2 and [parallel][parallel]1 in a new way:

Theorem 1.4. Let V [member of] L1 [intersection] L2. Then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.4)

Proof. Let r be arbitrary. Then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

On the other hand:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, as a function on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and so is [absolute value of V(y)]/[absolute value of x-y]. Cauchy-Schwartz then implies:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As a result, for any r, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This is optimized by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which yields (1.4).

Remark.

Theorem 1.4 (with a different constant) can be proven alternately by using (1.3), for:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (optimizing k). Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which with (1.3) implies a result of the same type as (1.4).

One can ask to what extent V [member of] R is weaker than V [member of] L3/2. Recall that:

Definition. f[member of] (Lp)W (weak Lp) if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some c.

We discuss these spaces in Appendix 1 of this section.

Theorem 1.5. Let V(x) be central [i.e. V(x) = V([absolute value of x])] and monotone [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then V [member of] R implies V [member of] (L3/2)W + L∞.

Proof. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If V is bounded it is in L∞, so suppose V -> [+ or -] ∞ at r = 0. Let rt be defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] implies [absolute value of V(r)] > t. Then V [member of] R implies:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Monotonicity and centralness are not really essential. What is really important is that V blows up without wiggling too much and without blowing up on "thin sets." Specifically, let us define:

Definition. A potential, V(x) will be called non-nasty if and only if there are constants to, λ, E so that, for every t > to, there is an rt

with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, for each fixed t, we picture the schematic behavior shown on preceding page.

The proof of Theorem 1.5 can be modified easily to yield:

Theorem 1.6. Let V [member of] R and suppose V is non-nasty. Then V[member of] (L3/2)W + L∞. Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] diverges at worst logarithmically as Λ -> ∞, If V [member of] R [intersection] L1is non-nasty, then V [member of] Lp for any 1 ≤ p < 3/2.

* * *

One last (and rather trivial) relation to the Lp spaces will be needed:

Theorem 1.7. Let V [member of] R + L∞and let V be of bounded support (i.e. for some R, [absolute value of x] > R implies V(x) = 0). Then V [member of] R [intersection] L1. In particular, any V [member of] R + L∞is locally L1.

Proof. Since any L∞ function of compact support is in R (it is in L3/2), we can suppose without loss that V [member of]. Since [parallel]V[parallel]R< ∞, we know [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some x (by Fubini's theorem). Thus, since [absolute value of x-y]2< ([absolute value of x] + R)2 where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Appendix 1 to 1.1. The weak Lp spaces

Since most physicists are unfamiliar with the weak Lp spaces (but have a passing acquaintance with the usual Lp spaces), we present a brief discussion of their properties.

The distribution function, mf(t)(or just m(t)) for a measurable function f on a measure space (M, μ)is defined by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.5)

An elementary computation proves:

Lemma 1.8. f [member of] Lp(μ) if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (Stieltjes integral)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(The boundary terms can be argued away.)

The weak Lpspaces are defined by:

Definition. f [member of] (Lp)W if and only if mf(t) ≤ C/tp for some c < ∞.

Thus Lp [subset] (Lp)W, but there are f [member of] [Lp]W with ∫∞o tp-1 m(t) dt logarithmically divergent (at 0 and/or ∞) so that the inclusion is strict. For example (Lp)W (R) includes functions like x-1/p The real value of the weak spaces is that various inequalities for f [member of] Lp extend to f [member of](Lp)W. Thus integrals that a priori might be logarithmically divergent turn out to converge.

There is a very simple characterization of (Lp)W contained implicitly in the work of Calderon, Lions and Peetre, and Stein and Weiss:

Lemma 1.9. Let po< p < p1. Then f [member of] (Lp)Wif and only if, there exist Co, C1so that, for all λ, f = f0,λ + f1, λwith

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. An elementary manipulation, see e.g. Nelson.

The idea behind the proof is that, in the language of Section 1.1, when f [member of] (Lp)W, f>[member of] Lq for q < p and f<[member of]Lq for q > p.

Appendix 2 to 1.1. The Sobolev Inequality

Let us discuss the basic inequalities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for f [member of] Lp(Rn), h [member of] Lr (Rn) and

1/p + 1/r + λ/n = 2; λ < n.

