Schrödinger operators, with application to quantum mechanics and global geometry. (English) Zbl 0619.47005

Cycon, H. L.; Froese, R. G.; Kirsch, W.; Simon, B.

Springer Study edition. Texts and Monographs in Physics. Berlin etc.: Springer-Verlag. ix, 319 pp.; DM 56.00 (1987).

This book started as a course by the fourth author in Thurnau (FRG) during the summer of 1982. Since then it has been further developed and rearranged by the authors, so as to constitute a comprehensive and synthetic monograph on the theory of Schrödinger operators. Recent progress and applications made in the last decade by various mathematical physicists are strongly emphasized so that the monograph can be read by graduate students as well as researchers who want to learn about the latest developments in the fields.
It covers, in particular, multiparticle quantum mechanics including bound states of Coulomb systems and scattering theory, quantum mechanics with electric and magnetic fields, Schrödinger operators with random and almost periodic potentials (connected to localization problems in disordered systems) and, finally, Schrödinger operator methods in differential geometry are used to prove the Morse inequalities and the index theorem (with a connection to super-symmetric ideas).

Reviewer: M.Combescure

MSC:

47A40	Scattering theory of linear operators
47B80	Random linear operators
47-02	Research exposition (monographs, survey articles) pertaining to operator theory
81U05	\(2\)-body potential quantum scattering theory
47A35	Ergodic theory of linear operators
81U10	\(n\)-body potential quantum scattering theory
82B44	Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60H25	Random operators and equations (aspects of stochastic analysis)
47A10	Spectrum, resolvent
58J20	Index theory and related fixed-point theorems on manifolds
58J50	Spectral problems; spectral geometry; scattering theory on manifolds

Keywords:

multiparticle quantum mechanics; bound states of Coulomb systems; scattering theory; quantum mechanics with electric and magnetic fields; Schrödinger operators with random and almost periodic potentials; localization problems; Morse inequalities; index theorem; super-symmetric ideas