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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
[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
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