A third order operator with periodic coefficients on the real line

Badanin, A.; Korotyaev, E.

doi:10.1090/S1061-0022-2014-01313-1

A third order operator with periodic coefficients on the real line
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by A. V. Badanin and E. L. Korotyaev
Translated by: N. B. Lebedinskaya

St. Petersburg Math. J. 25 (2014), 713-734

DOI: https://doi.org/10.1090/S1061-0022-2014-01313-1

Published electronically: July 18, 2014

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Abstract:

The operator $i\partial ^3+i \partial p+i p\partial +q$ with 1-periodic coefficients $p,q\in L_{\mathrm {loc}}^1(\mathbb {R})$ is considered on the real line. The following results are obtained: 1) the spectrum of this operator is absolutely continuous, covers the entire real line, and has multiplicity one or three; 2) the spectrum of multiplicity three is bounded and expressed in terms of real zeros of a certain entire function; 3) the Lyapunov function, analytic on a 3-sheeted Riemann surface, is constructed and investigated.

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