Eigenvalues of the Hill Equation to Any Order in the Adiabatic Limit

Abstract

To study the time‐dependent linear oscillator, Lewis has recently introduced an auxiliary function w. One of the advantages of this function is that, in the adiabatic limit, a formal expansion of w in ε is possible (ε characterizing the slowness of the time variation). We show that, in this adiabatic limit, the eigenvalues of the Hill equation can be very easily deduced from w. Moreover, the computation of the ε²ⁿ‐order solution is much simpler if we use the Chandrasekhar method of higher invariants, which is shown to be equivalent to the Lewis expansion. Compact formulas, easy to handle on a computer, are obtained, and the method, which must be considered as a generalization to higher order of the WKB solution, is finally tested on the Mathieu equation.