Asymptotic Estimates for the Adiabatic Invariance of a Simple Oscillator

Let u be a solution of the differential equation $\ddot u + \varphi ^2 u = 0$ with slowly varying coefficient $\varphi ^2 (\varepsilon t)$. Let $r^2 = \varphi (\varepsilon t)u^2 + \varphi ^{ - 1} (\varepsilon t)\dot u^2 $. Then r is an approximate or “adiabatic” invariant of u in the sense that $\dot r = O(\varepsilon )$. J. E. Littlewood [1] has shown that $r^2 ( + \infty ) - r^2 ( - \infty ) = O(\varepsilon ^2 )$ for all $n > 0$ under the hypotheses that $\varphi > 0$, $\varphi $ has positivelimits as $t \to \pm \infty $, and $\varphi ^{(n)} \in L_1 ( - \infty ,\infty )$ for all $n > 0$. The purpose of this paper is to obtain an upper estimate for $r^2 ( + \infty ) - r^2 ( - \infty )$ under Littlewood’s hypotheses. The main result is that the rate of decrease of $r^2 (\infty ) - r^2 ( - \infty )$ as $\varepsilon \to 0^ + $ is determined by the rate of growth of $\| {\varphi ^{(m)} } \|_1 $ as $m \to \infty $. It is shown that if $\| {\varphi ^{(m)} } \|_1 = O\{ \exp h(m)\} $, where $m\log m = o(h(m))$, then for any $\delta > 0$, \[ \frac{{r^2 (\infty )}} {{r^2 ( - \infty )}} - 1 = O\left\{ {\exp - h^ * \left( {\log \varepsilon ^{ - 1 + \delta } } \right)} \right\}, \] where $h^ * $ is the convex conjugate function of h\[ h^ * (x) = \mathop {\max }\limits_y \{ {xy - h( y )} \}. \]