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Precise estimates for tunneling and eigenvalues near a potential barrier. (English) Zbl 0668.34022
The authors study the semiclassical Schrödinger operator $$P=-h^ 2(d^ 2/dx^ 2)+V(x)$$ on $$L^ 2({\mathbb{R}})$$ with a real analytic potential V presenting a barrier between two potential wells $$(V(0)=V'(0)=0$$, $$V''(0)<0$$, $$\liminf_{| x| \to \infty}V(x)>0)$$. This operator has a discrete spectrum near 0 (the potential barrier). Asymptotic expansions as h tends to 0 of the eigenvalues near 0 and of the difference between two successive eigenvalues (splitting) are given. In addition to known results new estimates of the coefficients in these expansions are established. In order to prove the results exact solutions of the eigenvalue equation are constructed along the lines of A. Voros [Ann. Inst. H. Poincaré 29, 211-338 (1983; Zbl 0526.34046)] and J. Ecalle, and their WKB expansion (the “exact” WKB method) is used.
Reviewer: J.Weidmann

##### MSC:
 34L99 Ordinary differential operators 47E05 General theory of ordinary differential operators 34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
##### Keywords:
Asymptotic expansions; eigenvalues; WKB expansion; WKB method
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##### References:
 [1] Ecalle, J, Cinq applications des fonctions résurgentes, (1984), prépublications d’Orsay [2] Ford, K.W; Hill, D.L; Wakeno, M; Wheeler, J.A, Quantum effect near a barrier maximum, Ann. physics, 7, 239-258, (1959) · Zbl 0086.22406 [3] Grigis, A, Sur l’équation de Hill analytique, (), exposé n∘ 16 · Zbl 0567.34023 [4] Helffer, B; Robert, D, Puits de potentiel généralisés et asymptotique semiclassique, Ann. inst. H. Poincaré, 41, 291-331, (1984) · Zbl 0565.35082 [5] Helffer, B; Sjöstrand, J, Multiple wells in the semiclassical limit, I, Comm. partial differential equations, 337-408, (1984) · Zbl 0546.35053 [6] Martinez, A, Estimation de l’effet tunnel pour le double puits, II, etats hautement excités, (), exposé n∘ 15 · Zbl 0666.35069 [7] Sibuya, Y, Global theory of a second order linear ordinary differential equation with a polynomial coefficient, (1975), North-Holland Amsterdam · Zbl 0322.34006 [8] Simon, B, Instantons, double wells and large deviations, () · Zbl 0529.35059 [9] Sjöstrand, J, Singularités analytiques microlocales, (1982), Astérisque, S.M.F · Zbl 0524.35007 [10] Voros, A, The return of the quartic oscillator. the complex W.K.B. method, Ann. inst. H. Poincaré, 29, 211-338, (1983) · Zbl 0526.34046
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