Zeros of eigenfunctions of some anharmonic oscillators

Alexandre Eremenko[1]; Andrei Gabrielov[1]; Boris Shapiro[2]

  • [1] Purdue University West Lafayette, IN 47907-2067 (USA)
  • [2] Stockholm University Stockholm, S-10691 (Sweden)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 2, page 603-624
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Eremenko, Alexandre, Gabrielov, Andrei, and Shapiro, Boris. "Zeros of eigenfunctions of some anharmonic oscillators." Annales de l’institut Fourier 58.2 (2008): 603-624. <http://eudml.org/doc/10326>.

@article{Eremenko2008,
abstract = {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.},
affiliation = {Purdue University West Lafayette, IN 47907-2067 (USA); Purdue University West Lafayette, IN 47907-2067 (USA); Stockholm University Stockholm, S-10691 (Sweden)},
author = {Eremenko, Alexandre, Gabrielov, Andrei, Shapiro, Boris},
journal = {Annales de l’institut Fourier},
keywords = {Eigenfunctions; meromorphic functions; distribution of zeros; second order differential operator; eigenfunctions},
language = {eng},
number = {2},
pages = {603-624},
publisher = {Association des Annales de l’institut Fourier},
title = {Zeros of eigenfunctions of some anharmonic oscillators},
url = {http://eudml.org/doc/10326},
volume = {58},
year = {2008},
}

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
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 2
SP - 603
EP - 624
AB - 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.
LA - eng
KW - Eigenfunctions; meromorphic functions; distribution of zeros; second order differential operator; eigenfunctions
UR - http://eudml.org/doc/10326
ER -

References

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