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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
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