## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Simplicity of the lowest eigenvalue of non-commutative harmonic oscillators and the Riemann scheme of a certain Heun’s differential equation

Masato Wakayama

#### Abstract

The non-commutative harmonic oscillator (NcHO) is a special type of self-adjoint ordinary differential operator 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 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.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 89, Number 6 (2013), 69-73.

Dates
First available in Project Euclid: 31 May 2013

https://projecteuclid.org/euclid.pja/1370004861

Digital Object Identifier
doi:10.3792/pjaa.89.69

Mathematical Reviews number (MathSciNet)
MR3079292

Zentralblatt MATH identifier
1278.34099

#### Citation

Wakayama, Masato. Simplicity of the lowest eigenvalue of non-commutative harmonic oscillators and the Riemann scheme of a certain Heun’s differential equation. Proc. Japan Acad. Ser. A Math. Sci. 89 (2013), no. 6, 69--73. doi:10.3792/pjaa.89.69. https://projecteuclid.org/euclid.pja/1370004861

#### References

• S. Haroche and J.-M. Raimond, Exploring the quantum, Oxford Graduate Texts, Oxford Univ. Press, Oxford, 2006.
• 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.
• M. Hirokawa and F. Hiroshima, Absence of energy level crossing for the ground state energy of the Rabi model. (Preprint).
• F. Hiroshima and I. Sasaki, Multiplicity of the lowest eigenvalue of non-commutative harmonic oscillators, Kyushu J. Math. (to appear).
• R. Howe and E.-C. Tan, Nonabelian harmonic analysis, Universitext, Springer, New York, 1992.
• 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.
• T. Ichinose and M. Wakayama, On the spectral zeta function for the noncommutative harmonic oscillator, Rep. Math. Phys. 59 (2007), no. 3, 421–432.
• 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.
• 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.
• 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.
• 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.
• H. Ochiai, Non-commutative harmonic oscillators and Fuchsian ordinary differential operators, Comm. Math. Phys. 217 (2001), no. 2, 357–373.
• H. Ochiai, Non-commutative harmonic oscillators and the connection problem for the Heun differential equation, Lett. Math. Phys. 70 (2004), no. 2, 133–139.
• A. Parmeggiani, On the spectrum and the lowest eigenvalue of certain non-commutative harmonic oscillators, Kyushu J. Math. 58 (2004), no. 2, 277–322.
• A. Parmeggiani, On the spectrum of certain noncommutative harmonic oscillators, Ann. Univ. Ferrara Sez. VII Sci. Mat. 52 (2006), no. 2, 431–456.
• A. Parmeggiani, On the spectrum of certain non-commutative harmonic oscillators and semiclassical analysis, Comm. Math. Phys. 279 (2008), no. 2, 285–308.
• A. Parmeggiani, Spectral theory of non-commutative harmonic oscillators: an introduction, Lecture Notes in Mathematics, 1992, Springer, Berlin, 2010.
• A. Parmeggiani and A. Venni, On the essential spectrum of certain non-commutative oscillators. (Preprint).
• 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).
• A. Parmeggiani and M. Wakayama, Non-commutative harmonic oscillators. I, Forum Math. 14 (2002), no. 4, 539–604.
• A. Parmeggiani and M. Wakayama, Non-commutative harmonic oscillators. II, Forum Math. 14 (2002), no. 5, 669–690.
• 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.
• A. Ronveaux (eds.), Heun's differential equations, Oxford Science Publications, Oxford Univ. Press, New York, 1995.
• S. Yu. Slavyanov and W. Lay, Special functions, Oxford Mathematical Monographs, Oxford Univ. Press, Oxford, 2000.
• S. Taniguchi, The heat semigroup and kernel associated with certain non-commutative harmonic oscillators, Kyushu J. Math. 62 (2008), no. 1, 63–68.
• M. Wakayama, Correspondence between eigenfunctions of non-commutative harmonic oscillators and holomorphic solutions of Heun's differential equations. (in preparation).
• 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.