Eigenfunctions of the Laplacian on a Riemann Manifold

Steven Rosenberg’s The Laplacian on a Riemannian Manifold is one of the best books I have ever worked my way though. Some years back I needed to learn about the heat kernel at a deeper level than I had had occasion to before, with particular flavors of analysis and differential geometry dominant, and I was very fortunate to find this book. It taught me a decent amount of Riemannian geometry, some Hodge theory, and de Rham cohomology, all ultimately in the service of the heat equation, its kernel, and its operator, built around the Laplacian. These studies also went a long way to disclose to me something ever so central, namely, the attendant spectral theory of the Laplacian carried out in an ultimately rather sophisticated Hilbert space setting. With these fond memories in place, and with the awareness of the importance of this material, cutting across many borders including the one separating mathematics and physics, I note right off that Zelditch’s focus in the book under review is excellent: yes, indeed, it has to be about eigenfunctions — as regards the physics connection, just think about the mathematician’s favorite way to do quantum mechanics.

In contrast to Rosenberg’s very tight and dense approach, Zelditch pitches a very different game. But I think his choice of pitches is excellent: he starts with a relatively leisurely and expansive Introduction (his first chapter, actually) that starts off asking the question, “What are eigenfunctions and why are they useful?” and then explores, among other things, quantum mechanics: a very good way to motivate all this, even while for us mathematicians other contexts are at least equally important.

It’s also noteworthy that Zelditch closes this chapter, and all the (fourteen) chapters of the book, with a bibliography pertaining to the preceding material. This obviously facilitates further study and mastery at a deeper level: if your fancy is caught by something in a particular chapter, just leaf ahead a bit and follow the corresponding threads. The bibliography to the introductory chapter, for example, sports entries by Bohr, Dirac, Fefferman, Feynman, Hadamard, Hörmander, Lindenstrauss, Riesz (M), Sarnak, Schrödinger, and Weyl: what a line-up to start the book with.

Still taking care to lay a sound and solid foundation Zelditch’s Chapter 2 is devoted to “Geometric preliminaries,” and we are taken from symplectic geometry (and linear algebra: a good idea) to pseudo-differential operators. Again, there follows a very extensive and useful bibliography.

Well, chapter 3 is titled “Main results,” and Zelditch starts it off as follows: “To clarify the organization of this monograph we give a rapid survey of the relatively new results that form the main content of this monograph.” And a little later we read: “All of the results concern the oscillation and concentration of an orthonormal basis of eigenfunctions. Oscillation is measured by matrix elements or by restricted matrix elements. Concentration is measured by restricted matrix elements or by \(L^p\) norms of restrictions to hypersurfaces or thin tubes around hypersurfaces.” This is very good, of course: Zelditch is very explicit about what he wants to focus on, and, right off the bat, he tells the reader something technical in a very accessible and informative manner.” Kudos.

This said, the next ten chapters get into the thick of it in no uncertain terms, but, as already indicated, in an accessible fashion. It’s hard-core stuff: a load of hard analysis, of course, but also very tantalizing special foci. For example, Chapter 7 is concerned with “Lagrangian distributions and Fourier integral operators,” and Chapter 11 is about “Quantum integrable systems.” Regarding the latter, Zelditch says that “[r]oughly speaking, there are only two types of dynamical systems whose eigenfunctions are well understood: the ergodic ones and the integrable ones … In this section we study eigenfunctions in the integrable case.” The focus falls on QCI (= “quantum completely integrable”) Laplacians, and he makes reference to “flat tori, surfaces of revolution, ellipsoids, [and] Liouville tori” and then says “There are many further examples if one considers [rather than Laplacians] quantum integrable Schrödinger operators, but for the sake of brevity we only consider Laplacians here.” Fair enough.

A word or two about the style and nature of this book. To be sure it’s analysis with bells on, but here’s what the back-cover says: “This book … provides complete proofs of some model results, but more often than not it gives informal and intuitive explanations or proofs of fairly recent results. It conveys inter-related themes and results and offers up-to-date comprehensive treatment of this important active area of research.” This is of course consonant with Zelditch’s choice to include extensive local bibliographies, as well as sundry motivational reflections, so to speak.

Thus, in Eigenfunctions of the Laplacian on a Riemannian Manifold, we have a very serious work of scholarship, covering (if you’ll pardon the pun) quite a spectrum of analysis (largely of the hard kind, as opposed to the soft kind), and useful to many audiences, from advanced students to experienced insiders. It’s also distinctive as a springboard to any number of deeper studies flowing from the material in the book’s individual chapters. It promises to be a very valuable resource.

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