## On the spectrum of the Wannier-Stark operator.(English. Russian original)Zbl 1071.81041

St. Petersbg. Math. J. 16, No. 3, 561-581 (2005); translation from Algebra Anal. 16, No. 3, 171-200 (2004).
In the paper it is considered the one-dimensional Schrödinger operator $H=-\frac{d^2}{dx^2}-Fx+p(x)$ in $$L_2(\mathbb{R}_+)$$ with domain dom$$(H)=\{ \varphi: \varphi'$$ is absolutely continuous, $$\varphi(0)=0$$, supp$$(\varphi)$$ is bounded, $$-\varphi''+p\varphi\in L_2(\mathbb{R}_+) \}.$$ Here the potential $$p(x)$$ is a periodic real valued function such that $$p(x)\in L_1[0,1]$$. It is proved that $$H$$ is essentially selfadjoint. By $$H_d$$ it is denoted the closure of $$H$$.
The result of the author’s previous paper [St. Petersbg. Math. J. 14, 119–145 (2003; Zbl 1044.81041)] is generalized. This can be seen putting $r(l)=e^{i\frac{3\pi}{4}}\frac{1}{\sqrt{2F}}\int_0^1p(t)e^{2i\pi lt}\,dt, \qquad w(l)=\int_0^1\int_0^tp(y)p(t)e^{2i\pi lt}\,dy\,dt.$ The main result of the paper is the following
Theorem. If $$p(x)$$ satisfies the conditions $\sum_{l=1}^{\infty}| r(l)| l^{-1}<\infty, \qquad \sum_{l=1}^{\infty}| w(l)| l^{-1}<\infty,$ then the absolute continuous spectrum of $$H_d$$ fills the real axis.
It should be noted that the proof of the theorem is based on the results of the mentioned paper. In the proof also the methods of the following papers are used: [V. S. Buslaev, Differential operators and Spectral Theory, 45–57 (1999; Zbl 0935.34072)], [P. Deift and R. Killip, Commun. Math. Phys. 189, 341–347 (1999; Zbl 0934.34075), D. J. Gilbert and D. B. Pearson, J. Math. Anal. Appl. 128, 30–56 (1987; Zbl 0666.34023), Y. Last and B. Simon, Invent. Math. 135, 329–367 (1999; Zbl 0931.34066)].

### MSC:

 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 47E05 General theory of ordinary differential operators 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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### References:

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