Operator spectral states

Abstract

In chemical physics, especially quantum mechanics, one encounters a myriad of phenomenological models based upon formal operator manipulations and calculations leading to spectral conclusions. Yet often the exact nature of such spectral results is unclear mathematically. On the other side, mathematical clarification can lead to greater physical insight. Two important illustrations of this symbiotic relationship are von Neumann's placing of the quantum mechanics of Schrödinger, Heisenberg, and others, into a Hilbert space framework, and Schwartz's rendering of the Dirac ket and delta notation into the theory of distributions, some thirty years later. With chemical physics now actively treating time-symmetry breakings into nonunitary evolutions, complex resonances, computations concerning the spectrum as caused by underlying chaotic dynamical systems, it seems useful to gather together here elements of operator spectral theory to provide a mathematical framework into which such spectral results can be placed or from which they may be viewed. A particularly useful, clarifying, but little-known device, is that of operator state diagram, which I will develop rather fully in this paper. All operator dualities and hence much of operator theory may be systematically portrayed by these diagrams. From these diagrams, I then discuss operator spectral states, scattering states available from the essential spectrum of an operator, spectral states of shift operators, and rigged Hilbert space spectral states. Only a few examples can be given here, for the rigorous working out of the state diagrams of a given new class of operators requires some care. Nonetheless, I hope that I will have provided some helpful clarifications of the instances given. Once the approach is seen, it is clear what needs to be done for other applications. Finally, I turn to what I shall call regular spectral states and probabilistic spectral states. These are new notions. Briefly, my view is that nature prefers regular spectral states, whereas we (humankind) prefer more specific spectral states. To reinforce this ansatz, I connect it to all other sections of this paper. A rule of probabilistically preferred spectral states is seen as an instance of the more general regularization principle which is an extensive irreversibility law. I include in each section a principle. This is meant to help the beginner come away with useful general notions, and it also permits one to express personal viewpoints.