# Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann

E. H. Lieb
B. Simon
A. S. Wightman
Pages: 478
https://www.jstor.org/stable/j.ctt13x134j

1. Front Matter
(pp. i-iv)
(pp. v-vi)
3. INTRODUCTION
(pp. vii-ix)
ELLIOTT LIEB, BARRY SIMON and ARTHUR WIGHTMAN

This volume is dedicated to Valentine Bargmann on the occasion of his retirement as Professor of Mathematical Physics at Princeton University.

Valentine Bargmann was born in Berlin, Germany on April 6, 1908. He studied at the University of Berlin from 1926 to 1933. He moved to Zurich on Hitler’s rise to power and wrote his doctor’s thesis at the University under the guidance of Gregor Wentzel. On the completion of his degree, Bargmann emigrated to the United States. (That flat statement is correct, but does not evoke the temper of the times. Bargmann received a five-year German passport in 1931,...

4. PUBLICATIONS
(pp. x-2)
5. THE INVERSE r-SQUARED FORCE: AN INTRODUCTION TO ITS SYMMETRIES
(pp. 3-18)
Henry D. I. Abarbanel

The Coulomb or Kepler problem of solving for the motion of a point particle in an inverse square force has remained fundamental and fascinating from the earliest developments of the modern era of science. In a very underspoken sense one may view the achievements of Copernicus, Kepler, Brahe – the giants on whose shoulders Newton stood – and Newton as being the unfolding and solution of this very problem. Remaining underspoken we may note that the techniques and ideas developed in this endeavor had supremely broad implications.

In this article we wish to give a pedagogical development of the symmetry...

6. CERTAIN HILBERT SPACES OF ANALYTIC FUNCTIONS ASSOCIATED WITH THE HEISENBERG GROUP
(pp. 19-82)
Donald Babbitt

This survey is concerned with certain Hilbert spaces of analytic functions which arise with a certain realization of the boson creation and annihilation operator as operators on a Hilbert space. In particular, let${{\mathfrak{F}}_{1}}$be the entire functions on C with the following property: if$\text{f}=\sum\limits_{\text{m}}{{{\alpha }_{\text{m}}}{{\text{z}}^{\text{m}}}}$then${{\sum{\text{m}!\left| {{\alpha }_{\text{m}}} \right|}}^{2}}\textless\infty$. Define an inner product on${{\mathfrak{F}}_{1}}$by:

$(\text{f,g})=\sum\limits_{\text{m}}{\text{m}!{{{\bar{\alpha }}}_{\text{m}}}{{\beta }_{\text{m}}}}$

where$\text{g=}\sum\nolimits_{\text{m}}{{{\beta }_{\text{m}}}{{\text{z}}^{\text{m}}}}$. Define the creation and annihilation operators A* and A as follows:

$\text{A}*\text{f}=\text{zf}$

and

$\text{Af}=\frac{\text{d}}{\text{dz}}\text{f}.$

(We will leave the domain unspecified for now.) We see that A* and A satisfy the boson canonical commutation relations (C.C.R.) i.e.

$[\text{A,A}*]=1$

and at least...

7. LOWER BOUND FOR THE GROUND STATE ENERGY OF THE SCHRÖDINGER EQUATION USING THE SHARP FORM OF YOUNG’S INEQUALITY
(pp. 83-90)
John F. Barnes, Herm Jan Brascamp and Elliott H. Lieb

Professor Bargmann has always been interested in basic problems in quantum mechanics. In this paper, dedicated to him, we apply some modern technology to an old problem.

We shall be primarily interested in studying the lowest, or ground state, eigenvalue, E0, of the Schrodinger operator (Һ2/2m = 1)

$\text{H}=-\Delta +\text{V(x)}.$

H acts on L2(Rn). Normally it is difficult to obtain a lower bound to E0, although an upper bound can easily be obtained from the variational principle. We shall obtain the former by using a recent sharpening and generalization of Young’s inequality, in the case that V satisfies the condition

$\text{I}(\alpha )\equiv \int{\exp [-\alpha \text{V(x)}]{{\text{d}}^{\text{n}}}\text{x}\textless\infty }$ (1)...

8. ALTERNATIVE THEORIES OF GRAVITATION
(pp. 91-106)
Peter G. Bergmann

Attempts to modify Einstein’s general theory of gravitation go back at least to H. Weyl,¹ who replaced the Riemannian metric structure of the space-time manifold by a conformal metric so as to create room for a 1-form that might play the role of electromagnetic potentials. Since then, there have been numerous attempts to change the geometric structure that forms the basis of general relativity, as well as the assumed dynamical laws, so as to achieve a variety of purposes. One of these purposes has been to enrich the geometric structure in order to account for the variety of interactions observed...

