Puits multiples en mécanique semi-classique. IV: Étude du complexe de Witten. (Multiple wells in semi-classical limit. IV: Investigation of the Witten complex). (French) Zbl 0597.35024

[For Part III see the preceding review.]
The paper is devoted to a beautiful analytical proof of an improved version of Morse inequalities based on the geometry of the Laplace operator. Using some recent own results [ibid. 9, 337-408 (1984; Zbl 0546.35053)], the authors essentially complete and improve the original method due to E. Witten [J. Differ. Geom. 17, 661-692 (1982; Zbl 0499.53056)]: Let M be a manifold of dimension n with Riemann metric dx, f be a Morse function on M. Then a complex \(E: 0\to E^ 0\to...\to E^ n\to 0\) is constructed such that dim \(E^ k\) is the number of critical points of f of index k and dim \(H^ k(E)\) is the k-th Betti number.
The construction is based on the Hodge-de Rham theory to the Laplace operator \(\Delta_ f=d^*_ fd_ f+d_ fd^*_ f\) where \(d_ f=e^{-f/h} hd e^{f/h}\) (h is a small parameter) and the Agmon’s metric \(| \text{grad} f| dx\). The sought complex E appears if \(h\to 0\), after some subtle deformation argument near the critical points of f. Geometrical parts of the proofs are presented with details but the analytical reasonings are rather dense.
Reviewer: J.Chrastina


35J10 Schrödinger operator, Schrödinger equation
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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[1] Abraham R., Foundations of Mechanics (1978)
[2] Berger, Lecture Notes in Mathematics 194
[3] Bott R, Bull. A.M.S. 7 (2) pp 331– · Zbl 0505.58001
[4] Helffer B., Comm. in P.D.E 9 (4) pp 337– · Zbl 0546.35053
[5] Helffer B., A paraitre Annales de 1’I.H.P. (1984)
[6] Helffer B., Aparaitre Mathernatische Nachrichten (1984)
[7] Henniart G., Séminaire Bourbaki 36éme année 617 (1983)
[8] Lions Magenés, Problèmes aux limites non homogénes et applications
[9] Melrose R., Elliptic operators on manifolds · Zbl 0384.35052
[10] Milnor J., Annals of Mathematics studies n{\(\deg\)} 51 (1963)
[11] Milnor J., Lectures on h-cobordism (1965) · Zbl 0161.20302
[12] de Rham G., Variétés différentielles (1960)
[13] Simon, B. 1983.Semi-classical analysis of low lying eigenvalues 1, Vol. 3, 323–326. Ann. I.H.P.
[14] Simon, B. 1983.Instantons, double wells and large deviations, Vol. 8, 323–326. Hull. A.M.S. · Zbl 0529.35059
[15] Simon B., Tunneling,Annals of Math (1984)
[16] Smale S., Hull. Arner. Math. Soc. 73 pp 747– (1967)
[17] Von Westenholz C., Differentiable k dynamical systems
[18] Witten E., J. Of Differential Geometry, 17 pp 661– (1982) · Zbl 0900.53036
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