Analytic Continuation of Eigenvalues of a Quartic Oscillator

We consider the Schrödinger operator on the real line with even quartic potential x 4 + α x 2 and study analytic continuation of eigenvalues, as functions of parameter α . We prove several properties of this analytic continuation conjectured by Bender, Wu, Loeffel and Martin. 1. All eigenvalues ar...

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Vydáno v:	Communications in mathematical physics Ročník 287; číslo 2; s. 431 - 457
Hlavní autoři:	Eremenko, Alexandre, Gabrielov, Andrei
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer-Verlag 01.04.2009 Springer
Témata:	Classical and Quantum Gravitation Complex Systems Exact sciences and technology Functions of a complex variable Mathematical analysis Mathematical and Computational Physics Mathematical methods in physics Mathematical Physics Mathematics Other topics in mathematical methods in physics Partial differential equations Physics Physics and Astronomy Quantum Physics Relativity Theory Sciences and techniques of general use Several complex variables and analytic spaces Theoretical Cell Decomposition Meromorphic Function Riemann Surface Analytic Continuation Anharmonic Oscillator Complex function Eigenvalues Eigenfunctions Analytic functions Analytic continuation Mathematical physics
ISSN:	0010-3616, 1432-0916
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Abstract	We consider the Schrödinger operator on the real line with even quartic potential x 4 + α x 2 and study analytic continuation of eigenvalues, as functions of parameter α . We prove several properties of this analytic continuation conjectured by Bender, Wu, Loeffel and Martin. 1. All eigenvalues are given by branches of two multi-valued analytic functions, one for even eigenfunctions and one for odd ones. 2. The only singularities of these multi-valued functions in the complex α -plane are algebraic ramification points, and there are only finitely many singularities over each compact subset of the α -plane.
AbstractList	We consider the Schrödinger operator on the real line with even quartic potential x 4 + α x 2 and study analytic continuation of eigenvalues, as functions of parameter α . We prove several properties of this analytic continuation conjectured by Bender, Wu, Loeffel and Martin. 1. All eigenvalues are given by branches of two multi-valued analytic functions, one for even eigenfunctions and one for odd ones. 2. The only singularities of these multi-valued functions in the complex α -plane are algebraic ramification points, and there are only finitely many singularities over each compact subset of the α -plane.
Author	Eremenko, Alexandre Gabrielov, Andrei
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Cites_doi	10.1007/978-3-662-06842-7 10.1006/aphy.1997.5737 10.5802/aif.1326 10.1016/0003-4916(70)90240-X 10.1088/0305-4470/31/14/001 10.1090/ulect/038 10.1007/978-3-642-66282-9 10.1088/0305-4470/34/28/305 10.1007/978-3-642-58016-1 10.1007/978-3-540-38361-1 10.1088/0305-4470/38/27/005 10.2307/2373998 10.1103/PhysRev.184.1231 10.5802/aif.2362 10.1007/BF02547780 10.1016/0375-9601(87)90456-7 10.1307/mmj/1163789921 10.1002/qua.560210103 10.1063/1.532206 10.1088/0305-4470/33/48/314 10.1090/conm/355/06453 10.24033/bsmf.1095 10.1007/s00220-002-0706-3 10.1016/0375-9601(93)90153-Q
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Keywords	Cell Decomposition Meromorphic Function Riemann Surface Analytic Continuation Anharmonic Oscillator Complex function Eigenvalues Eigenfunctions Analytic functions Analytic continuation Mathematical physics
