Butterfly in the quantum world. The story of the most fascinating quantum fractal. With contributions by Douglas Hofstadter (Q2833064)

The book is based on the papers by \textit{P. G. Harper} [Proc. Phys. Soc., Sect. A 68, 874--878 (1955; Zbl 0065.23708)] and the 1976 paper by \textit{D. R. Hofstadter} [``Energy level and wave functions of Bloch electrons in rational and irrational magnetic fields'', Phys. Rev. B 14, No. 6, 2239--2249 (1976; \url{doi:10.1103/PhysRevB.14.2239})]. The spectra of a selfadjoint operator (Harper model) in the Hilbert space \(\ell_2({\mathbb Z})\) provide Cantor-like spectra and the butterfly fractal. Both the underlying mathematics and physics are outlined in detail in the book.NEWLINENEWLINEThe author first provides an introduction to fractals, emphasizing the Cantor set, Apollonian gasket and the Hofstadter set. Chapter 2 looks at number theory and scaling in connection with Cantor-like spectra. Both rational and irrational numbers are introduced together with continued fraction expansion and the golden mean number. On the physics side the topological aspects of the quantum Hall effect and the Berry phase are studied in detail. Other systems with Cantor-like spectra are also discussed.NEWLINENEWLINEThe historical development of the butterfly fractals is outlined in detail. D. R. Hofstadter himself contributed to the book so there is a first hand account about the history of the butterfly fractal. Stimulating images, illustrations and poems are included, making this book an enjoyable read. The book can be recommended for students in physics as well as mathematics. Researchers in solid state physics would also be benefit from the book. An extensive list of references on these topics is also provided.