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

Abstract : We compute an explicit formula for the integrated density of states of the periodic Airy-Schrödinger operator on the real line. The potential of this Schrödinger operator is periodic, continuous and piecewise affine. For this purpose, we study precisely the spectrum of the Schrödinger operator whose potential is the restriction of the periodic Airy-Schrödinger 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-Schrödinger 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.
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Cited literature [20 references]

https://hal.archives-ouvertes.fr/hal-02964056
Contributor : Hakim Boumaza <>
Submitted on : Wednesday, October 14, 2020 - 11:01:27 AM
Last modification on : Tuesday, October 20, 2020 - 3:56:15 PM

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• HAL Id : hal-02964056, version 1

### Citation

Hakim Boumaza, Olivier Lafitte. Integrated density of states: from the finite range to the periodic Airy-Schrödinger operator. 2020. ⟨hal-02964056⟩

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