A number of perplexing situations which have arisen during the course of development of quantum mechanics have a very nice interpretation in terms of the complex poles and other singularities of the spectral density. Foremost among these is the question of the convergence of perturbation theory, let alone its very meaning in the context of a continuous spectrum.
Perturbation theory, even in the variant forms in which it is sometimes presented, inevitably consists of formulas which describe a purely real process carried out on purely real data. Naturally the procedure involved cannot account for the migration of a pole away from the real axis out into the complex plane. Moreover, it seems to be the general rule that the movement of the poles out to infinity from their real unperturbed values depends exponentially on the perturbation parameter, in a way which introduces an essential singularity at zero strength. Power series expressions for the perturbed eigenvalues cannot work, leaving asymptotic formulas as the best for which we can hope.
Symmetry and degeneracy do not play any important role in one-dimensional problems, but the spacing of the eigenvalues and relative location of the complex poles constitute an important consideration in assessing these characteristics of separable higher dimensional problems. It is therefore interesting to consider the implications of the Weyl-Titchmarsh theory for symmetry and degeneracy for those potentials amenable to analysis through separation of variables. The theory of the universal symmetry group, for example, was based on the construction of ladder operators for each of the separation coordinates.
Certainly any separation equations which lead to purely bound states will be amenable to a Sturm-Liouville treatment which can be extended in the limit to a complete orthonormal family of eigenfunctions. As this family is naturally ordered by the size of the eigenvalues and the number of nodes of the eigenfunctions, there is not much difficulty to imagine ladder operators for the assemblage, both raising and lowering the eigenvalue. Continuum states will create complications, particularly if the m function has singularities other than poles near the real axis, but in general terms can be fitted into the same scheme.
However, it will hardly be expected that there will be any regular relationship among the eigenvalues. Whenever such a relationship might be discovered, it would certainly be possible to modify it slightly by a small modification in the potential. Thus, the orderly spectra should constitute at the most a very small sample among the possible spectra.
Dealing with continuous spectra, which realism insists must occur in the preponderant majority of cases, we would expect the eigenvalues to be replaced by the complex poles in the spectral density function, which are likely to be countable in number, although not directly associated with a complete orthonormal set of eigenfunctions. Especially when the poles lie very close to the real axis, they will constitute the closest approach to an assortment of eigenvalues which we are likely to obtain. By a more careful analysis it can be made plausible that here, too, the actual procession of poles can just about be made to order, making it a vain hope that there would be some general principle of nature which would regulate the location or spacing of the poles.
It is not excluded that such regularity might dominate the equations describing fundamental processes, such as the direct interactions of elementary particles, but it seems hopeless to expect it to persist in the equations summarizing more complicated configurations.
In a realistic multidimensional problem, the way it which these poles, be they real or complex, are to be combined once they arise from the separated constituents of Schrödinger's equation will depend on the individual cases. That is, the expression for the complete eigenvalue will depend rather much on the separation constants which arise, how they enter into the separated equations, and how the total energy is functionally dependent on them. Nevertheless it would seem that any degeneracies or approximate degeneracies which might exist would have an overwhelming tendency to be statistical accidents. Supporting this view is the observation already made that slightly different separation potentials would have slightly different spectra, and moreover these spectra could be slightly different in any way which we might choose. It is not even necessary to speculate on the modifications which might lead to the same result of slightly deviating spectra, but through a nonseparable modification to the potential. Therefore the distribution of energies, and hence apparent degeneracies or near degeneracies, would seem to be alterable in a completely arbitrary manner.
It is therefore clear that it is not the mere existence of ladder operators, capable of mapping one eigenvalue into another, which is the determining factor in establishing accidental degeneracy. It is necessary that each of the individual separation spectra has a strict regularity, and that the raising of eigenvalues in one separation coordinate can be exactly balanced by a lowering in another.
Such reasoning, it would seem, completely dissipates any further hope for a universal symmetry group. At the same time it points a direct finger at the relationship between unitary and canonical transformations as seen from the theory of operators on Hilbert space.
If we have lost a possible application of group theory in quantum mechanics through the loss of the universal group, we have at least gained a new source of interest through the intervention of the symplectic group in such an interesting way in the space of boundary conditions, much as the unitary group enters into the space of wave functions.
Some new perspectives in the relationship of symmetry groups and dynamical groups to the spectrum of the Hamiltonian emerge from the complex approach to eigenvalue theory One of these arises in Regge pole theory, because the angular momentum is treated as a complex variable, rather than being restricted to the eigenvalues of the total angular momentum. Presumably there exists a complex eigenvalue theory for the angular momentum operators as well as for the Hamiltonian, with a corresponding representation theory. Some studies have been made of this phenomenon, which requires the infinite-dimensional nonunitary representations of the three-dimensional rotation group. It is a speculative area, just as the attempts to fit the nonphysical eigenfunclions of the Hamiltonian into a Hilbert space theory are somewhat peripheral to the main lines of quantum theory. Nevertheless some interesting relationships may repay still further study.