9. GENERALIZED WRONSKIAN RELATIONS: A NOVEL APPROACH TO BARGMANN-EQUIVALENT AND PHASE-EQUIVALENT POTENTIALS
(pp. 107-150)
F. Calogero

In this paper we derive a generalization of the usual wronskian theorem, providing relations between the values of the solutions of linear second-order differential equations, and of their first derivatives, evaluated at the two ends of an interval of values of the independent variable, and an integral, over the same interval, of the solutions themselves multiplied by appropriate coefficients derived directly from the differential equations. These relations are then shown to provide a flexible and powerful tool to extract information about the solutions of the differential equation, by using them in the context of the nonrelativistic quantal scattering problem based...

10. OLD AND NEW APPROACHES TO THE INVERSE SCATTERING PROBLEM
(pp. 151-168)
Freeman J. Dyson

When I agreed to write a chapter about the inverse scattering problem, I thought I would produce a systematic review article describing the historical development of the subject from its origins in the two classic papers of Bargmann.1,2However, I soon discovered that the review article which I wanted to write had already been written. The article of Faddeev,³ written in 1959, is a masterpiece of exposition, and it is fortunately available in a good English translation by Seckler.⁴ So far as I can discover, the Diplomarbeit of Steinmann,⁵ on the inverse scattering problem for periodic potentials, is the only...

11. A FAMILY OF OPTIMAL CONDITIONS FOR THE ABSENCE OF BOUND STATES IN A POTENTIAL
(pp. 169-194)
V. Glaser, A. Martin, H. Grosse and W. Thirring

A standard problem in Schrödinger’s theory is to obtain conditions on a potential in order to guarantee that this potential has at most one or n bound states. For instance, Jost and Pais,(1)Bargmann(2)Birman and Schwinge(3)have shown that a spherically symmetric potential such that

$\int{{{\text{V}}^{-}}}\text{(r)rdr}\textless 1$ (1)

where V> 0 is the attractive part of the potential, with units such that 2m/ħ2= 1, has no bound states. Similarly, it is also known(4)that if

$\sup {{\text{r}}^{2}}{{\text{V}}^{-}}(\text{r})\textless 1/4$ (2)there is no bound state. Bargmann has also obtained that in the state of angular momentum$\ell$, the number of bound states${{\nu }_{\ell }}$...

12. SPINNING TOPS IN EXTERNAL FIELDS
(pp. 195-208)
Sergio Hojmant and Tullio Regge

The Bargmann Michel Telegdi (1959) equations solve completely the problem of determining the precession of the spin of a particle with magnetic moment from its translational motion in a homogeneous field. Therefore, any theory which endeavors to describe the motion of a like spinning particle in an arbitrary field has to reduce to the BMT equations in the appropriate limit. One such a theory in Lagrangian form has been proposed by one of us (T. R.) in collaboration with A. Hanson (1974).

We would like to discuss briefly here the connection, as well as the differences, between this theory and...

13. MEASURES ON THE FINITE DIMENSIONAL SUBSPACES OF A HILBERT SPACE: REMARKS TO A THEOREM BY A. M. GLEASON
(pp. 209-228)
Res Jost

The theme of this note has its origin in George Mackey’s [4] foundations of quantum mechanics. He introduces the notion of a (positive, normed) σ-additive measure on the lattice of closed subspaces of a separable Hilbert space. A. M. Gleason [2] succeeded in a famous paper to prove that Mackey’s measures are in fact exactly the statistical operators of Johnny von Neumann. In spite of my admiration for Gleason’s proof, I could not help feeling that somehow the true essence of the problem had not yet been uncovered. This paper is the result of an attempt to get rid of...

14. THE FROISSART BOUND AND CROSSING SYMMETRY
(pp. 229-238)
N. N. Khuri

There are several logical chains that connect Axiomatic Quantum Field Theory to physical and measurable quantities. One of the best known and best established is the chain that leads from the axioms to the Froissart bound.¹ This gives, among other bounds, an upper bound for the total cross section for the collision of two particles which is of the form:

${{\sigma }_{\text{tot}}}(\text{s})\le \text{C}{{(\text{lns})}^{2}}$ (1)

where s is the square of the total center of mass energy, and C = (π/μ2) in the case of pion-pion scattering withμbeing the mass of the pion.

15. INTERTWINING OPERATORS FOR SL(n, R)
(pp. 239-268)
A. W. Knapp and E. M. Stein

In 1947 Bargmann [1] derived the list of the irreducible unitary representations of the group G = SL(2, R) of real two-by-two matrices of determinant one. For the most part, he grouped these into three series – the principal series,¹ the discrete series,² and the complementary series.¹ He showed that only the first two series are needed for the analysis of L2(G) and gave a number of formulas that can be interpreted [7] as the Plancherel formula for G. The principal series representations were realized in L2of the circle, and he observed that one representation of that form was...

