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

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
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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} 

• http://mi.mathnet.ru/eng/mmj428
• http://mi.mathnet.ru/eng/mmj/v11/i3/p473

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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.
2. Eremenko A., Gabrielov A., “Two-Parametric Pt-Symmetric Quartic Family”, J. Phys. A-Math. Theor., 45:17 (2012), 175206
3. Eremenko A., Gabrielov A., “Quasi-Exactly Solvable Quartic: Real Algebraic Spectral Locus”, J. Phys. A-Math. Theor., 45:17 (2012), 175205
4. Steinmetz N., “Complex Riccati Differential Equations Revisited”, Ann. Acad. Sci. Fenn. Ser. A1-Math., 39:2 (2014), 503–511
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
6. Shapiro B., “on Evgrafov-Fedoryuk'S Theory and Quadratic Differentials”, Anal. Math. Phys., 5:2 (2015), 171–181
7. R. Giachetti, V. Grecchi, “Level crossings in a $PT$-symmetric double well”, J. Phys. A-Math. Theor., 49:10, SI (2016), 105202
8. R. Giachetti, V. Grecchi, “Bender–Wu singularities”, J. Math. Phys., 57:12 (2016), 122109
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
10. Steinmetz N., “Nevanlinna Theory, Normal Families, and Algebraic Differential Equations”, Nevanlinna Theory, Normal Families, and Algebraic Differential Equations, Universitext, Springer, 2017, 1–235