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Unbounded Self-Adjoint Operators on Hilbert Space

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MAA Review

In graduate functional analysis one typically encounters a cast of characters that includes unitary operators and self-adjoint (or Hermitian) operators on a Hilbert space. Both types of operator are of central importance because they set the stage for a lot of structure built around a collection of very famous and important theorems. In other words, they figure dramatically into the enterprise of laying out a theoretical framework that is not only exceedingly elegant in its own right and provide the point of departure for explorations and applications into allied areas.

One example of the latter, of unsurpassed historical importance, is that of quantum mechanics, with much functional analysis having been designed by John von Neumann (building on Methoden der Mathematische Physik by Courant and Hilbert) so as to address the need of this burgeoning new physics of the 1920s and ’30s.

There is a notorious sticking point when passing from graduate functional analysis (as per, for instance, Barbara MacCluer’s Elementary Functional Analysis) to its putative application in a special context like quantum mechanics. The prevailing operators on a quantum mechanical Hilbert space of states are often only densely defined unbounded operators, replete with their own behavior patterns and inner life, particularly concerning the attendant spectral theory. And this is where the book under review comes in: we read in the author’s “Preface and Overview” that its

prerequisites … are the basics in functional analysis and … the theory of bounded Hilbert space operators as covered by a standard one semester course in functional analysis, together with a good working knowledge of measure theory.

Thus, the book should be tailor-made for a second-year graduate student on the interface between functional analysis and quantum mechanics and the material springing forth from this.

Still in his Preface Schmüdgen presents the following concise and on-target rationale for what he is about to present:

Self-adjoint operators are fundamental objects in mathematics and in quantum physics. The spectral theorem states that any self-adjoint operator T has an integral representation T =∫λdE(λ) with respect to some unique spectral measure E. This gives the possibility to define functions f(T) =∫f(λ)dE(λ) of the operator and to develop a functional calculus as a powerful tool for applications. The spectrum of a self-adjoint operator is always a subset of the reals. In quantum physics it is postulated that each observable is given by a self-adjoint operator T. The spectrum of T is then the set of possible measured values of the observable, and for any vector x [in the underlying Hilbert space of states] and subset M [of the reals] the number <E(M)x,x> is the probability that measured value in the state x lies in the set M. If T is the Hamilton[ian] … the one-parameter unitary group t → e^itT describes the quantum dynamics.

Surely this is one of the best thumb-nail sketches of quantum mechanics in the game, and this bodes very well for what is to follow.

What follows is a six-part exposition of the subject, arranged as follows. Part I deals with closed operators, Part II with spectral theory, Part III with “special topics” (most notably 1-parameter groups and tensor products of Hilbert spaces (needed for more complex quantum systems)), Part IV with perturbation theory, Part V with forms and operators, and, finally, Part VI with the important theme of the “self-adjoint extension theory of symmetric operators.” There are a host of useful appendices added, including material on more prosaic functional analysis (and measure theory), Fourier analysis, and Sobolev spaces. A particularly pleasing feature of the book is found in its “References” section, namely, a list of “classical articles” that contains some true gems and clearly comes down hard on the side of mathematicians’ ways of doing QM: no Heisenberg, Schrödinger, Pauli, or (even) Dirac, but a lot of von Neumann and Weyl. Schmüdgen also includes a good number of exercises in his book: it’s obviously poised to make a solid impact both as a scholarly work and as sound pedagogy.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

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