16. INEQUALITIES FOR THE MOMENTS OF THE EIGENVALUES OF THE SCHRÖDINGER HAMILTONIAN AND THEIR RELATION TO SOBOLEV INEQUALITIES
(pp. 269-304)
Elliott H. Lieb and Walter E. Thirring

Estimates for the number of bound states and their energies, ej≤ 0, are of obvious importance for the investigation of quantum mechanical Hamiltonians. If the latter are of the single particle form H = −Δ + V(x) in Rn, we shall use available methods to derive the bounds

${{\sum\limits_{\text{j}}{\left| {{\text{e}}_{\text{j}}} \right|}}^{\gamma }}\le {{\text{L}}_{\gamma ,\text{n}}}\int{{{\text{d}}^{\text{n}}}\text{x}\left| \text{V(x)} \right|_{-}^{\gamma +\text{n}/2},\gamma >\max (0,1-\text{n}/2).}$ (1.1)

Here,${{\left| \text{V(x)} \right|}_{-}}=-\text{V(x)}$V(x) ≤ 0 and is zero otherwise.

Of course, in many-body theory, one is more interested in Hamiltonians of the form$-\sum\limits_{\text{i}}{{{\Delta }_{\text{i}}}+\sum\limits_{\text{i}>\text{j}}{\text{v(}{{\text{x}}_{\text{i}}}\text{-}{{\text{x}}_{\text{j}}})}}$. It turns out, however, that the energy bounds for the single particle Hamiltonian yield a lower bound for the kinetic energy, T, of N fermions...

17. ON THE NUMBER OF BOUND STATES OF TWO BODY SCHRÖDINGER OPERATORS – A REVIEW
(pp. 305-326)
Barry Simon

Given a measurable function V on Rn, consider the operator −Δ + V on L2(Rn). Under wide circumstances, this operator is known to be essentially self-adjoint on$\text{C}_{0}^{\infty }({{\text{R}}^{\text{n}}})$(see [1] for a review) and under more general circumstances, it can be defined as a sum of quadratic forms [2, 3, 4]. Physic ally, it represents the Hamiltonian (energy) operator of the particles in nonrelativistic quantum mechanics after the center of mass motion has been removed. For this reason, −Δ + V is called a two-body Schrödinger operator. We will denote by N(V) the dimension of the spectral projection for −Δ...

18. QUANTUM DYNAMICS: FROM AUTOMORPHISM TO HAMILTONIAN
(pp. 327-350)
Barry Simon

Every student of quantum mechanics learns in a first course that quantum dynamics is governed by the Schrödinger equation iħψ= Hψ. However, even professional quantum mechanics who have delved into the axiomatic foundations of quantum theory are sometimes unaware of the full chain of argument leading from the primitive version of dynamics as a one-parameter continuous (or measurable) group of automorphisms of the axiomatic structure to the Schrödinger equation. Our goal in this note is to put down in one place this full chain of argument. We expect the experts will find nothing new here and we do not...

19. SEMICLASSICAL ANALYSIS ILLUMINATES THE CONNECTION BETWEEN POTENTIAL AND BOUND STATES AND SCATTERING
(pp. 351-422)
John Archibald Wheeler

Not only on the Galapagos Islands has a single bird species separated into now distinct species that no longer interbreed. A similar speciation is close to completion in scattering theory. In that territory, what are the two birds that once intermingled but now so often turn the other way when they see each other? The crimson bird with the bright plumage of mathematical exactitude occupies one ecological niche; the black bird of semiclassical approximation methods, another. They get their nourishment in the same domain of nonrelativistic scattering physics. There two interacting centers, of reduced massμ= M1M2/(M1+...

20. INSTABILITY PHENOMENA IN THE EXTERNAL FIELD PROBLEM FOR TWO CLASSES OF RELATIVISTIC WAVE EQUATIONS
(pp. 423-460)
A. S. Wightman

A relativistic wave equation is a system of partial differential equations of the form

$({{\beta }^{\mu }}{{\partial }_{\mu }}+\rho )\psi (\text{x})=0$ (1)

whereψstands for a function on space-time with N componentsψi, i = 1,…, N, and theβμ,μ= 0, 1, 2, 3 andρare N×N numerical matrices satisfying

$\begin{matrix} {{\text{S}}_{1}}({{\text{A}}^{-1}}){{\beta }^{\mu }}{{\text{S}}_{2}}(\text{A})=\Lambda {{(\text{A})}^{\mu }}_{\nu }{{\beta }^{\nu }} \\ {{\text{S}}_{1}}({{\text{A}}^{-1}})\rho {{\text{S}}_{2}}(\text{A})=\rho .\\ \end{matrix}$ (2)

Here A → S1(A) and A → S2(A) are two representations of the group SL(2, C).

The external field problem for the wave equation (1) is the problem of solving

$[{{\beta }^{\mu }}{{\partial }_{\mu }}+\rho +\text{B(x)}]\psi (\text{x})=0$ (1′)

where B(x) is some N x N matrix valued function of space-time.

In the context of relativistic field theory there are...

21. Back Matter
(pp. 461-461)