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Puits multiples en mécanique semi-classique. VI: Cas des puits sous- variétés. (Multiple wells in semi-classical mechanics. VI: The case of submanifold wells). (French) Zbl 0648.35027
[For Part V see the authors, Current topics in partial differential equations, Pap. dedic. S. Mizohata occas. 60th. Birthday, 133-186 (1986; Zbl 0628.35024).]
The authors continue their subtle studies in Part V concerning the tunneling effect between submanifold wells to give a strong fundament for Witten’s proof of degenerate Morse inequalities; cf. the same authors, Part IV [Commun. Partial Differ. Equations 10, 245-340 (1985; Zbl 0597.35024)]. The problem consists in the B.K.W. asymptotics of the fine kind \(E_ 0+E_ 1h+E_ 2h^ 2+{\mathcal O}(h^ 3)\) for the eigenvalue of the operator \(-h^ 2\Delta +V_ 0+hV_ 1\) on a compact Riemannian manifold. Here \(E_ 0=\min V_ 0\), \(V_ 0^{-1}(E_ 0)=\Gamma\) is a submanifold (or a union of submanifolds) uniformly degenerate in the sense that \((\Delta +V_ 1)\phi =E_ 1=const\). on \(\Gamma\) (where \(\phi =\phi (x)\) is the Agmon distance to \(\Gamma\) arising from the Agmon metric \((V_ 0-E_ 0)dx^ 2)\), and \(E_ 2\) appears as the minimal eigenvalue of certain auxiliary elliptical operator.
Reviewer: J.Chrastina

MSC:
35J10 Schrödinger operator, Schrödinger equation
35P05 General topics in linear spectral theory for PDEs
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35Q99 Partial differential equations of mathematical physics and other areas of application
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References:
[1] J.M. Bismut , The Witten complex and the degenerate Morse Inequalities . Journal of differential Geometry , t. 23 , n^\circ 3 , 1986 , p. 207 - 241 . MR 852155 | Zbl 0608.58038 · Zbl 0608.58038
[2] R. Bott , Lectures on Morse theory, old and new . B. A. M. S. , t. 7 , n^\circ 2 , 1982 . MR 663786 | Zbl 0505.58001 · Zbl 0505.58001 · doi:10.1090/S0273-0979-1982-15038-8
[3] J. Chazarain , Formule de Poisson pour les variétés Riemanniennes . Inv. Math. , t. 24 , 1974 , p. 65 - 82 . MR 343320 | Zbl 0281.35028 · Zbl 0281.35028 · doi:10.1007/BF01418788 · eudml:142270
[4] J. Chazarain , A. Piriou , Introduction à la théorie des E. D. P. Linéaires , Gauthier-Villars . Zbl 0446.35001 · Zbl 0446.35001
[5] H. Cycon , R. Froese , W. Kirsch , B. Simon , Topics in the theory of Schrödinger operators . Livre en préparation (à paraître chez Springer ). · Zbl 0619.47005
[6] A. El Soufi , X.P. Wang , Quelques remarques sur la méthode de Witten ; cas du théorème de Poincaré Hopf et d’une formule d’Atiyah-Bott . Publications de l’institut Fourier , 1986 .
[7] B. Helffer , J. Sjöstrand [1] Multiple wells in the semi-classical limit. I . Comm. in P. D. E. , t. 9 , ( 4 ), 1984 , p. 337 - 408 . MR 740094 | Zbl 0546.35053 · Zbl 0546.35053 · doi:10.1080/03605308408820335
[8] Puits multiples en limite semi-classique. II. Interaction moléculaire - symétries - perturbations . Annales de l’I. H. P. (Section Physique théorique) , t. 42 , n^\circ 2 , 1985 , p. 127 - 212 . Numdam | MR 798695 | Zbl 0595.35031 · Zbl 0595.35031 · numdam:AIHPA_1985__42_2_127_0 · eudml:76277
[9] Multiple wells in the semi-classical limit. III. Non resonant wells . Math. Nachrichte , t. 124 , 1985 , p. 263 - 313 . Zbl 0597.35023 · Zbl 0597.35023 · doi:10.1002/mana.19851240117
[10] Puits multiples en limite semi-classique. IV. Étude du complexe de Witten . Comm. in P.D.E. , t. 10 , ( 3 ), 1985 , p. 245 - 340 . MR 780068 | Zbl 0597.35024 · Zbl 0597.35024 · doi:10.1080/03605308508820379
[11] Puits multiples en limite semi-classique. V. Étude des mini-puits. Current Topics in Partial Differential Equations . Kinokuniya Company Ltd ., Tokyo , p. 133 - 186 . (Volume en l’honneur de S. Mizohata). Zbl 0628.35024 · Zbl 0628.35024
[12] Exposé à l’X (Janvier 1986 ) Séminaire d’équations aux dérivées partielles .
[13] M. Hirsh , Differential topology . Graduate texts in Math. , n^\circ 33 , Berlin - Springer , 1976 . MR 448362 | Zbl 0356.57001 · Zbl 0356.57001 · doi:10.1007/978-1-4684-9449-5
[14] B. Simon , [1] Semi-classical analysis of low lying eigenvalues. I. Non degenerate minima: Asymptotic expansions . Ann. Inst. H. Poincaré , t. 38 , 1983 , p. 295 - 307 . Numdam | MR 708966 | Zbl 0526.35027 · Zbl 0526.35027 · numdam:AIHPA_1983__38_3_295_0 · eudml:76200
[15] Semi-classical analysis of low lying eigenvalues. II. Tunneling . Annals of Math. , t. 120 , 1984 , p. 89 - 118 . MR 750717 | Zbl 0626.35070 · Zbl 0626.35070 · doi:10.2307/2007072
[16] Communication personnelle (Mars 1984 ).
[17] E. Witten , Supersymmetry and Morse theory . J. of differential geometry , t. 17 , 1982 , p. 661 - 692 . MR 683171 | Zbl 0499.53056 · Zbl 0499.53056
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