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Operators on Hilbert Space

Kemble's Fundamental Principles of Quantum Mechanics [6], one of the first and certainly the most scholarly of the early books on quantum mechanics to be published in the United States, despaired of Weyl's theory, commenting ``The problem has been treated bv Weyl in a basic paper which unfortunately involves an elaborate mathematical technique and makes difficult reading for the non-specialist.'' By and large the general tendency has been to use Dirac's symbolism, and to treat all wave functions as though they were normalizable. even when such was not the case.

Schwartz's theory of distributions nowadays provides the opportunity to legitimize most of this work from the rigorous mathematical point of view, although it does not seem to have contributed much additional physical insight.

The use of a theory operators on Hilbert space has sometimes engendered the feeling that the problem of specifying boundary conditions has been sidestepped. In reality it is only recast in a different form, but by a mechanism - the specification of a domain - which merits attention.

A Hilbert space theory of differential operators is complicated by the fact that differentiability and square integrability are really two quite different concepts. Integrability is a global characteristic whereas differentiability is a local property. Thus Hilbert space includes many functions which have mathematically unpleasant aspects, such as lacking derivatives or being discontinuous. Yet at the same time it lacks numerous functions considered to be important, such as the powers or the complex exponentials.

As a result we frequently find that the quantum mechanical operators can be applied in a meaningful way to functions which do not belong to Hilbert space, whereas at other times it happens that they cannot be applied to some of the legitimate members of Hilbert space.

The situation is a fundamental one, and the result is that for every operator on a Hilbert space there must he specified its ``domain,'' which is to say. the collection of functions to which we propose to apply it. It is through the definition of the domain that the boundary conditions of differential operators frequently enter, often without there being overt recognition of the fact. The selection of a domain for an operator is of the utmost importance; if the domain is too small, the operator may not be adequately defined, but if an attempt is made to make the domain too large, the operator may simply be impossible to define. Moreover, the Hermiticity of an operator may depend on the proper selection of its domain. A final precaution, one which is too often overlooked, is the verification that two or more operators, when they are employed in a common calculation, either have a common domain or the calculation purports only to refer to the domain which they share in common. Nevertheless, once there is an adequate realizalion of the limitations of Hilbert space theory, there is a vast reservoir of rigorously demonstrated mathematical results waiting to assist calculations and the endeavor to solve Schrödinger's equation. Hilbert space theory and differential equation theory can be used in harmony, without the necessity to give up such things as continuum wave functions because of their nonnormalizability.

At the beginning of this century the theory of differential operators was already rather extensively developed, especially with regard to the classification of the singularities which could occur in the solulions as a consequence of singularities in the coefficients, or of the infinitude of the interval of integration. But we might say that the theory of differential operators commenced with Hermann Weyl's investigation of the relationship between square integrability of the solutions and the boundary conditions to which the equation was subject. These results appeared in the Mathematische Annalen of 1910 [7].

Although Weyl's article was frequently cited in succeeding years, it was limited to ordinary second-order singular differential equations, and did not seem to motivate any great amount of further study for some 30 years. Then, E. C. Titchmarsh began a systematic analysis of the properties of eigenfunction expansions according to the solutions of differential equations, which he published in numerous papers in the British mathematical journals. Eventually these papers formed the skeleton of his two-volume treatise [8] on eigenfunction expansions which was published after the war. The first time that Weyl's theory was available in a popular English language textbook seems to have been with the publication of Coddington and Levinson's authoritative Theory of Ordinary Differential Equations [9] in 1955.

At present, with the appearance of Dunford and Schwartz's massive three-volume Linear Operators [10], whose bibliography alone exceeds 100 pages containing nearly 2000 references, we have a completely overwhelming, albeit encyclopedic compendium of the whole subject. Nor should it be overlooked that there is also a considerable Russian literature, whose growth commenced mainly in the 1940's.

A comprehensive and straightforward theory of partial differential equations is still hard to come by, in spite of the importance of these equations and the vast amount of theoretical and computational effort which has been expended on them over the years. Fortunately there is one exception, which practically speaking is a very important one, whereby it is possible to refer the analysis of separable partial differential equations to systems of ordinary differential equations, which are much more tractable.

