Memoirs of the American Mathematical Society 2013; 97 pp; softcover Volume: 221 ISBN10: 0821875604 ISBN13: 9780821875605 List Price: US$69 Individual Members: US$41.40 Institutional Members: US$55.20 Order Code: MEMO/221/1040
 The solution to the KohnSham equation in the density functional theory of the quantum manybody problem is studied in the context of the electronic structure of smoothly deformed macroscopic crystals. An analog of the classical CauchyBorn rule for crystal lattices is established for the electronic structure of the deformed crystal under the following physical conditions: (1) the band structure of the undeformed crystal has a gap, i.e. the crystal is an insulator, (2) the charge density waves are stable, and (3) the macroscopic dielectric tensor is positive definite. The effective equation governing the piezoelectric effect of a material is rigorously derived. Along the way, the authors also establish a number of fundamental properties of the KohnSham map. Table of Contents  Introduction
 Perfect crystal
 Stability condition
 Homogeneously deformed crystal
 Deformed crystal and the extended CauchyBorn rule
 The linearized KohnSham operator
 Proof of the results for the homogeneously deformed crystal
 Exponential decay of the resolvent
 Asymptotic analysis of the KohnSham equation
 Higher order approximate solution to the KohnSham equation
 Proofs of Lemmas 5.3 and 5.4
 Appendix A. Proofs of Lemmas 9.3 and 9.9
 Bibliography