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Phys.19973812612661840896.3405110.1063/1.5322061997JMP....38.6126D1483488 FedoryukM.Asymptotic Analysis1993New YorkSpringer0782.34001 SibuyaY.Global theory of a second order linear ordinary differential equation with a polynomial coefficient1975AmsterdamNorth Holland0322.34006 Simon, B.: The anharmonic oscillator: a singular perturbation theory. In: Cargése Lectures in Physics, Vol. 5, New York: Gordon and Breach, 1972, pp. 383–414 StoïlovS.Leçons sur les principes topologiques de la théorie des fonctions analytiques1956ParisGauthier-Villars KatoT.Perturbation theory for linear operators1976Berlin-New YorkSpringer-Verlag0342.47009 DelabaereE.PhamF.Resurgent methods in semi-classical asymptotics, Annales de l’InstPoincaré, Sect. A1999711940977.340531704654 BenderC.WuT.Anharmonic oscillatorPhys. Rev. (2)19691841231126010.1103/PhysRev.184.12311969PhRv..184.1231B260323 DoreyP.DunningC.TateoR.Spectral equivalences, Bethe ansatz equations and reality properties in PT-symmetric quantum mechanicsJ. Phys. A200134567957040982.8102110.1088/0305-4470/34/28/3052001JPhA...34.5679D1857169 Loeffel, J., Martin, A.: Propriétés analytiques des niveaux de l’oscillateur anharmonique et convergence des approximants de Pade. In: Cargése Lectures in Physics, Vol. 5, New York: Gordon and Breach, 1972, pp. 415–429 ShinKwang C.Eigenvalues of PT-symmetric oscillators with polynomial potentialsJ. Phys. A200538614761661079.3406310.1088/0305-4470/38/27/0052005JPhA...38.6147S2166739 BakkenI.A multiparameter eigenvalue problem in the complex planeAmer. J. Math.1977995101510440379.3402110.2307/2373998508244 BenderC.TurbinerA.Analytic continuation of eigenvalue problemsPhys. Lett. A1993173644244610.1016/0375-9601(93)90153-Q1993PhLA..173..442B1204365 Shin, Kwang C.: Schrödinger type eigenvalue problems with polynomial potentials: asymptotics of eigenvalues. http://arXiv.org/list/math.SP/0411143v1, 2004 EremenkoA.Exceptional values in holomorphic families of entire functionsMichigan Math. J.20065436876960537379810.1307/mmj/11637899212280501 NevanlinnaR.Über Riemannsche Flächen mit endlich vielen WindungspunktenActa Math.19325829537310.1007/BF025477801555350 Eremenko, A.: Geometric theory of meromorphic functions. In: In the tradition of Ahlfors–Bers III, Contemp. Math. 355, Providence, RI, Amer. Math. Soc., 2004, pp. 221–230. (Expanded version available at http://www.math.purdue.edu/~eremenko/dvi/mich.pdf) Gurarii, V., Matsaev, V., Ruzmatova, N.: Asymptotic behavior of solutions of second-order ordinary differential equation in the complex domain, and the spectrum of an anharmonic oscillator. In: Analytic methods in probability theory and operator theory, Kiev: Naukova Dumka, 1990, pp. 145–154 LandoS.ZvonkinA.Graphs on surfaces and their applications2004Berlin-Heidelberg-New YorkSpringer1040.05001 SimonB.Large order and summability of eigenvalue perturbation theory: a mathematical overviewIntl. J. Quantum Chem.19822132510.1002/qua.560210103 Nevanlinna, F.: Über eine Klasse meromorpher Funktionen. 7 Congr. Math. Scand., Oslo, 1929 Ahlfors, L.: Lectures on quasiconformal mappings. Second edition, Providence, RI: Amer. Math. Soc., 2007 JuliaG.Sur le domain d’existence d’une fonction implicite définie par une relation entière G(x, y) = 0Bull. Soc. Math. France1926542637JFM 52.0327.051504890 ShinKwang C.On the reality of the eigenvalues for a class of PT-symmetric operatorsCommun. Math. Phys.20022295435641017.3408310.1007/s00220-002-0706-32002CMaPh.229..543S SimonB.Coupling constant analyticity for the anharmonic oscillatorAnn. Phys.1970587613610.1016/0003-4916(70)90240-X1970AnPhy..58...76S DrapeE.