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Instable equilibrium in semi-classical limit. (Équilibre instable en régime semi-classique.) (French) Zbl 0880.34084

Families of eigenfunctions associated with the families of eigenvalues of the dimension-one Schrödinger equation of the form \[ P\psi_h= \Biggl(-{1\over 2} h^2 {d^2\over dx^2}+ V(x)\Biggr)\phi_h= E(h)\phi_h, \] where \(h\to 0\) for energy values \(E(h)\) near the local maximum of \(V(x)\) (four potentials \(V\) are used), are studied. The results relate to the concentration of the eigenfunctions and to the form of the energy spectra near the energy critical values.
A formula describing the microlocal concentration \(\mu_h\) is obtained. The quantification of energy values by the Bohr-Sommerfeld conditions is discussed. Propagation of singularities is studied. The form parameter of the energy spectra is shown. The results are obtained in collaboration with Y. Colin de Verdière and B. Parisse [Commun. Partial Differ. Equations 19, No. 9-10, 1553-1563 (1994; Zbl 0819.35116); Ann. Inst. Henri Poincaré, Phys. Théor. 61, No. 3, 347-367 (1994; Zbl 0845.35076)].
Reviewer: V.Burjan (Praha)

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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