Generalized phase-integrals for linear homogeneous ODEs

Using a surprising result for the Wronskian of solutions with a common factor we show that all of the linearly independent solutions of linear-homogeneous ODEs have a simple form in a generalized phase-integral representation. This allows the generalization of WKB-like expansions to higher-order differential equations in a way that extends the usual phase-integral methods. This work clarifies the internal structure of phase-integral representations as being discrete transforms over the quasiphases of the linearly independent ODE solutions and hence clarifies the structure of solutions to linear ODEs. Consider the stationary Schrödinger equation y ′′(x)+ R(x)y(x) = 0. (1) The WKB approximation for large R(x) may be carried out in the following formal manner [1]: Introduce a small parameter into equation (1) y ′′(x)+ R(x) 2 y(x) = 0. (2) Next, take the ansatz solution to this new equation to have the pure exponential form y(x) = exp [ ± i ∫ x dx κ(x) ] . (3) Finally, expanding κ(x) in powers of κ(x) = κ0(x)+ κ1(x)+ κ2(x)+ · · · (4) and substituting this into (2) yields the familiar result to first order y±(x) ∼ 1 R(x)1/4 exp [ ± i ∫ x dx √ R(x) ]