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A product formula for the eigenfunctions of a quartic oscillator

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journal contribution
posted on 17.03.2015 by Martin Hallnas, Edwin Langmann
We consider the Schrödinger operator on the real line with an even quartic potential. Our main result is a product formula of the type. ψk(x)ψk(y)=∫Rψk(z)K(x,y,z)dz for its eigenfunctions ψk. The kernel function K is given explicitly in terms of the Airy function Ai(x), and it is positive for appropriate parameter values. As an application, we obtain a particular asymptotic expansion of the eigenfunctions ψk.

Funding

This work was supported by the Göran Gustafsson Foundation (grant No. GGS 1221) and the Swedish Research Council (VR) under contract No.621-2010-3708.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Journal of Mathematical Analysis and Applications

Citation

HALLNAS, M. and LANGMANN, E., 2015. A product formula for the eigenfunctions of a quartic oscillator. Journal of Mathematical Analysis and Applications, 426 (2), pp. 1012-1025.

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/

Publication date

2015

Notes

This article was published in Journal of Mathematical Analysis and Applications [© Elsevier] and the definitive version is available at: http://dx.doi.org/10.1016/j.jmaa.2015.02.014

ISSN

0022-247X

eISSN

1096-0813

Language

en

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