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Geometric tools of the adiabatic complex WKB method. (English) Zbl 1070.34124

The authors study the asymptotic behavior of the one-dimensional Schrödinger equation \[ -\frac{d^2\psi}{dx^2}\,(x)+V(x)\psi(x)+W(\varepsilon x)\psi(x)=E\psi(x), \] \(x\in\mathbb R\), where \(\varepsilon\) is small positive parameter, and \(V(x)\) a real-valued periodic function, with \(V(x+1)=V(x)\). It is assumed that \(V\in L^{2}_{\text{loc}}\) and that \(\zeta\mapsto W(\zeta)\) is analytic in some neighborhood of \(\mathbb R\subset\mathbb C\). Here, the authors present a new geometrical approach which simplifies the complexity of the computations connected with their asymptotic method [see the authors, Asymptotic Anal. 27, 219–264 (2001; Zbl 1001.34082)].

MSC:

34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent)
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)

Citations:

Zbl 1001.34082
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