Asymptotics of the spectrum of Heun's equation and the exact WKB analysis (Q2744105)

The author begins with Heun's equation of the form NEWLINE\[NEWLINE\Biggl({d^2\over dx^2}+ \sum^4_{j=1} {\alpha_j\over x- a_j} {d\over dx}+ {h-lx\over (x- a_1)(x- a_2)(x- a_3)}\Biggr) \omega(x)= 0,NEWLINE\]NEWLINE where \(\alpha_j\), \(h\) and \(l\) are some constants. Its characteristic exponents at each singular point are NEWLINE\[NEWLINE\begin{pmatrix} x=a_j & x=\infty\\ 0 & (\alpha-1+ \sqrt{(\alpha- 1)^2+ 4l})/2\\ 1-\alpha_j & (\alpha- 1-\sqrt{(\alpha- 1)^2+ 4l})/2\end{pmatrix},\qquad \alpha= \sum^3_{j=1} \alpha_j.NEWLINE\]NEWLINE The form of the solution is then taken to be \(\omega(x)= (x- s_1)^{\beta_1}(x- s_2)^{\beta_2}\widetilde\omega(x)\), where \(s_j\) is a singular point of Heun's equation, \(\beta_j\) is one of its characteristic exponents at \(x= s_j\), and \(\widetilde\omega(x)\) is a holomorphic function near \(x=s_1,s_2\). The Heun equation and the form of the solution is then transformed, respectively, giving us NEWLINE\[NEWLINE\Biggl(-{d^2\over dx^2}+ \eta^2(Q_0(x)+ \eta^{-2}Q_2(x))\Biggr) \psi=0,NEWLINE\]NEWLINE where the form of the solution was transformed to be NEWLINE\[NEWLINE\psi(x)= (x- a_1)^{\alpha_1/2}(x- a_2)^{\alpha_2/2}(x- a_3)^{\alpha_3/2} \omega(x).NEWLINE\]NEWLINE Numerical results as well as graphics are included in the paper. The paper is very well presented including many of the details surrounding the problem.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00055].