On page 126 of this book we encounter what the author calls the RAGE theorem, named for its discoverers, Ruelle, Amrein, Gorgescu, and Enß, and presented as “a dynamical characterization of the continuous and pure point spectra” of a self-adjoint operator on a Hilbert space. The parent theorem precipitating RAGE is a classical theorem of Norbert Wiener dealing with the limit at infinity of the Cesàro time average of the Fourier transform of a finite complex Borel measure on R. Wiener’s theorem is a thing of great beauty and its surprisingly simple proof, turning on the Lebesgue dominated convergence theorem, is an exercise in Fourier analysis that has both classical and modern overtones.
So, with clear evidence in place of the author’s sense of humor as well as his good taste, we encounter in this snapshot a decent representation of what the book under review is all about. There is a dominance of Fourier and functional analysis in the game, which has to do with quantum mechanics (QM), of course, from the viewpoint of Schrödinger rather than Heisenberg. This approach is probably the more palatable one for us mathematicians, given the attendant toolkit: it’s all beautiful and, ultimately, familiar mathematics. Of course there’s something of a historical inversion in play here in that this toolkit, indeed functional analysis itself, owes so much to quantum mechanics in the first place; however, it’s all too hard to sort out such chronological priorities: it’s largely a question of the observer’s viewpoint and more than likely he’s still uncertain when all is said and done. (Sorry… Well, not so much…)
In any case, Teschl’s Mathematical Methods of Quantum Mechanics (with Applications to Schrödinger Operators) makes for a very exciting march through huge chunks of functional analysis, with QM always on the scene. Says the author: “The present text… gives a brief but rather self-contained introduction to the mathematical methods of quantum mechanics with a view towards applications to Schrödinger operators … built around the spectral theorem as the central object.” And how could it be otherwise, given the famous relationship between observables and eigenstates? I’m afraid I cannot resist citing something here from p. 43 of Lectures on Quantum Mechanics for Mathematics Students, by Faddeev-Yakubovskii, which I reviewed elsewhere in this venue: “[t]he set of eigenvalues of an observable A coincides with the set of possible results of a measurement of this observable… [Furthermore, when we] take the state space to be a complex Hilbert space… the observables [are] self-adjoint operators acting in this space.” Very beautiful stuff, no?
But back to Teschl’s book. It’s a very nice one, and, to boot, is distinguished by a number of unusual moves, specifically the author’s exclusive emphasis on unbounded operators (which is a both practical and reasonable move, in my opinion) and his avoidance of Riesz representation when getting at spectral measures — something he motivates well: see p. xi. Otherwise Mathematical Methods of Quantum Mechanics (with Applications to Schrödinger Operators is split up into two parts, a 160 page treatment of functional analysis from Banach and Hilbert spaces to compact and self-adjoint operators, followed by almost 100 pages on the promised Schrödinger operators. (An example of a Schrödinger operator is a self-adjoint linear transformation on a Hilbert space along the lines of e.g. (the negative of) the Laplacian.)
A few more observations are in order: on p. 23 we meet the BLT theorem (B for Banach?), on p. 150 the KLMN theorem (Kato, Lions, Lax, Milgram, Nelson), and on p. 252, in Problem 12.1, we encounter the important but oft-neglected method of stationary phase in the context of a nice exercise, supplied with an excellent hint. And this says something else about the book: the exercises should please any one with even mild functional analytic tendencies.
There are useful and informative biographical notes at the end of the book, as well as appendix slyly called “Almost everything about Lebesgue integration.” Puns notwithstanding, it’s a very good book.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.