Asymptotic Behavior of Stability Regions for Hill’s Equation

The asymptotic behavior of the solutions of Hill’s equation $u'' + [ E - \lambda ^2 q( x ) ]u = 0$ is determined for large positive real values of the coupling constant $\lambda ^2 $ and large real values of the energy E. The locations and widths of the stability bands and instability gaps are found. The band widths are shown to decrease exponentially as $\lambda $ increases when $\lambda^{ - 2} E$ lies between the minimum and maximum values of the periodic potential $q( x )$. The gap widths decrease exponentially with $\lambda $ when $\lambda ^{ - 2} E$ is greater than the maximum of $q( x )$. For $\lambda ^{ - 2} E$ asymptotically equal to the maximum of $q( x )$, the width of the nth band is asymptotically half the width of the nth gap. The exponentially small band and gap widths are related to the exponentially small transmission and reflection coefficients, associated with one period of $q( x )$. The present results extend previous ones of Meixner and Schäfke, Harrell, and the authors, in which $\lambda ^{ - 2} E$ was near the minimum of $q( x )$.