Customer Reviews

Introduction to Spectral Theory in Hilbert Space (Dover Books on Mathematics)

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Helmberg's book provides a superb introduction to Hilbert Spaces and Spectral theory. It's an exceptionally clear, careful, yet concise exposition for anyone who is, as the author states: "interested in the topic but lacks the time or desire to fill in gaps ... or to work through an inspiring set of exercises considered to form an integral part of the text (p. viii)". (I personally dislike books that make exercises integral to the exposition as this makes self-study more difficult than necessary.)

I rank this book as beginning graduate level or for math majors, advanced undergraduate. Prerequisites are minimal for the topic but include standard calculus, linear algebra at the level of Axler's Linear Algebra Done Right, basic analysis, basic topology (metric space, inner product space, normed space) and a little group theory. Other books I like include Rynne & Youngson Linear Functional Analysis (Springer Undergraduate Mathematics Series) and the Wiley classic Kreyszig Introductory Functional Analysis with Applications. Helmberg is a bit more advanced than Rynne / Youngson or Kreyszig. Both Kreyszig and Rynee / Youngson provide more mathematical background than Helmberg. Note that Kreyzsig devotes chapter 11 to use of unbounded linear operators in quantum mechanics. Also, if you're more interested in quantum mechanics applications, I'd check out Jordan's Linear Operators for Quantum Mechanics (Dover Books on Mathematics). Jordon, however, is a very concise, graduate level text and expects a great deal of mathematical maturity.

Even though the exposition is kind to the reader, the pace is nice: Hilbert spaces on p. 23; spectrum of a linear operator p.157. Ex appendices and back matter, book is only 309 pp. but covers: Ch.1: Concept of a Hilbert Space [1-35]; Ch. 2: Specific geometry of Hilbert space [36-70]; Ch. 3: Bounded linear operators [71-116]; Ch. 4: General theory of linear operators [117-176]; Ch. 5: Spectral analysis of compact linear operators [177-218]; Ch. 6: Spectral analysis of bounded linear operators [219-287]; Ch. 7: Spectral analysis of unbounded selfadjoint linear operators [288-309].

Anyone seriously interested in the mathematical details of the Hilbert Space formalism of Quantum Mechanics could profit from reading Helmberg. The book is very nicely produced, with large enough font for aging eyes. And since it's Dover, the price is right. Note that the original publication date was 1969 but hey, the math has not changed.

So before you buy some expensive book on this topic, you owe it to yourself to check out Helmberg: you can get it and still afford those Lattes you need to read it!