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Strong resonant tunneling, level repulsion and spectral type for one-dimensional adiabatic quasi-periodic Schrödinger operators. (English) Zbl 1112.47038

The authors consider a family of one-dimensional Schrödinger operators \[ (H_{z,\varepsilon }\psi )(x)=-\psi''(x)+(V(x-z)+\alpha \cos (\varepsilon x))\psi (x), \] where \(\alpha >0\), \(V\in \L_2^{\text{loc}}(\mathbb R)\) is a non-constant 1-periodic function, \(\varepsilon\) is a small positive number such that \(2\pi /\varepsilon\) is irrational, and \(z\) is a real parameter. Let \(E\) belong to a spectral gap of the periodic operator \(H_0=-d^2/dx^2 +V(x)\), while the spectral window \([E-\alpha ,E+\alpha ]\) contains the gap. In this energy, region the spectrum of \(H_{z,\varepsilon }\) is localized near two sequences of quantized energy values; each of the sequences is “generated” by one of the ends of the neighboring spectral bands of \(H_0\).
The paper is devoted to the resonant case where two of the “interacting energies” are very close or even coincide. The authors perform a detailed study of the nature and location of the spectrum.

MSC:

47E05 General theory of ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47N50 Applications of operator theory in the physical sciences
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47A10 Spectrum, resolvent
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