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
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
@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}, zbl = {1155.34043}, mrnumber = {2410384}, language = {en}, url = {https://www.numdam.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://www.numdam.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://www.numdam.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://www.numdam.org/articles/10.5802/aif.2362/
[1] A note on the zeros of solutions
[2] The Schrödinger equation, Kluwer, Dordrecht, 1991 | MR | Zbl
[3] Über die Darstellung Riemannscher Flächen durch Streckenkomplexe, Deutsche Math., Volume 1 (1936), pp. 805-824
[4] Asymptotique et analyse spectrale de l’oscillateur cubique, Université de Nice (2002) (Ph. D. Thesis)
[5] On the Sturm-Liouville problem for complex cubic oscillator, Asymptot. Anal., Volume 40 (2004) no. 3-4, pp. 211-324 | MR | Zbl
[6] High energy eigenfunctions of one-dimensional Schrödinger operators with polynomial potentials (Preprint arXiv:math-ph/0703049)
[7] Nevanlinna functions with real zeros, Illinois J. Math., Volume 49 (2005) no. 3-4, pp. 1093-1110 | MR | Zbl
[8] Distribution of values of meromorphic functions, Nauka, Moscow, 1970 (English translation to appear in AMS) | MR
[9] Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators, Comm. Math. Phys., Volume 153 (1993), pp. 117-146 | DOI | MR | Zbl
[10] Lectures on ordinary differential equations, Addison-Wesley, Menlo Park, CA, 1969 | MR | Zbl
[11] Ordinary differential equations in the complex domain, John Wiley and Sons, New York, 1976 | MR | Zbl
[12] Lie algebras, cohomology and new applications in quantum mechanics, Contemp. Math., 160, Amer. Math. Soc., Providence, RI, 1994 | MR | Zbl
[13] Über die Herstellung transzendenter Funktionen als Grenzwerte rationaler Funktionen, Acta Math., Volume 55 (1930), pp. 259-276 | DOI | MR
[14] Über Riemannsche Flächen mit endlich vielen Windungspunkten, Acta Math., Volume 58 (1932), pp. 295-373 | DOI | MR
[15] Eindeutige analytische Funktionen, 2-te Aufl., Springer, Berlin-Göttingen-Heidelberg, 1953 | MR | Zbl
[16] Quasi-exactly-solvable spectral problems and conformal field theory, Contemp. Math., 160, Amer. Math. Soc., Providence, RI, 1994 | MR | Zbl
[17] Global theory of a second order linear ordinary differential equation with a polynomial coefficient, North-Holland Publishing Co., Amsterdam-Oxford, 1995 | MR | Zbl
[18] Sur certains polynômes qui vérifient une équation differentielle linéaire du second ordre et sur la théorie des fonctions de Lamé, Acta Math., Volume 6 (1885), pp. 321-326 | DOI | MR
[19] Œuvres complètes, 1, Springer, Berlin, 1993 | Zbl
[20] Eigenfunction expansions associated with second order differential equations, 1, Clarendon Press, Oxford, 1946 | Zbl
[21] Quasi-exactly-solvable problems and
[22] Lie algebras and linear operators with invariant subspaces, Contemp. Math., 160, Amer. Math. Soc., Providence, RI, 1994 | MR | Zbl
[23] Anharmonic oscillator and double well potential: approximating eigenfunctions, Letters in Math. Phys., Volume 74 (2005), pp. 169-180 | DOI | MR | Zbl
[24] Spectral singularities and the quasi-exactly solvable problem, Phys. Lett. A, Volume 126 (1987), pp. 181-183 | DOI | MR
[25] Quasi-exactly solvable models in quantum mechanics, Inst. of Physics Publ., Bristol, 1994 | MR | Zbl
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