Scattering theory. Rev. ed. (English) Zbl 0697.35004

Lax, Peter D.; Phillips, Ralph S.

Pure and Applied Mathematics, 26. Boston etc.: Academic Press, Inc. xii, 309 p. $ 44.95 (1989).

Pages 1–273, 307–309 of this monograph contain a page by page reprint of the well-known first edition which appeared more than 20 years ago. Up to a few minor revisions the text is unchanged; for a review of the first edition see Zbl 0186.16301. For the benefit of the reader, the authors took the opportunity of this revised edition to add an epilogue (pp. 275–299) and some epilogue references (pp. 299–305) which outline the wealth of new results discovered in the intervening years.
Section 1 of the epilogue surveys scattering properties dependent on the geometry of the obstacle. In particular, the authors review the solution of the compactness conjecture from pp. 155–157 of the present book and they discuss the dependence of the eigenvalues of the generator \(B\) on the geometry of the obstacle and the asymptotic distribution of eigenvalues.
Section 2 deals with scattering theory for automorphic functions. Here, the authors survey recent work on the eigenvalue problem of the automorphic Laplacian which starts from the corresponding wave equation and from scattering theory. In particular, reference is made to Selberg’s eigenvalue conjecture. The reviewer would like to add the reference: A. Selberg, Harmonic analysis. In: A. Selberg: Collected Papers, Vol. I, pp. 626–674. Berlin: Springer-Verlag (1989; Zbl 0675.10001).
Section 3 deals with symmetric hyperbolic systems, and the final Section 4 covers applications and extensions.
Many students and researchers will welcome this revised and updated reprint of a classic.

Reviewer: Jürgen Elstrodt (Münster)

MSC:

35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
47-02	Research exposition (monographs, survey articles) pertaining to operator theory
35P25	Scattering theory for PDEs
47A40	Scattering theory of linear operators
58J50	Spectral problems; spectral geometry; scattering theory on manifolds
11F03	Modular and automorphic functions
11F72	Spectral theory; trace formulas (e.g., that of Selberg)

Keywords:

geometry of the obstacle; compactness conjecture; Selberg’s eigenvalue conjecture; symmetric hyperbolic systems; scattering theory for automorphic functions; eigenvalue problem of automorphic Laplacian