On a complete description of the Stokes geometry for higher order ordinary differential equations with a large parameter via integral representations
The exact WKB analysis for higher order is well known to be extremely difficult. The authors focuse on an integral solution to the integral equation NEWLINE\[NEWLINE\Biggl({d^3\over dx^3}+ (c_1 x+ d_1)\eta^2{d\over dx}+ (c_0x+ d_0)\eta^3\Biggr) \psi(x)= 0NEWLINE\]NEWLINE with its integral solution, NEWLINE\[NEWLINE\psi(x)= \int_y C(\zeta)^{-1}\exp(\eta(x\zeta+ g(\zeta))) d\zeta,NEWLINE\]NEWLINE where \(g(\zeta)\) is a multi-valued function that satisfies the differential equation \({dg\over d\zeta}= {D(\zeta)\over C(\zeta)}\). The function \(C(\zeta)= c_1\zeta+ c_0\) is linear, and the function \(D(\zeta)= \zeta^3+ d_1\zeta+ d_0\) is cubic. The authors then make the following conjecture: Let \(\alpha(x)\) and \(\beta(x)\) be saddle points for the integral equation given above and suppose that a steepest descent path \(\gamma\) connects \(\alpha(x)\) and \(\beta(x)\). Then the variation of \(\arg(\zeta+{c_0\over c_1})\) along the path \(\gamma\) is smaller than \(2\pi\), i.e., there exists a constant \(\theta_0(x)\) for which \(|\arg(\zeta+{c_0\over c_1})- \theta_0(x)|< 2\pi\) holds for \(\zeta\) in \(\gamma\).NEWLINENEWLINENEWLINEThis conjecture when it is true, would mean that for each fixed \(x\) a steepest descent segment connecting the two saddle points should lie on one preferred branch of the logarithmic singularity contained in \(f\). The authors then explore under what conditions this conjecture holds. The linearity of \(C(\zeta)\) is in fact a crucial ingredient for the study since in general the conjecture is not true.NEWLINENEWLINENEWLINEThe paper offers the reader a very good introduction to the complexity of the problem surrounding WKB analysis.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00055].
scientific article; zbMATH DE number 1648112