Finally, and most important, the concept of square integrability should lose some of its mysterious aloofness. If it is seen to be an auxiliary concept rather than a first principle, the process of integrating the Schrödinger equation or the Dirac equation becomes an ordinary exercise in differential equation theory. Thus the use of differential equations in physics should be no different than in any other branch of engineering.
One point which perhaps is not as much appreciated as it should be, and which emerges from a careful theory of differential equations, is the way in which the eigenfunctions of the differential equation operator may be used to form an eigenfunction expansion of Hilbert space without ever being required to belong to Hilbert space themselves. Only in this way can we have continuum wave functions, and at the same time a probabilistic interpretation with normalizable functions, which evidently have to be wave packets, and not individual eigenfunctions. Such is the compromise between probability theory and differential equation theory which seems to be required.
Much remains incomplete in this survey of quantization as an eigenvalue problem, particularly since we have fostered the idea that an ``eigenvalue problem'' consists in selecting those solutions of a differential equation which look ``interesting.'' At least in one dimension it is not too difficult to produce an assortment of interesting possibilities from which to make a selection. It is also possible to make some well-founded evaluations of the situation prevailing for partial differential equations, when more than one dimension has to be considered.
Square integrability is a valid quantizing principle, but must be applied to the total wave function. Ordinary continuum wave functions are not expected to be square integrabte, and the probability interpretation must be abandoned in favor of probability currents or of relative probability. Only if the growth of the wave function is still too strong for these localized requirements must it be rejected.
Finiteness as a quantizing principle seems to make its appearance in separated equations, and is related to the requirement that the total wave function have some acceptable probabilistic interpretation.
The total wave function for a separable equation is a product of individual wave functions, which leads to a specialized form of Green's formula. The one-dimensional formula can produce a finite bracket from very large arguments because it involves a difference of a product, whereas the bracket for separated equations involves the arguments directly as factors. Hence, there is an additional requirement for their finiteness which is not relevant to an isolated one-dimensional wave equation. The difficulty lies in the more complicated structure of the multidimensional bracket, which violates our naive assumptions. It is not unlike the way in which an element of area transforms as a vector product of the edge elements, rather than the simple product which we tend to assume carelessly.
The one instance where square integrability rather than finiteness is decisive occurs in the radial equation of the Dirac hydrogen atom. To fit this into the explanation just given would require us to investigate Green's formula for the separable Dirac equation, which is surely somewhat different from the Schrödinger Green's formula.
Directly or indirectly, the insights which produced quantum mechanics have already sustained the. mathematical sciences for 50 years, with every indication of continuing to exert their influence. What began as a mixture between a calculus for combining tables of spectral lines and a differential equation generalizing classical mechanics in the direction of wave optics has dominated our entire thinking about physical reality, and has still not ceased shaping and guiding much of the world's intellectual effort. The theory of operators on Hilbert space, not to mention the theory of rings of operators, was rudely forced beyond the mathematicians' convenient restrictions to bounded operators and continuous mappings by the necessity of solving the Schrödinger equation for realistic systems. The foundations of probability theory had to be given a sharper form before accurate statements could be made about the probabilistic interpretation of the wave function and about the theory of measurement. In recent years numerical methods have been refined and computer technology stimulated, in part to integrate the Schrödinger equation. Even the arcane lore of group representation theory, which flourished at the beginning of the century, was resurrected and adapted for continuous groups to allow progress in the quantum mechanics of many-particle systems. Yet even after all this activity, our understanding of quantization as an eigenvalue problem has barely begun.
Acknowledgements
Major portions of the planning and preparation of the manuscript were carried out at the Quantum Chemistry Institute of Uppsala University. The excellent hospitality of the Institute, and particularly of the Director, Professor P.-O. Löwdin, as well as extensive discussions of the subject matter with Erkki Brändas and Michael Hehenberger have been indispensable in its preparation. Support for the visit was received from the Swedish Natural Sciences Research Council. Some additional assistance was provided by the Aerospace Research Laboratories (AFSC). USAF (Grant AFOSR 73-2546).
The Mexican Institute for Nuclear Energy has particularly supported the computer graphics used in the illustrations, as well as extensive numerical studies of the Titchmarsh-Weyl theory.
The article is dedicated lo the memory of Professor Solomon Lefschetz. in appreciation of his efforts to advance the theory ol differential equations, at RIAS, and in Mexico.