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Spectral properties of the massless relativistic quartic oscillator

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journal contribution
posted on 29.11.2017, 15:52 by Samuel O. Durugo, Jozsef Lorinczi
An explicit solution of the spectral problem of the non-local Schrodinger operator obtained as the sum of the square root of the Laplacian and a quartic potential in one dimension is presented. The eigenvalues are obtained as zeroes of special functions re- lated to the fourth order Airy function, and closed formulae for the Fourier transform of the eigenfunctions are derived. These representations allow to derive further spectral properties such as estimates of spectral gaps, heat trace and the asymptotic distribution of eigenvalues, as well as a detailed analysis of the eigenfunctions. A subtle spectral effect is observed which manifests in an exponentially tight approximation of the spectrum by the zeroes of the dominating term in the Fourier representation of the eigenfunctions and its derivative.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Journal of Differential Equations

Volume

264

Issue

5

Pages

3775 - 3809

Citation

DURUGO, S.O. and LORINCZI, J., 2018. Spectral properties of the massless relativistic quartic oscillator. Journal of Differential Equations, 264 (5), pp. 3775-3809.

Publisher

© Elsevier

Version

AM (Accepted Manuscript)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Acceptance date

27/11/2017

Publication date

2017-12-08

Notes

This paper was accepted for publication in the journal Journal of Differential Equations and the definitive published version is available at https://doi.org/10.1016/j.jde.2017.11.030

ISSN

0022-0396

Language

en