New Stokes' line in WKB theory

The WKB theory for differential equations of arbitrary order or integral equations in one dimension is investigated. The rules previously stated for the construction of Stokes' lines for Nth-order differential equations, N> or =3, or integral equations are found to be incomplete because these rules lead to asymptotic forms of the solutions that depend on path. This paradox is resolved by the demonstration that new Stokes' lines can arise when previously defined Stokes' lines cross. A new formulation of the WKB problem is given to justify the new Stokes' lines. With the new Stokes' lines, the asymptotic forms can be shown to be independent of path. In addition, the WKB eigenvalue problem is formulated, and the global dispersion relation is shown to be a functional of loop integrals of the action