Über die Darstellung Riemannscher Flächen durch StreckenkomplexeDeutsche Math.19361805824 Ushveridze, A.: Quasi-exactly solvable models in quantum mechanics. Bristol and Philadelphia: Inst. of Phys. Publ., 1994 Goldberg, A., Ostrovskii, I.: Distribution of values of meromorphic functions, Moscow: Nauka, 1970, (in Russian. English translation: Value Distribution of Meromorphic Functions, Providence, RI: Amer. Math. Soc., 2008) NevanlinnaR.Eindeutige analytische Funktionen1953BerlinSpringer0050.30302 BenderC.BoettcherS.Quasi-exactly solvable quartic potentialJ. Phys. A: Math. Gen.199831L273L2770929.3407410.1088/0305-4470/31/14/0011621102 Kwang C. Shin (663_CR29) 2005; 38 663_CR20 R. Nevanlinna (663_CR27) 1953 E. Delabaere (663_CR7) 1997; 261 S. Lando (663_CR23) 2004 M. Fedoryuk (663_CR17) 1993 663_CR24 Y. Sibuya (663_CR31) 1975 E. Delabaere (663_CR8) 1997; 38 B. Simon (663_CR32) 1970; 58 663_CR25 R. Nevanlinna (663_CR26) 1932; 58 663_CR18 G. Julia (663_CR21) 1926; 54 E. Delabaere (663_CR10) 1999; 71 A. Eremenko (663_CR15) 2006; 54 T. Kato (663_CR22) 1976 S. Bochner (663_CR6) 1948 E. Delabaere (663_CR11) 2000; 33 A. Eremenko (663_CR16) 2008; 58 663_CR1 C. Bender (663_CR3) 1998; 31 B. Simon (663_CR34) 1982; 21 C. Bender (663_CR5) 1969; 184 663_CR33 E. Delabaere (663_CR9) 1993; 43 Kwang C. Shin (663_CR30) 2002; 229 663_CR37 663_CR14 I. Bakken (663_CR2) 1977; 99 663_CR28 C. Bender (663_CR4) 1993; 173 R. Gunning (663_CR19) 1965 A. Turbiner (663_CR36) 1987; 126 A P. Dorey (663_CR12) 2001; 34 S. Stoïlov (663_CR35) 1956 E. Drape (663_CR13) 1936; 1
References_xml	– reference: DelabaereE.PhamF.Unfolding the quartic oscillatorAnn. Physics199726121802180977.3405210.1006/aphy.1997.57371487700 – reference: DrapeE.Über die Darstellung Riemannscher Flächen durch StreckenkomplexeDeutsche Math.19361805824 – reference: ShinKwang C.Eigenvalues of PT-symmetric oscillators with polynomial potentialsJ. Phys. A200538614761661079.3406310.1088/0305-4470/38/27/0052005JPhA...38.6147S2166739 – reference: BakkenI.A multiparameter eigenvalue problem in the complex planeAmer. J. Math.1977995101510440379.3402110.2307/2373998508244 – reference: Nevanlinna, F.: Über eine Klasse meromorpher Funktionen. 7 Congr. Math. Scand., Oslo, 1929 – reference: BenderC.WuT.Anharmonic oscillatorPhys. Rev. (2)19691841231126010.1103/PhysRev.184.12311969PhRv..184.1231B260323 – reference: BenderC.BoettcherS.Quasi-exactly solvable quartic potentialJ. Phys. A: Math. Gen.199831L273L2770929.3407410.1088/0305-4470/31/14/0011621102 – reference: KatoT.Perturbation theory for linear operators1976Berlin-New YorkSpringer-Verlag0342.47009 – reference: BenderC.TurbinerA.Analytic continuation of eigenvalue problemsPhys. Lett. A1993173644244610.1016/0375-9601(93)90153-Q1993PhLA..173..442B1204365 – reference: DelabaereE.PhamF.Resurgent methods in semi-classical asymptotics, Annales de l’InstPoincaré, Sect. A1999711940977.340531704654 – reference: SimonB.Coupling constant analyticity for the anharmonic oscillatorAnn. Phys.1970587613610.1016/0003-4916(70)90240-X1970AnPhy..58...76S – reference: Eremenko, A.: Geometric theory of meromorphic functions. In: In the tradition of Ahlfors–Bers III, Contemp. Math. 355, Providence, RI, Amer. Math. Soc., 2004, pp. 221–230. (Expanded version available at http://www.math.purdue.edu/~eremenko/dvi/mich.pdf) – reference: DoreyP.DunningC.TateoR.Spectral equivalences, Bethe ansatz equations and reality properties in PT-symmetric quantum mechanicsJ. Phys. A200134567957040982.8102110.1088/0305-4470/34/28/3052001JPhA...34.5679D1857169 – reference: NevanlinnaR.