## 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 $$-\psi''+p(x)\psi+q(\epsilon x)\psi=E\psi$$, $$\epsilon\to 0$$.

### MSC:

 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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### References:

 [1] Buslaev V. S., Alg. Anal. 23 pp 1– (1989) [2] Buslaev V. S., Math. J. 1 pp 287– (1990) [3] Buslaev V. S., Usp. Mat. Nauk 42 pp 77– (1987) [4] DOI: 10.1016/0003-4916(82)90213-5 [5] DOI: 10.1007/BF01212445 · Zbl 0651.47006 [6] DOI: 10.1007/BF02099175 · Zbl 0743.35053 [7] DOI: 10.1007/BF02099501 · Zbl 0737.34060 [8] DOI: 10.1007/BF02102626 · Zbl 0770.47030 [9] DOI: 10.1007/BF01206948 · Zbl 0493.47009 [10] DOI: 10.1007/BF02100105 · Zbl 0851.34078 [11] DOI: 10.1007/BF01017921 · Zbl 0557.34053 [12] Firsova N. E., Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 51 pp 183– (1975) [13] Marchenko V. A., Mat. Sb. 97 pp 540– (1975)
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