## Journal of the Mathematical Society of Japan

### Integral transformation of Heun's equation and some applications

Kouichi TAKEMURA

#### Abstract

It is known that the Fuchsian differential equation which produces the sixth Painlevé equation corresponds to the Fuchsian differential equation with different parameters via Euler's integral transformation, and Heun's equation also corresponds to Heun's equation with different parameters, again via Euler's integral transformation. In this paper we study the correspondences in detail. After investigating correspondences with respect to monodromy, it is demonstrated that the existence of polynomial-type solutions corresponds to apparency of a singularity. For the elliptical representation of Heun's equation, correspondence with respect to monodromy implies isospectral symmetry. We apply the symmetry to finite-gap potentials and express the monodromy of Heun's equation with parameters which have not yet been studied.

#### Article information

Source
J. Math. Soc. Japan Volume 69, Number 2 (2017), 849-891.

Dates
First available in Project Euclid: 20 April 2017

https://projecteuclid.org/euclid.jmsj/1492653649

Digital Object Identifier
doi:10.2969/jmsj/06920849

#### Citation

TAKEMURA, Kouichi. Integral transformation of Heun's equation and some applications. J. Math. Soc. Japan 69 (2017), no. 2, 849--891. doi:10.2969/jmsj/06920849. https://projecteuclid.org/euclid.jmsj/1492653649

#### References

• P. Boalch, From Klein to Painleve via Fourier, Laplace and Jimbo Proc. London Math. Soc. (3), 90 (2005), 167–208.
• F. Gesztesy and R. Weikard, Treibich–Verdier potentials and the stationary (m)KdV hierarchy, Math. Z., 219 (1995), 451–476.
• M. Inaba, K. Iwasaki and M.-H. Saito, Backlund transformations of the sixth Painleve equation in terms of Riemann–Hilbert correspondence, Int. Math. Res. Not., 2004 (2004), 1–30.
• K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painleve. A modern theory of special functions. Aspects of Mathematics, E16. Friedr. Vieweg & Sohn, Braunschweig, 1991.
• A. Ishkhanyan and K.,A. Suominen, New solutions of Heun's general equation, J. Phys. A, 36 (2003), L81–L85.
• A.,Ya. Kazakov, Monodromy of Heun Equations with Apparent Singularities, Intern. J. Theoret. Math. Phys., 3 (2013), 617–639.
• A.,Ya. Kazakov and S.,Yu. Slavyanov, Integral relations for special functions of the Heun class, Theoret. and Math. Phys., 107 (1996), 733–739.
• A.,Ya. Kazakov and S.,Yu. Slavyanov, Euler integral symmetries for a deformed Heun equation and symmetries of the Painlevé PVI equation, Theoret. and Math. Phys., 155 (2008), 721–732.
• A. Khare and U. Sukhatme, Complex periodic potentials with a finite number of band gaps, J. Math. Phys., 47 (2006), 062103, 22 pp.
• D.,P. Novikov, Integral transformation of solutions of a Fuchs-class equation that corresponds to the Okamoto transformation of the Painleve VI equation, Theoret. and Math. Phys., 146 (2006), 295–303.
• S P. Novikov, A periodic problem for the Korteweg-de Vries equation, Functional Anal. Appl., 8 (1974), 236–246.
• A. Ronveaux (ed.), Heun's differential equations, Oxford Science Publications, Oxford University Press, Oxford, 1995.
• S.,N.,M. Ruijsenaars, Hilbert–Schmidt operators vs. integrable systems of elliptic Calogero–Moser type III. The Heun case, SIGMA Symmetry Integrability Geom. Methods Appl., 5 (2009), paper 049, 21 pp.
• A.,V. Shanin and R.,V. Craster, Removing false singular points as a method of solving ordinary differential equations, Euro. J. Appl. Math., 13 (2002), 617–639.
• A.,O. Smirnov, Elliptic solitons and Heun's equation, The Kowalevski property, 287–305, CRM Proc. Lecture Notes, 32, Amer. Math. Soc., Providence (2002).
• K. Takemura, The Heun equation and the Calogero–Moser–Sutherland system I: the Bethe Ansatz method, Comm. Math. Phys., 235 (2003), 467–494.
• K. Takemura, The Heun equation and the Calogero–Moser–Sutherland system II: the perturbation and the algebraic solution, Electron. J. Differential Equations, 2004 (2004), no. 15, 1–30.
• K. Takemura, The Heun equation and the Calogero–Moser–Sutherland system III: the finite gap property and the monodromy, J. Nonlinear Math. Phys., 11 (2004), 21–46.
• K. Takemura, The Heun equation and the Calogero–Moser–Sutherland system IV: the Hermite–Krichever Ansatz, Comm. Math. Phys., 258 (2005), 367–403.
• K. Takemura, The Heun equation and the Calogero–Moser–Sutherland system V: generalized Darboux transformations, J. Nonlinear Math. Phys., 13 (2006), 584–611.
• K. Takemura, On the Heun equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366 (2008), no. 1867, 1179–1201.
• K. Takemura, Integral representation of solutions to Fuchsian system and Heun's equation, J. Math. Anal. Appl., 342 (2008), 52–69.
• K. Takemura, Middle convolution and Heun's equation, SIGMA Symmetry Integrability Geom. Methods Appl., 5 (2009), paper 040, 22 pp.
• K. Takemura, Integral transformation and Darboux transformation of Heun's differential equation, Nonlinear and modern mathematical physics, AIP Conference Proceedings, 1212, Amer. Inst. Phys., New York, 2010, 58–65.
• A. Treibich and J.-L. Verdier, Revetements exceptionnels et sommes de 4 nombres triangulaires, Duke Math. J., 68 (1992), 217–236.
• G. Valent, An integral transform involving Heun functions and a related eigenvalue problem, SIAM J. Math. Anal., 17 (1986), 688–703.