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Unbounded Self-Adjoint Operators on Hilbert Space
Publisher: Springer (2012)
Details: 432 pages, Hardcover
Series: Graduate Texts in Mathematics
Topics: Operator Theory, Quantum Mechanics
MAA Review[Reviewed by Michael Berg, on 09/13/2012]
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
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:
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.