Abstract
We prove the simplicity and analyticity of the eigenvalues of the cubic oscillator Hamiltonian,
, for β in the cut plane \({\mathcal{C}_c:=\mathcal{C}\backslash \mathcal{R}_-}\). Moreover, we prove that the spectrum consists of the perturbative eigenvalues {E n (β)} n ≥ 0 labeled by the constant number n of nodes of the corresponding eigenfunctions. In addition, for all \({\beta \in \mathcal{C}_c, E_n(\beta)}\) can be computed as the Stieltjes-Padé sum of its perturbation series at β = 0. This also gives an alternative proof of the fact that the spectrum of H(β) is real when β is a positive number. This way, the main results on the repulsive PT-symmetric and on the attractive quartic oscillators are extended to the cubic case.
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References
Alvarez G.: Bender-Wu branch points in the cubic oscillator. J. Phys. A 28(16), 4589–4598 (1995)
Bender C.M., Boettcher S.: Real spectra in non-hermitian Hamiltonian having PT symmetry. Phys. Rev. Lett. 80, 5243 (1998)
Bender C.M., Weniger E.J.: Numerical evidence that the perturbation expansion for a non-Hermitian PT-symmetric Hamiltonian is Stieltjes. J. Math. Phys. 42(5), 2167–2183 (2001)
Buslaev V., Grecchi V.: Equivalence of unstable anharmonic oscillators and double wells. J. Phys. A Math. Gen. 26, 5541–5549 (1993)
Caliceti E.: Distributional Borel summability of odd anharmonic oscillators. J. Phys. A: Math. Gen. 33, 3753–3770 (2000)
Caliceti E., Graffi S., Maioli M.: Perturbation theory of odd anharmonic oscillators. Commun. Math. Phys. 75, 51 (1980)
Caliceti E., Maioli M.: Odd anharmonic oscillators and shape resonances. Ann. Inst. Henri Poincaré XXXVIII(2), 175–186 (1983)
Davydov A.: Quantum Mechanics. Pergamon Press, London (1965)
Delabaere E., Pham F.: Eigenvalues of complex Hamitonians with PT symmetry I. Phys. Lett. A. 250, 25 (1998)
Delabaere E., Pham F.: Eigenvalues of complex Hamitonians with PT symmetry II. Phys. Lett. A. 250, 29 (1998)
Delabaere E., Trinh D.T.: Spectral analysis of the complex cubic oscillator. J. Phys. A: Math. Gen. 33, 8771–8796 (2000)
Dorey P., Dunning C., Tateo R.: Spectral equivalence, Bethe ansatz equations, and reality properties in PT-symmetric quantum mechanics. J. Phys. A 34(28), 5679–5704 (2001)
Eremenko A., Gabrielov A.: Analytic continuation of eigenvalues of a quartic oscillator. Commun. Math. Phys. 287(2), 431–457 (2009)
Eremenko A., Gabrielov A., Shapiro B.: High energy eigenfunctions of one-dimensional Schrödinger operators with polynomial potential. Comput. Methods Funct. Theory 8, 513–529 (2008)
Grecchi V., Maioli M., Martinez A.: Padé summability for the cubic oscillator. J. Phys. A: Math. Theor. 42, 425208 (2009)
Grecchi V., Maioli M., Martinez A.: The top resonances of the cubic oscillator. J. Phys. A: Math. Theor. 43, 474027 (2010)
Harrell E.M. II, Simon B.: The mathematical theory of resonances whose widths are exponentially small. Duke Math. 47(4), 845–902 (1980)
Kato, T.: Perturbation Theory for Linear Operators. Berlin-Heidelberg-Newyork: Springer-Verlag, 1976
Loeffel J.-J., Martin A., Simon B., Wightman A.: Padé approximants and the anharmonic oscillator. Phys. Lett. B 30, 656–658 (1969)
Loeffel, J.-J., Martin A.: Propriétés analytiques des niveaux de l’oscillateur anharmonique et convergence des approximants de Padé. Proceedings of R.C.P. n. 25, Strasbourg, 1970
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Vol. II. New-York: Academic Press, 1975
Shin K.C.: On the reality of the eigenvalues for a class of PT-symmetric operators. Commun. Math. Phys. 229, 543–564 (2002)
Sibuya, Y.: Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient. Amsterdam: North-Holland, 1975
Simon B.: Coupling constant analyticity for the anharmonic oscillator. Ann of Phys. 58, 76–136 (1970)
Stieltjes, T.J.: Recherche sur les fractions continues. Ann. Fac. Sci. Univ. Toulouse 1re série, tome 8, no. 4, J1–J22 (1894)
Trinh, D.T.: Asymptotique et analyse spectrale de l’oscillateur cubique, PhD Thesis 2002, Nice (France)
Voros A.: The return of the quartic oscillator. Ann. Inst. Henri Poincaré, Section A XXXIX(3), 211–338 (1983)
Wall, H.S.: Analytic Theory of Continued Fractions. Princeton, NJ: D. Van Nostrand Company, Inc., (1948)
Zinn-Justin, J., Jentschura, U.D.: Imaginary cubic perturbation: numerical and analytic study. J. Phys. A: Math. Theor. 43, 425301 (2010) (29pp)
Zinn-Justin J., Jentschura U.D.: Order-dependent mappings: Strong-coupling behavior from weak-coupling expansions in non-Hermitian theories. J. Math. Phys. 51, 072106 (2010)
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Communicated by B. Simon
Partly supported by Università di Bologna, Funds for Selected Research Topics.
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Grecchi, V., Martinez, A. The Spectrum of the Cubic Oscillator. Commun. Math. Phys. 319, 479–500 (2013). https://doi.org/10.1007/s00220-012-1559-z
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DOI: https://doi.org/10.1007/s00220-012-1559-z