## 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

### MSC:

 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

### Citations:

Zbl 0597.35023; Zbl 0546.35053; Zbl 0499.53056
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### References:

 [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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