The earliest proof of this result is due to Hardy and Littlewood in the n = 1 case (see also Hardy et al, eq. 3-82, pg. 288) and to Sobolev who reduced the general case to the n = 1 case. Du Plessis later found a completely elementary trick to reduce any n to n = 1 (see Bers et al, pp. 242-243). There is an elegant proof due to Stein and Weiss which uses a strengthened Marcinkiewicz interpolation theorem. We present a simple proof using two interpolation formulae:

Marcinkiewicz interpolation theorem (see Marcinkiewicz, Zygmund). Let pi< qi, i = 1, 2. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let q1 ≠ q2. Then for any is 0 < t < 1, T: Lp -> Lq is a bounded map where

1/p = t/p1 + [(1-t)/p2]; 1/q = t/q1 + [(1-t)/q2]. (1.5)

Hunt's Interpolation Theorem (Hunt, Calderon, Lions and Peetre, Stein and Weiss).

Let q1 ≠ q2, p1 ≠ p2. Let:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], bounded, i = 1, 2.

Then for 0 < t < 1, T: (Lp)W -> (Lq)W is bounded where p and q are given by (1.5).

For a general discussion of interpolation formulae, see Nelson; for the Marcinkiewicz theorem, also see Zygmund, pp. 111-120. We call Hunt's theorem by that name (it is not standard) since it is a special case of a theorem of R. Hunt, This case can be proven easily from Lemma 1.9.

Convention. For the remainder of this appendix, p and p' (or q, q', etc.) etc.) will stand for a conjugate pair of indices, i.e. p-1 + (p')-1 = 1.

Lemma 1.10. Let f [member of] Lp. If g [member of] Lp', then f * g [member of] L∞, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If f [member of] Lp and g [member of] L1, then f * g [member of] Lpand [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. If g [member of] Lp' by Hölder's inequality:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which proves the first statement. Let g [member of] L1 and h [member of] Lp' be arbitrary. Then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus f * g [member of] Lp and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark.

By a third interpolation theorem (the Reisz-Thorin theorem) which we haven't mentioned (or alternately by combining our two interpolation theorems), we can prove Lp * Lq [subset] Ls if p-1 + q-1 > 1; p- 1 + q-1 = 1 + s-1 using Theorem 1.10. This result is Young's theorem.

Lemma 1.11. Let f [member of] Lp. Let g [member of] (Lq)Wwhere 1 < q < p'. Then f * g [member of] (Ls)Wwhere p-1 + q-1 = 1 + s-1.

Proof. This follows from Hunt's interpolation theorem and Lemma 1.10.

Lemma 1.12. Let f [member of] Lp. Let g [member of] (Lq)Wwhere 1 < q < p' < ∞. Then f * g [member of] Lswhere p-1 + q-1 = 1 + s-1.

Proof. Fix g [member of] (Lq)W. Then * g:Lp -> (Ls)W where

s-1 = p-1 + (q-1) < p-1.

Thus *g increases the index so Marcinkiewicz can be applied. Given p and q obeying the conditions, we find p1, p2 with 1 < p1< p < p2< q' and apply the Marcinkiewicz theorem using [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

pi< si, p1< p < p2

to conclude the proof of the lemma.

Remark.

Using Hunt's theorem, we could prove now a weak Young's theorem, i.e. (Lp)W * (Lq)W [subset] (Ls)W.

Lemma 1.12 is equivalent to:

Theorem 1.13. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with p-1 + q-1 + r-1 = 2; 1 < p, q, r < ∞.

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. p-1 + q-1 > 1 so 1 < q < p' < ∞. Thus f*g [member of] Lr',

Corollary 1.14. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let f [member of] Lp (Rn), h [member of] Lr (Rn)

Then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Let g(z) = 1/[absolute value of z]λ. Then mg(t), volume of a sphere of radius [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus g [member of](Ln/λ)W and Theorem 1.13 implies the result.

1.2. The p-space form

In this section, we will examine the various p-space forms of the Rollnik condition. These connections between the p-space and x-space forms and, in particular, the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] appear to have been first noted explicitly in the physics literature by Ghirardi and Rimini [42]. Our treatment differs from theirs mainly in its rigor (and thereby in its pedantry).

(Continues...)

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Table of Contents

• Frontmatter, pg. i
• Introduction, pg. vii
• Contents, pg. xiii
• Chapter I. The Rollnik Condition, pg. 3
• Chapter II. The Hamiltonian Chapter, pg. 31
• Chapter III. Bound States, pg. 79
• Chapter IV. Time-Dependent Scattering Theory, pg. 94
• Chapter V. Time-Independent Scattering Theory, pg. 132
• Chapter VI. Analytic Scattering Theory, pg. 146
• Chapter VII. Multiparticle Systems, pg. 174
• Appendix. Some Mathematical Background, pg. 201
• References, pg. 223
• List of Symbols, pg. 241
• Index, pg. 243