Spectral Theory and Applications

Several textbooks on Spectral Theory have been reviewed in the MAA Reviews in the past. The peculiarity of this small volume is that it collects lecture notes from mini-courses given at a Summer School, so it is not, strictly speaking, a textbook. Each chapter is a short, rapid excursion in some particular topic, for example, quantum mechanics or spectral geometry. To a large extent, the chapters can be read independently from each other. The reader (perhaps a graduate student) could then catch a glimpse of a topic, with a minimal investment of time.

For example, the chapter From classical to quantum mechanics provides an outline of the role of spectral theory in quantum mechanics. The sections on Lagrangian submanifolds and the Hamilton-Jacobi equations lack some details, so I suspect that some readers may feel lost there. In any case, the reader determined to deepen her understanding should then refer to more comprehensive references, like Quantum Theory for Mathematicians by Hall.

The applications mentioned in the title cover mostly differential equations, quantum mechanics, and spectral geometry. Applications to other fields, like graph theory or number theory, are not covered and can be found in other books, like Spectral Graph Theory by Chung, or the books by Einsiedler and Ward.

The chapter I enjoyed the most is Spectral Geometry by Y. Canzani, for the conversational style, the amount of explicit, illuminating examples and the historical notes and anecdotes; definitely worth reading.

Fabio Mainardi (fabio.mainardi@rd.nestle.com) is a mathematician working as a senior data scientist at Nestlé Research. His mathematical interests are number theory, functional analysis, discrete mathematics and probability.