Puits multiples en limite semi-classique. II: Interaction moléculaire. Symétries. Perturbation. (Multiple wells in the semi-classical limit. II: Molecular interaction. Symmetry. Perturbation). (French) Zbl 0595.35031

Helffer, B.; Sjöstrand, J.

Ann. Inst. Henri Poincaré, Phys. Théor. 42, 127-212 (1985).

[For part I, see Commun. Partial Differ. Equations 9, 337-408 (1984; Zbl 0546.35053).]
The authors continue the study of the splitting of eigenvalues of the Schrödinger operator \(P=-h^ 2\Delta +V(x)+E_ 0\) initiated in Part I using the same method: Assuming the set \(V^{-1}(]-\infty,0])\) to be a disjoint union of a finite number of compact connected sets \(U^ j\) \((j=1,...,N\); the wells), the Dirichlet problem to each separate well (the reference problem) is studied and the eigenvalues of P are described by the eigenvalues of the reference problems up to certain exponentially small corrections.
In the present paper, the remainder estimates are improved using the interactions between wells and families of wells, the one-dimensional case of a double well \((V(x)=V(-x)\), \(V(x)>0\) if \(x\neq \pm a\) \((a>0)\), \(V(a)=V(-a)=0\), \(V''(a)>0\), \(V^{-1}(]-\infty,\epsilon])\) is compact for some \(\epsilon >0)\) is thoroughly analysed, influence of finite groups of symmetries on the derived results with concrete applications to particular examples, perturbations for the Dirichlet problem in the case of one well \((V(x_ 0)>0\) if \(x\neq x_ 0)\), perturbations for several wells in one dimension are studied.
The paper is very dense and the numerous results cannot be adequately mentioned here. They are partly related to earlier work by E. M. Harrel, B. Simon, G. Jona-Lasinio, E. Scoppola and others.

Reviewer: J.Chrastina

MSC:

35J10	Schrödinger operator, Schrödinger equation
58J32	Boundary value problems on manifolds
35P15	Estimates of eigenvalues in context of PDEs
35P99	Spectral theory and eigenvalue problems for partial differential equations

Keywords:

spectral geometry; group symmetries; Schrödinger operator; Dirichlet problem; well

Citations:

Zbl 0546.35053

References:

[1]	S. Agmon , Lectures on exponential decay of solutions of second order elliptic equations , Math. Notes , t. 29 , Princeton University Press . Zbl 0503.35001 · Zbl 0503.35001
[2]	J.M. Combes , P. Duclos , R. Seiler , I. Krein’s formula and one dimensional multiple wells , J. of Functional Analysis , t. 52 , 1983 , p. 257 - 301 . II. Convergent expansions for tunneling . Comm. in Math. Physics , t. 82 , 1983 , p. 229 - 245 . Article | MR 707207 | Zbl 0562.47002 · Zbl 0562.47002 · doi:10.1016/0022-1236(83)90085-X
[3]	H. Donnelly , G-spaces, the asymptotic splitting of L2(M) into irreducibles . Math. Annalen. , t. 237 , 1978 , p. 23 - 40 . MR 506653 | Zbl 0379.53019 · Zbl 0379.53019 · doi:10.1007/BF01351556
[4]	Z. El Houakmi , Comportement asymptotique du spectre en présence de symétries . Thèse de 3e cycle à Nantes , Juin 1983 .
[5]	Gildener , PATRASCIOU , Phys. Rev. D. , t. 16 , n^\circ 2 , Juillet 1977 .
[6]	E.M. Harrell , On the rate of asymptotic eigenvalue degeneracy , Comm. Math. Phys. , t. 60 , 1978 , p. 73 - 95 . Article | MR 486764 | Zbl 0395.34023 · Zbl 0395.34023 · doi:10.1007/BF01609474
[7]	E.M. Harrell , Double wells , Comm. Math. Phys. , t. 75 , 1980 , p. 239 - 261 . Article | MR 581948 | Zbl 0445.35036 · Zbl 0445.35036 · doi:10.1007/BF01212711
[8]	B. Helffer , D. Robert , Étude du spectre pour un opérateur globalement elliptique dont le symbole de Weyl présente des symétries : I. Action des groupes finis (à paraître à Amer. J. of Math. ). II. Action des groupes compacts (Preprint). MR 761584 · Zbl 0605.58042
[9]	B. Helffer , D. Robert , Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles . J. of Functional Analysis , t. 53 , n^\circ 3 , 1983 , p. 246 - 268 . MR 724029 | Zbl 0524.35103 · Zbl 0524.35103 · doi:10.1016/0022-1236(83)90034-4
[10]	B. Helffer , D. Robert , Puits de potentiel généralisés et asymptotique semi-classique . Annales de l’I. H. P. , Vol. 41 , n^\circ 3 , 1984 , p. 291 - 331 . Numdam | MR 776281 | Zbl 0565.35082 · Zbl 0565.35082
[11]	B. Helffer , J. Sjöstrand , Multiple wells in the semi-classical limit I . Comm. in P.D.E. , t. 9 , n^\circ 4 , 1984 , p. 337 - 408 . MR 740094 | Zbl 0546.35053 · Zbl 0546.35053 · doi:10.1080/03605308408820335
[12]	B. Helffer , J. Sjöstrand , Multiple wells in the semi-classical limit III. Interaction through non-resonant wells (à paraître Mathematische Nachrichte ). Zbl 0597.35023 · Zbl 0597.35023 · doi:10.1002/mana.19851240117
[13]	G. Jona-Lasinio , F. Martinelli et E. Scoppola , New approach in the semi-classical limit of quantum mechanics I. Multiple tunnelings in one dimension . Comm. Math. Phys. , t. 80 , 1981 , p. 223 . Article | MR 623159 | Zbl 0483.60094 · Zbl 0483.60094 · doi:10.1007/BF01213012
[14]	L. Landau , E. Lifchitz , Mécanique quantique, théorie non relativiste , Éditions Mir , Moscou , 1966 . Zbl 0144.47605 · Zbl 0144.47605
[15]	Morgan , B. Simon , Behaviour of molecular potential energy curves for large nuclear separations , International Journal of Quantum chemistry , t. XVII , 1980 , p. 1143 - 1166 .
[16]	M. Reed , B. Simon , Methods of modern mathematical physics , t. 4 , 1978 , Academic Press , New York . Zbl 0401.47001 · Zbl 0401.47001
[17]	J.P. Serre , Représentation linéaire des groupes finis ( Herman ), 1967 . Zbl 0189.02603 · Zbl 0189.02603
[18]	B. Simon , Semi-classical analysis of low lying eigenvalues I . Ann. Inst. Poincaré , t. 38 , 1983 , p. 295 - 307 . Numdam | Zbl 0526.35027 · Zbl 0526.35027
[19]	B. Simon , Instantons, double wells and large deviations . Bull. AMS. , t. 8 , 1983 , p. 323 - 326 . Article | MR 684899 | Zbl 0529.35059 · Zbl 0529.35059 · doi:10.1090/S0273-0979-1983-15104-2
[20]	B. Simon , Semi-classical analysis of low lying eigenvalues II, Tunneling ( Annals of Mathematics , 1984 ). Zbl 0626.35070 · Zbl 0626.35070 · doi:10.2307/2007072