Imagrinary parts of Stark-Wannier resonances. (English) Zbl 1001.34075

Summary: We consider a one-dimensional Stark-Wannier Hamiltonian, \(H=-d^2/dx^2+p(x)-\epsilon x\), \(x\in\mathbb R\), where \(p\) is a smooth periodic, finite-gap potential, and \(\epsilon>0\) is small enough. We compute rigorously the imaginary parts of the spectral resonances. For this purpose we develop some related elements of the adiabatic approach to the equations of the form \(-\psi''+p(x)\psi+q(\epsilon x)\psi=E\psi\), \(\epsilon\to 0\).


34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
34L05 General spectral theory of ordinary differential operators
81U05 \(2\)-body potential quantum scattering theory
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