[Zéros des fonctions propres de certains oscillateurs anharmoniques]
On étudie les zéros complexes des fonctions propres d’opérateurs différentiels linéaires du second ordre avec des potentiels polynomiaux réels pairs. Pour les potentiels de degré , on montre que tous les zéros de toutes les fonctions propres appartiennent à la réunion de l’axe réel et l’axe imaginaire. Pour les potentiels de degré , on classifie les fonctions propres ayant un nombre fini de zéros et on montre que, dans ce cas aussi, tous les zéros sont réels ou imaginaires purs.
We study complex zeros of eigenfunctions of second order linear differential operators with real even polynomial potentials. For potentials of degree 4, we prove that all zeros of all eigenfunctions belong to the union of the real and imaginary axes. For potentials of degree 6, we classify eigenfunctions with finitely many zeros, and show that in this case too, all zeros are real or pure imaginary.
Keywords: Eigenfunctions, meromorphic functions, distribution of zeros
Mot clés : fonctions propres, fonctions méromorphes, distribution de zéros
Eremenko, Alexandre 1 ; Gabrielov, Andrei 1 ; Shapiro, Boris 2
@article{AIF_2008__58_2_603_0,
author = {Eremenko, Alexandre and Gabrielov, Andrei and Shapiro, Boris},
title = {Zeros of eigenfunctions of some anharmonic oscillators},
journal = {Annales de l'Institut Fourier},
pages = {603--624},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {58},
number = {2},
year = {2008},
doi = {10.5802/aif.2362},
mrnumber = {2410384},
zbl = {1155.34043},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2362/}
}
TY - JOUR AU - Eremenko, Alexandre AU - Gabrielov, Andrei AU - Shapiro, Boris TI - Zeros of eigenfunctions of some anharmonic oscillators JO - Annales de l'Institut Fourier PY - 2008 SP - 603 EP - 624 VL - 58 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2362/ DO - 10.5802/aif.2362 LA - en ID - AIF_2008__58_2_603_0 ER -
%0 Journal Article %A Eremenko, Alexandre %A Gabrielov, Andrei %A Shapiro, Boris %T Zeros of eigenfunctions of some anharmonic oscillators %J Annales de l'Institut Fourier %D 2008 %P 603-624 %V 58 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2362/ %R 10.5802/aif.2362 %G en %F AIF_2008__58_2_603_0
Eremenko, Alexandre; Gabrielov, Andrei; Shapiro, Boris. Zeros of eigenfunctions of some anharmonic oscillators. Annales de l'Institut Fourier, Tome 58 (2008) no. 2, pp. 603-624. doi : 10.5802/aif.2362. https://aif.centre-mersenne.org/articles/10.5802/aif.2362/
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