A complete understanding of Schrödinger operators is a necessary prerequisite for unveiling the physics of nonrelativistic quanturn mechanics. Furthermore recent research shows that it also helps to deepen our insight into global differential geometry. This monograph written for both graduate students and researchers summarizes and synthesizes the theory of Schrödinger operators emphasizing the progress made in the last decade by Lieb, Enss, Witten and others. Besides general properties, the book covers, in particular, multiparticle quantum mechanics including bound states of Coulomb systems and scattering theory, quantum mechanics in constant electric and magnetic fields, Schrödinger operators with random and almost periodic potentials and, finally, Schrödinger operator methods in differential geometry to prove the Morse inequalities and the index theorem.

Self-Adjointness.- Lp-Properties of Eigenfunctions, and All That.- Geometric Methods for Bound States.- Local Commutator Estimates.- Phase Space Analysis of Scattering.- Magnetic Fields.- Electric Fields.- Complex Scaling.- Random Jacobi Matrices.- Almost Periodic Jacobi Matrices.- Witten's Proof of the Morse Inequalities.- Patodi's Proof of the Gauss-Bonnet-Chern Theorem and Superproofs of Index Theorems.