Density of states oscillations for magnetic Schrödinger operators. (English) Zbl 0778.35089

Differential equations and mathematical physics, Proc. Int. Conf., Birmingham/AL (USA) 1990, Math. Sci. Eng. 186, 295-345 (1992).
[For the entire collection see Zbl 0728.00011.]
This text gives a detailed account for some recent improvement of some results of Helffer-Sjöstrand [B. Helffer and J. Sjöstrand, Ann. Inst. Henri Poincaré, Phys. Théor. 52, No. 4, 303- 375 (1990; Zbl 0715.35070)] devoted to the study of the de Haas-van Alphen effect. It is also complementary with other surveys written recently by the author [Microlocal analysis and applications, Lect. Notes Math. 1495, 237-332 (1991; Zbl 0761.35090) and Magnetic Schrödinger and effective Hamiltonians, Lecture at the Summer School on Math. Phys. and PDE, Braşov/Roumania (August-September 1989) (to appear)].
The general object is the study of the magnetic Schrödinger operator \[ P_{B,V}=\sum^ 3_{j=1}(D_{x_ j}+A_ j(x))^ 2+V(x)\text{ on } \mathbb{R}^ 3, \] with \(D_{x_ j}=i^{-1}\partial_{x_ j}\), where \(V\) is smooth and periodic with respect to some lattice, and \(A_ k(x)=(1/2)\Sigma_ jb_{j,k}x_ j\), with \(b_{j,k}=-b_{k,j}\) assumed to be constant and small. In particular, it is proved in the case of the weak \(B\) regime the appearance of peaks in the density of states measure.
But this survey presents also new results for the branching situation in the \(C^ \infty\) case, with proof of existence of normal forms which are (not only for the de Haas-Van Alphen effect) basic for many problems in semi-classical analysis where critical values of the Hamiltonian play an important role. In particular this permits to improve the results of C. Gérard and A. Grigis in J. Differ. Equations 72, No. 1, 149- 177 (1988; Zbl 0668.34022).
Reviewer: B.Helffer (Paris)


35Q40 PDEs in connection with quantum mechanics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
47F05 General theory of partial differential operators
35S05 Pseudodifferential operators as generalizations of partial differential operators