# Quantum Mechanics for Hamiltonians Defined as Quadratic Forms

Barry Simon
Pages: 262
https://www.jstor.org/stable/j.ctt13x123j

1. Front Matter
(pp. i-vi)
2. INTRODUCTION
(pp. vii-xii)

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 attempt 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...

(pp. xiii-2)
4. CHAPTER I THE ROLLNIK CONDITION
(pp. 3-30)

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

$\int {\frac{{\left| {V(x)} \right|\left| {V(y)} \right|}}{{{{\left| {x - y} \right|}^2}}}} {d^3}x{d^3}y\langle \infty$(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:

${\left\| V \right\|^2}_R = \frac{{\left| {V\left. {\left( x \right)} \right|\left| {V\left. {\left( y \right)} \right|} \right.} \right.}}{{\left| {x - {{\left. y \right|}^2}} \right.}}{d^3}x{d^3}y.$(1.2)

In Section 1.2, we will show that R with${\rm{|| |}}{{\rm{|}}_R}$is a complete normed vector space. We remark that the significance of (1.1) is that it assures us that${V^{\frac{1}{2}}}{\left( {E - {H_o}} \right)^{ - 1}}{V^{\frac{1}{2}}}$is a Hilbert-Schmidt operator in the limit (and for any other this aspect of the Rollnik condition will be considered in...

5. CHAPTER II THE HAMILTONIAN
(pp. 31-78)

From a fundamental point of view, the basic dynamical object in a quantum mechanical system is the propagator, U(t), which is a one parameter group of unitary operators.¹ On the other hand, the objects one obtains from classical mechanics are the Hamiltonian and its dynamics as determined by the Schrödinger equation. Stone’s theorem (Theorem A.21)² provides the connection between the two approaches; on its authority we know that there is a one-to-one correspondence between unitary oneparameter groups and self-adjoint operators via:

$U\left( t \right) = {e^{ - iHt}}$

The connection with Schrodinger’s equation is provided by:

$i\frac{d}{{dt}}\left( {U\left( t \right)} \right.\left. \Psi \right) = HU\left( t \right)\Psi$

for$\Psi \varepsilon D\left( H \right)$

Since Stone’s theorem deals with with self-adjointness...

6. CHAPTER III BOUND STATES
(pp. 79-93)

In this chapter, we will study bound states for Hamiltonians of the type defined in Chapter II. Even if bound states were not important in and of themselves, some of the material of this chapter would be essential in scattering theory. For, as we have already seen, the kernel${R_k}\left( {x,y} \right)$defined in (1-17) arises naturally in several places. A central property of this kernel is that${\left( {1 - {R_k}} \right)^{ - 1}}$exists [when${V_\varepsilon }R + {\left( {{L^\infty }} \right)_\varepsilon }$] whenever E lies in the canonically cut plane and is not an eigenvalue of H, and this property can only be proven by a study of the bound states of...

7. CHAPTER IV TIME-DEPENDENT SCATTERING THEORY
(pp. 94-131)

Scattering theory is a subject with a long and complicated history. It is only in the last decade and a half that its results have been put in a mathematically precise form. These mathematical developments have been accompanied by clarifications in the physical foundations of the theory.¹ The result is a conceptual picture quite appealing from the pedagogical point of view. To put this picture into perspective, let us first summarize its historical development.

The time-honored approach to scattering via wave functions that are asymptotically of the form${e^{i\vec k}}^{.\vec r}{r^{ - 1}}f\left( \theta \right){e^{ikr}}$is of a nature that “wouldn’t convince an educated first grader.”²...

8. CHAPTER V TIME-INDEPENDENT SCATTERING THEORY
(pp. 132-145)

In Sections IV.1 through IV.4, we studied scattering theory by direct consideration of${e^{ + iHt}}{e^{ - i{H_o}t}}$. In contradistinction to such an approach, the “time-independent” method uses, as a starting point, formal equations for${\Omega ^{\underline + }}$and/or S, which no longer mention time explicitly. Actually, the method is used so differently in the hands of physicists and mathematicians that one is really dealing with two distinct programs when describing “time-independent scattering theory.”

The mathematical approach goes back at least as far as the work of Friedrichs [38] and involves a three step procedure: (1) the “derivation” of a formal equation for some scattering theoretic...

9. CHAPTER VI ANALYTIC SCATTERING THEORY
(pp. 146-173)

In this chapter, we discuss analyticity of the “scattering amplitude” T(k, k’) for the case where$V\varepsilon {L^1} \cap R$. The material we present is fairly standard and thus this chapter is primarily discursive. We include it for the following reasons:

First, one original motivation for studying R was that${L^1} \cap R$is the natural set of conditions for the validity of forward dispersion relations via an L² non-determinantal Fredholm theory approach (see Section VI.2). One contribution of this monograph is to justify from “first principles” the starting point for this standard argument. By combining suitable sections from Chapters II-VI, one can provide a...

10. CHAPTER VII MULTIPARTICLE SYSTEMS
(pp. 174-200)

We have thus far presented a more or less complete mathematical treatment of the physics of two-body quantum mechanical systems with Rollnik potentials. In this chapter, we should like to discuss some simple aspects of multiparticle systems with two body Rollnik interactions and no n-body forces (n > 2). We first note that:

THEOREM VII. 1. Let${H_0} = \sum\limits_{i = 1}^N {{{\frac{{{P_i}}}{{2{m_i}}}}^2}}$. Let$V = \sum\limits_{i > 1} {{V_{ij}}({q_i}} - {q_j})$where

each.${V_{ij}}\varepsilon R + {L^\infty }$Then there is a unique self-adjoint operatorH so that$Q(H) = Q({H_o})$and$\langle \Psi ,H\Psi \rangle = \langle \Psi ,{H_0}\Psi \rangle + \langle \Psi ,V\Psi >$

Proof. The uniqueness is a direct consequence of the fact that a selfadjoint operator is determined by its quadratic form. As in Chapter II,...

11. APPENDIX: SOME MATHEMATICAL BACKGROUND
(pp. 201-222)
12. REFERENCES
(pp. 223-240)
13. LIST OF SYMBOLS
(pp. 241-242)
14. INDEX
(pp. 243-244)