Über Riemannsche Flächen mit endlich vielen WindungspunktenActa Math.19325829537310.1007/BF025477801555350 – reference: Simon, B.: The anharmonic oscillator: a singular perturbation theory. In: Cargése Lectures in Physics, Vol. 5, New York: Gordon and Breach, 1972, pp. 383–414 – reference: StoïlovS.Leçons sur les principes topologiques de la théorie des fonctions analytiques1956ParisGauthier-Villars – reference: Ushveridze, A.: Quasi-exactly solvable models in quantum mechanics. Bristol and Philadelphia: Inst. of Phys. Publ., 1994 – reference: TurbinerA.UshveridzeA.Spectral singularities and the quasi-exactly solvable problemPhys. Lett.1987126 A1811831987PhLA..126..181T921178 – reference: EremenkoA.Exceptional values in holomorphic families of entire functionsMichigan Math. J.20065436876960537379810.1307/mmj/11637899212280501 – reference: Goldberg, A., Ostrovskii, I.: Distribution of values of meromorphic functions, Moscow: Nauka, 1970, (in Russian. English translation: Value Distribution of Meromorphic Functions, Providence, RI: Amer. Math. Soc., 2008) – reference: FedoryukM.Asymptotic Analysis1993New YorkSpringer0782.34001 – reference: JuliaG.Sur le domain d’existence d’une fonction implicite définie par une relation entière G(x, y) = 0Bull. Soc. Math. France1926542637JFM 52.0327.051504890 – reference: Ahlfors, L.: Lectures on quasiconformal mappings. Second edition, Providence, RI: Amer. Math. Soc., 2007 – reference: BochnerS.MartinW.Several Complex Variables1948Princeton, NJPrinceton University Press0041.05205 – reference: DelabaereE.PhamF.Resurgence de Voros et periodes de curbes hyperelliptiquesAnn. Inst Fourier19934311631990766.340321209700 – reference: NevanlinnaR.Eindeutige analytische Funktionen1953BerlinSpringer0050.30302 – reference: DelabaereE.TrinhD.T.Spectral analysis of the complex cubic oscillatorJ. Phys. A200033877187961044.8155510.1088/0305-4470/33/48/3142000JPhA...33.8771D1801468 – reference: Loeffel, J., Martin, A.: Propriétés analytiques des niveaux de l’oscillateur anharmonique et convergence des approximants de Pade. In: Cargése Lectures in Physics, Vol. 5, New York: Gordon and Breach, 1972, pp. 415–429 – reference: ShinKwang C.On the reality of the eigenvalues for a class of PT-symmetric operatorsCommun. Math. Phys.20022295435641017.3408310.1007/s00220-002-0706-32002CMaPh.229..543S – reference: Gurarii, V., Matsaev, V., Ruzmatova, N.: Asymptotic behavior of solutions of second-order ordinary differential equation in the complex domain, and the spectrum of an anharmonic oscillator. In: Analytic methods in probability theory and operator theory, Kiev: Naukova Dumka, 1990, pp. 145–154 – reference: SimonB.Large order and summability of eigenvalue perturbation theory: a mathematical overviewIntl. J. Quantum Chem.19822132510.1002/qua.560210103 – reference: DelabaereE.DillingerH.PhamF.Exact semiclassical expansions for one-dimensional quantum oscillatorsJ. Math. 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Snippet	We consider the Schrödinger operator on the real line with even quartic potential x 4 + α x 2 and study analytic continuation of eigenvalues, as functions of...
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SubjectTerms	Classical and Quantum Gravitation Complex Systems Exact sciences and technology Functions of a complex variable Mathematical analysis Mathematical and Computational Physics Mathematical methods in physics Mathematical Physics Mathematics Other topics in mathematical methods in physics Partial differential equations Physics Physics and Astronomy Quantum Physics Relativity Theory Sciences and techniques of general use Several complex variables and analytic spaces Theoretical
Title	Analytic Continuation of Eigenvalues of a Quartic Oscillator
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