Zeros of eigenfunctions of some anharmonic oscillators

Eremenko, Alexandre; Gabrielov, Andrei; Shapiro, Boris

Annales de l'Institut Fourier, Tome 58 (2008) no. 2, pp. 603-624.

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 $4$, 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 $6$, 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.

Classification : 34L40, 81Q10, 30D99
Keywords: Eigenfunctions, meromorphic functions, distribution of zeros
Mot clés : fonctions propres, fonctions méromorphes, distribution de zéros

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