On a regular singular point in the exact WKB analysis (Q2744104)

Here, the author considers the following equation near the origin: NEWLINE\[NEWLINE\Biggl(-\frac{d^{2}}{dx^{2}}+\eta^{2} \biggl(\frac{Q_{0}(x)}{x}+\eta^{-2}\frac{Q_{2}(x) }{x^{2}}\biggr)\Biggr)\psi= 0,\tag{1}NEWLINE\]NEWLINE where \(\eta\) denotes a large parameter and each \(Q_{j}(x)\), \(j= 0,2\), denotes a holomorphic function near the origin under the condition that \(Q_{0}(x)\) does not vanish at the origin. NEWLINENEWLINENEWLINEThe author's purpose is to determine the connection formula of the Borel sum of WKB solutions to (1) from the point of view of the exact WKB analysis. He reduces (1) to the following equation: NEWLINE\[NEWLINE\Biggl(-\frac{d^{2}}{dx^{2}}+\eta^{2} \biggl(\frac{1}{x}+\eta^{-2}\frac{\lambda}{x^{2}} \biggr)\Biggr)\psi= 0,NEWLINE\]NEWLINE receives for it a required result and generalizes the last on the initial equation.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00055].