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Mosc. Math. J., 2011, Volume 11, Number 3, Pages 473–503 (Mi mmj428)  

This article is cited in 10 scientific papers (total in 10 papers)

Singular perturbation of polynomial potentials with applications to $PT$-symmetric families

Alexandre Eremenko, Andrei Gabrielov

Purdue University, West Lafayette, IN, USA

Abstract: We discuss eigenvalue problems of the form $-w"+Pw=\lambda w$ with complex polynomial potential $P(z)=tz^d+\ldots$, where $t$ is a parameter, with zero boundary conditions at infinity on two rays in the complex plane. In the first part of the paper we give sufficient conditions for continuity of the spectrum at $t=0$. In the second part we apply these results to the study of topology and geometry of the real spectral loci of $PT$-symmetric families with $P$ of degree 3 and 4, and prove several related results on the location of zeros of their eigenfunctions.

Key words and phrases: singular perturbation, Schrödinger operator, eigenvalue, spectral determinant, $PT$-symmetry.

Full text: http://www.ams.org/.../abst11-3-2011.html
References: PDF file   HTML file

Bibliographic databases:

Document Type: Article
MSC: 34M35, 35J10
Language: English

Citation: Alexandre Eremenko, Andrei Gabrielov, “Singular perturbation of polynomial potentials with applications to $PT$-symmetric families”, Mosc. Math. J., 11:3 (2011), 473–503

Citation in format AMSBIB
\Bibitem{EreGab11}
\by Alexandre~Eremenko, Andrei~Gabrielov
\paper Singular perturbation of polynomial potentials with applications to $PT$-symmetric families
\jour Mosc. Math.~J.
\yr 2011
\vol 11
\issue 3
\pages 473--503
\mathnet{http://mi.mathnet.ru/mmj428}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2894426}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000300365900004}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Eremenko A., Gabrielov A., “Quasi-exactly solvable quartic: elementary integrals and asymptotics”, J. Phys. A, 44:31 (2011), 312001, 14 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. Eremenko A., Gabrielov A., “Two-Parametric Pt-Symmetric Quartic Family”, J. Phys. A-Math. Theor., 45:17 (2012), 175206  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Eremenko A., Gabrielov A., “Quasi-Exactly Solvable Quartic: Real Algebraic Spectral Locus”, J. Phys. A-Math. Theor., 45:17 (2012), 175205  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. Steinmetz N., “Complex Riccati Differential Equations Revisited”, Ann. Acad. Sci. Fenn. Ser. A1-Math., 39:2 (2014), 503–511  crossref  mathscinet  zmath  isi  scopus
    5. Tumanov S.N., Shkalikov A.A., “the Limit Spectral Graph in Semiclassical Approximation For the Sturm-Liouville Problem With Complex Polynomial Potential”, Dokl. Math., 92:3 (2015), 773–777  crossref  mathscinet  zmath  isi  elib  scopus
    6. Shapiro B., “on Evgrafov-Fedoryuk'S Theory and Quadratic Differentials”, Anal. Math. Phys., 5:2 (2015), 171–181  crossref  mathscinet  zmath  isi  scopus
    7. R. Giachetti, V. Grecchi, “Level crossings in a $PT$-symmetric double well”, J. Phys. A-Math. Theor., 49:10, SI (2016), 105202  crossref  mathscinet  zmath  isi  scopus
    8. R. Giachetti, V. Grecchi, “Bender–Wu singularities”, J. Math. Phys., 57:12 (2016), 122109  crossref  mathscinet  zmath  isi  scopus
    9. Tumanov S.N., Shkalikov A.A., “Eigenvalue Dynamics of a Pj -Symmetric Sturm-Liouville Operator and Criteria For Similarity to a Self-Adjoint Or a Normal Operator”, Dokl. Math., 96:3 (2017), 607–611  crossref  mathscinet  zmath  isi
    10. Steinmetz N., “Nevanlinna Theory, Normal Families, and Algebraic Differential Equations”, Nevanlinna Theory, Normal Families, and Algebraic Differential Equations, Universitext, Springer, 2017, 1–235  crossref  mathscinet  isi
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