Asymptotic behavior of the Heun equation and Heun functions.

*(Russian)*Zbl 0742.34004The paper deals with the “centenarian” Heun’s equation in the complex domain: (1)

$${w}^{\text{'}\text{'}}+p\left(z\right){w}^{\text{'}}+q\left(z\right)w=0,\phantom{\rule{1.em}{0ex}}p\left(z\right)=\sum _{j=1}^{3}{\alpha}_{j}/(z-{a}_{j}),\phantom{\rule{1.em}{0ex}}q\left(z\right)=(h-\ell z)/(z-{a}_{1})(z-{a}_{2})(z-{a}_{3})\xb7$$

Firstly, the author recalls the classification of the special cases of Heun’s equations and other basic notions, as a spectrum of Heun’s equation and Heun’s functions. Further, by the properties of the solution of (1), the spectrum ${\Sigma}$ of Heun’s equation is characterized. For example, $(h,\ell )\in {\Sigma}$ iff the equation (1) has such a solution $w\left(z\right)$, for which ${w}^{\text{'}}\left(z\right)/w\left(z\right)$ is a rational function. With help of the Heun’s surface function, the properties of the spectrum are studied. Finally, the WKB-estimates of the solutions of (1) are derived.

Reviewer: A.Klíč (Praha)

##### MSC:

34M99 | Differential equations in the complex domain |

34E20 | Asymptotic singular perturbations, turning point theory, WKB methods (ODE) |

34L20 | Asymptotic distribution of eigenvalues for OD operators |