The bulk of the partial differential equations which physicists, chemists, or engineers solve are separable. Other kinds are very much more difficult to resolve, and so do not receive so much attention. Yet, given the spherical symmetry inherent in many problems of interest, and given the applicability of the Hartree-Fock equations which originate in an approximation based on separability, it would suffice for the understanding of a sizable class of problems of practical importance to arrive at a good understanding of ordinary differential equations. Separable problems, at least, can then be resolved by combining all their constituents, once each has been worked out individually.

In a sense, the point of view presented here has already been worked out extensively during the past two decades in the guise of Regge pole theory, dispersion relations, analytic S-matrix theory, and related topics. Two characteristics of that work may explain the reason it is not more widely known. It was carried out mainly in those branches of high-energy physics which were primarily concerned with scattering theory, and for which bound states were of secondary importance, even somewhat undesirable. Additionally, the properties of angular momentum were inextricably mixed with the energy dependence of the wave functions. Indeed. Regge [11] began his program to apply the Watson transform to quantum mechanics knowing how successfully it had been used by Sommerfeld to sum the partial waves arising in the theory of the propagation of radio waves over a spherical earth. As the presence of angular momentum is the only possible consequence of separation in spherical coordinates, we cannot complain.

Returning to the contention that the architects of quantum mechanics must have been content with the sufficiency of their quantizing principles, we are substantially left with an evaluation of von Neumann's Mathematical Foundations of Quantum Mechanics [4]. There differential equations have been replaced by operators on Hilbert space, their boundary conditions subsumed in the selection of a domain. Quanitzation transpires in the Hermiticity of the Hamiltoman: to have this property the Hamiltonian operator must certainly be ``hypermaximal,'' which can become an issue for the singular potentials. Their naive Hamiltonian is not hypermaximal. To make it so requires a ``self-adjoint extension'' which amounts to restricting the domain. This implies an additional boundary condition, the same one which arises in the ``limit circle'' alternative for Weyl's theory of differential operators. Whether the extension has a physical significance does not concern the mathematical theory, so it is said that we should have no trouble with the singular potentials.

What about the continuum? We learn that the spectrum of an operator ${\cal H}$ is to be defined by the nonexistence of the resolvent, $({\cal H - \lambda I})^{-1}$. For finite matrices the inverse may simply not exist, the responsible vector is annihilated by $({\cal H - \lambda I})$ and is thus an eigenvector. For operators a second possibility is that the resolvent is not bounded. The former situation defines the discrete spectrum - the only possibility for finite matrices - but the second leads to the continuous spectrum. The would-be eigenvectors are afflicted with an infinite norm; they may well enough exist, but they do not belong to Hilbert space. Infinity, in Hilbert space, may contain some very respectable functions. Instead we must deal with approximations, which are not quite eigenvectors, but which do belong to Hilbert space. These are Weyl's eigendifferentials. So much for the continuum; our discomfiture lies in being deprived of those nonnormalizable eigenvectors.

The demand that the Hamiltonian be Hermitean is apparently an adequate quantizing principle; at any rate it was sufficiently salisfying in the historical development of quantum mechanics that the question lay dormant thereafter. However, it is to be seen that the emphasis had shifted subtly from the solution of a differential equation to an operator calculus. If the differential equation is fundamental, we at least have to know how to relate its solutions to physical processes, which to date has always been through a statistical interpretation of the wave function. If we are to use an operator calculus, we had better know how to choose the operator corresponding to a given physical process.

There seem to be opposites involved here: take the differential operator and make the best of its solutions, or take a theory of probabilities and try to find the best operator to fit the circumstances. Maybe there is a middle ground. Perhaps both of these extremes tend to impose constraints on a problem which has yet to be thought through in its entirety.

In either event, it is clear that the theory of operators has not yet been understood completely, and this in spite of the prodigious size of the more ambitious treatises. Nor has the theory of differential equations been explored to its conclusions. Nor is the popular understanding of either theme commensurable with then present states of development. With these thoughts in mind, it could be interesting to explore the relationship between a differential equation and its boundary conditions.

Operator theory would like to delegate the boundary conditions to the axiomatic preliminaries, which is to say that in any given instance they enter into the formalities wherein it is verified that the axioms are satisfied, so that we can proceed with the important business of mathematical deductions and proving theorems. Differential equation theory tends to be somewhat preoccupied with establishing the existence and uniqueness of solutions, the conditions which favor the establishment of boundary conditions, and similar matters.

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Next: Differential Equation Theory Up: Quantization as an Eigenvalue Problem Previous: Quantization