Zeros of eigenfunctions of some anharmonic oscillators
Annales de l'Institut Fourier, Volume 58 (2008) no. 2, pp. 603-624.

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.

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.

DOI: 10.5802/aif.2362
Classification: 34L40, 81Q10, 30D99
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

1 Purdue University West Lafayette, IN 47907-2067 (USA)
2 Stockholm University Stockholm, S-10691 (Sweden)
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Eremenko, Alexandre; Gabrielov, Andrei; Shapiro, Boris. Zeros of eigenfunctions of some anharmonic oscillators. Annales de l'Institut Fourier, Volume 58 (2008) no. 2, pp. 603-624. doi : 10.5802/aif.2362. http://www.numdam.org/articles/10.5802/aif.2362/

[1] Bank, S. A note on the zeros of solutions w +P(z)w=0 where P is a polynomial, Appl. Anal., Volume 25 (1987) no. 1-2, pp. 29-41 | DOI | MR | Zbl

[2] Berezin, F. A.; Shubin, M. A. The Schrödinger equation, Kluwer, Dordrecht, 1991 | MR | Zbl

[3] Drape, E. Über die Darstellung Riemannscher Flächen durch Streckenkomplexe, Deutsche Math., Volume 1 (1936), pp. 805-824

[4] Duc Tai Trinh Asymptotique et analyse spectrale de l’oscillateur cubique, Université de Nice (2002) (Ph. D. Thesis)

[5] Duc Tai Trinh On the Sturm-Liouville problem for complex cubic oscillator, Asymptot. Anal., Volume 40 (2004) no. 3-4, pp. 211-324 | MR | Zbl

[6] Eremenko, A.; Gabrielov, A.; Shapiro, B. High energy eigenfunctions of one-dimensional Schrödinger operators with polynomial potentials (Preprint arXiv:math-ph/0703049)

[7] Eremenko, A.; Merenkov, S. Nevanlinna functions with real zeros, Illinois J. Math., Volume 49 (2005) no. 3-4, pp. 1093-1110 | MR | Zbl

[8] Goldberg, A. A.; Ostrovskii, I. V. Distribution of values of meromorphic functions, Nauka, Moscow, 1970 (English translation to appear in AMS) | MR

[9] González-Lopéz, A.; Kamran, N.; Olver, P. Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators, Comm. Math. Phys., Volume 153 (1993), pp. 117-146 | DOI | MR | Zbl

[10] Hille, E. Lectures on ordinary differential equations, Addison-Wesley, Menlo Park, CA, 1969 | MR | Zbl

[11] Hille, E. Ordinary differential equations in the complex domain, John Wiley and Sons, New York, 1976 | MR | Zbl

[12] Kamran, N.; Olver, P. Lie algebras, cohomology and new applications in quantum mechanics, Contemp. Math., 160, Amer. Math. Soc., Providence, RI, 1994 | MR | Zbl

[13] Nevanlinna, R. Über die Herstellung transzendenter Funktionen als Grenzwerte rationaler Funktionen, Acta Math., Volume 55 (1930), pp. 259-276 | DOI | MR

[14] Nevanlinna, R. Über Riemannsche Flächen mit endlich vielen Windungspunkten, Acta Math., Volume 58 (1932), pp. 295-373 | DOI | MR

[15] Nevanlinna, R. Eindeutige analytische Funktionen, 2-te Aufl., Springer, Berlin-Göttingen-Heidelberg, 1953 | MR | Zbl

[16] Shifman, M. Quasi-exactly-solvable spectral problems and conformal field theory, Contemp. Math., 160, Amer. Math. Soc., Providence, RI, 1994 | MR | Zbl

[17] Sibuya, Y. Global theory of a second order linear ordinary differential equation with a polynomial coefficient, North-Holland Publishing Co., Amsterdam-Oxford, 1995 | MR | Zbl

[18] Stieltjes, T. 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] Stieltjes, T. Œuvres complètes, 1, Springer, Berlin, 1993 | Zbl

[20] Titchmarsh, E. Eigenfunction expansions associated with second order differential equations, 1, Clarendon Press, Oxford, 1946 | Zbl

[21] Turbiner, A. Quasi-exactly-solvable problems and sl (2) algebra, Comm. Math. Phys., Volume 118 (1988), pp. 467-474 | DOI | MR | Zbl

[22] Turbiner, A. Lie algebras and linear operators with invariant subspaces, Contemp. Math., 160, Amer. Math. Soc., Providence, RI, 1994 | MR | Zbl

[23] Turbiner, A. Anharmonic oscillator and double well potential: approximating eigenfunctions, Letters in Math. Phys., Volume 74 (2005), pp. 169-180 | DOI | MR | Zbl

[24] Turbiner, A.; Ushveridze, A. Spectral singularities and the quasi-exactly solvable problem, Phys. Lett. A, Volume 126 (1987), pp. 181-183 | DOI | MR

[25] Ushveridze, A. Quasi-exactly solvable models in quantum mechanics, Inst. of Physics Publ., Bristol, 1994 | MR | Zbl

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