Integrated density of states: from the finite range to the periodic Airy-Schroedinger operator

We compute an explicit formula for the integrated density of states of the periodic Airy-Schroedinger operator on the real line. The potential of this Schroedinger operator is periodic, continuous and piecewise affine. For this purpose, we study precisely the spectrum of the Schroedinger operator wh...

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Bibliographic Details
Published in:Journal of mathematical physics 2021-04, Vol.62
Main Authors: Boumaza, Hakim, Lafitte, Olivier
Format: Article
Language:English
Subjects:
Mathematical Physics
Mathematics
Spectral Theory
Online Access:Get full text
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Summary:We compute an explicit formula for the integrated density of states of the periodic Airy-Schroedinger operator on the real line. The potential of this Schroedinger operator is periodic, continuous and piecewise affine. For this purpose, we study precisely the spectrum of the Schroedinger operator whose potential is the restriction of the periodic Airy-Schroedinger potential to a finite number of periods. We prove that all the eigenvalues of the operator corresponding to the restricted potential are in the spectral bands of the periodic Airy-Schroedinger operator and none of them are in its spectral gaps. We count exactly the number of these eigenvalues in each of these spectral bands. Note that our results depend on a semiclassical parameter and are valid for values of it larger than explicit constants.
ISSN:0022-2488
DOI:10.1063/5.0015181