## 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.)

Zbl 1001.34082
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