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Simplicity of the lowest eigenvalue of non-commutative harmonic oscillators and the Riemann scheme of a certain Heun’s differential equation. (English) Zbl 1278.34099

Summary: The non-commutative harmonic oscillator (NcHO) is a special type of self-adjoint ordinary differential operators with non-commutative coefficients. In the present note, we aim to provide a reasonable criterion that derives the simplicity of the lowest eigenvalue of NcHO. It actually proves the simplicity of the lowest eigenvalue for a large class of structure parameters. Moreover, this note describes a certain equivalence between the spectral problem of the NcHO (for the even parity) and the existence of holomorphic solutions of Heun’s ordinary differential equations in a complex domain. The corresponding Riemann scheme allows us to give another proof to the criterion.

MSC:

34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
81S05 Commutation relations and statistics as related to quantum mechanics (general)
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References:

[1] S. Haroche and J.-M. Raimond, Exploring the quantum , Oxford Graduate Texts, Oxford Univ. Press, Oxford, 2006. · Zbl 1118.81001
[2] M. Hirokawa, The Dicke-type crossings among eigenvalues of differential operators in a class of non-commutative oscillators, Indiana Univ. Math. J. 58 (2009), no. 4, 1493-1535. · Zbl 1170.81025 · doi:10.1512/iumj.2009.58.3645
[3] M. Hirokawa and F. Hiroshima, Absence of energy level crossing for the ground state energy of the Rabi model. (Preprint).
[4] F. Hiroshima and I. Sasaki, Multiplicity of the lowest eigenvalue of non-commutative harmonic oscillators, Kyushu J. Math. (to appear). · Zbl 1276.35122 · doi:10.2206/kyushujm.67.355
[5] R. Howe and E.-C. Tan, Nonabelian harmonic analysis , Universitext, Springer, New York, 1992.
[6] T. Ichinose and M. Wakayama, Zeta functions for the spectrum of the non-commutative harmonic oscillators, Comm. Math. Phys. 258 (2005), no. 3, 697-739. · Zbl 1145.81031 · doi:10.1007/s00220-005-1308-7
[7] T. Ichinose and M. Wakayama, On the spectral zeta function for the noncommutative harmonic oscillator, Rep. Math. Phys. 59 (2007), no. 3, 421-432. · Zbl 1190.11047 · doi:10.1016/S0034-4877(07)80077-2
[8] K. Kimoto and M. Wakayama, Elliptic curves arising from the spectral zeta function for non-commutative harmonic oscillators and \(\Gamma_{0}(4)\)-modular forms, in The Conference on \(L\)-Functions , (eds. L. Weng and M. Kaneko), World Sci. Publ., Hackensack, NJ, 2007, pp. 201-218. · Zbl 1206.11109
[9] K. Kimoto and M. Wakayama, Spectrum of non-commutative harmonic oscillators and residual modular forms, in Noncommutative Geometry and Physics 3 (eds. G. Dito, M. Kotani, Y. Maeda, H. Moriyoshi, T. Natsume and S. Watamura), Keio COE Lecture Series on Mathematical Science, vol. 1, World Sci. Publ., Hackensack, NJ, 2013, pp. 237-267. · Zbl 1292.81079
[10] K. Kimoto and Y. Yamasaki, A variation of multiple \(L\)-values arising from the spectral zeta function of the non-commutative harmonic oscillator, Proc. Am. Math. Soc. 137 (2009), no. 8, 2503-2515. · Zbl 1260.11055 · doi:10.1090/S0002-9939-09-09881-5
[11] K. Nagatou, M. T. Nakao and M. Wakayama, Verified numerical computations for eigenvalues of non-commutative harmonic oscillators, Numer. Funct. Anal. Optim. 23 (2002), no. 5-6, 633-650. · Zbl 1016.65056 · doi:10.1081/NFA-120014756
[12] H. Ochiai, Non-commutative harmonic oscillators and Fuchsian ordinary differential operators, Comm. Math. Phys. 217 (2001), no. 2, 357-373. · Zbl 0982.34070 · doi:10.1007/s002200100362
[13] H. Ochiai, Non-commutative harmonic oscillators and the connection problem for the Heun differential equation, Lett. Math. Phys. 70 (2004), no. 2, 133-139. · Zbl 1129.34339 · doi:10.1007/s11005-004-4292-5
[14] A. Parmeggiani, On the spectrum and the lowest eigenvalue of certain non-commutative harmonic oscillators, Kyushu J. Math. 58 (2004), no. 2, 277-322. · Zbl 1134.34336 · doi:10.2206/kyushujm.58.277
[15] A. Parmeggiani, On the spectrum of certain noncommutative harmonic oscillators, Ann. Univ. Ferrara Sez. VII Sci. Mat. 52 (2006), no. 2, 431-456. · Zbl 1144.34055 · doi:10.1007/s11565-006-0030-5
[16] A. Parmeggiani, On the spectrum of certain non-commutative harmonic oscillators and semiclassical analysis, Comm. Math. Phys. 279 (2008), no. 2, 285-308. · Zbl 1168.47038 · doi:10.1007/s00220-008-0436-2
[17] A. Parmeggiani, Spectral theory of non-commutative harmonic oscillators: an introduction , Lecture Notes in Mathematics, 1992, Springer, Berlin, 2010. · Zbl 1200.35346
[18] A. Parmeggiani and A. Venni, On the essential spectrum of certain non-commutative oscillators. (Preprint). · Zbl 1285.81024
[19] A. Parmeggiani and M. Wakayama, Oscillator representations and systems of ordinary differential equations, Proc. Natl. Acad. Sci. USA 98 (2001), no. 1, 26-30 (electronic). · Zbl 1065.81058 · doi:10.1073/pnas.011393898
[20] A. Parmeggiani and M. Wakayama, Non-commutative harmonic oscillators. I, Forum Math. 14 (2002), no. 4, 539-604. · Zbl 1001.22012 · doi:10.1515/form.2002.025
[21] A. Parmeggiani and M. Wakayama, Non-commutative harmonic oscillators. II, Forum Math. 14 (2002), no. 5, 669-690. · Zbl 1020.22006 · doi:10.1515/form.2002.029
[22] A. Parmeggiani and M. Wakayama, Corrigenda and remarks to: “Non-commutative harmonic oscillators. I” [Forum Math. 14 (2002), no. 4, 539-604], Forum Math. 15 (2003), no. 6, 955-963. · Zbl 1001.22012
[23] A. Ronveaux (eds.), Heun’s differential equations , Oxford Science Publications, Oxford Univ. Press, New York, 1995. · Zbl 0847.34006
[24] S. Yu. Slavyanov and W. Lay, Special functions , Oxford Mathematical Monographs, Oxford Univ. Press, Oxford, 2000.
[25] S. Taniguchi, The heat semigroup and kernel associated with certain non-commutative harmonic oscillators, Kyushu J. Math. 62 (2008), no. 1, 63-68. · Zbl 1146.60049 · doi:10.2206/kyushujm.62.63
[26] M. Wakayama, Correspondence between eigenfunctions of non-commutative harmonic oscillators and holomorphic solutions of Heun’s differential equations. (in preparation).
[27] D. Zagier, Integral solutions of Apéry-like recurrence equations, in Groups and symmetries , 349-366, CRM Proc. Lecture Notes, 47 Amer. Math. Soc., Providence, RI, 2009. · Zbl 1244.11042
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