Exponential solutions of $y^{”}+(r-q)y=0$ and the least eigenvalue of Hill’s equation

Abstract:

It is shown that if $q$ is a nonnegative continuous function on $[0,\infty )$ such that for some positive constants $A$ and $L$, \[ \lim \inf \limits _{x \to \infty } \int _x^{x + A} {{q^{1/2}}(t)dt} > AL,\] then $y'' + (r - q)y = 0$ has an exponentially increasing solution and an exponentially decreasing solution whenever the uniform norm of the continuous function $r$ satisfies $||r|{|_\infty } < {[L/(AL + 1)]^2}$. A refinement of the proof is used to show that for all sufficiently large values of $k$ the least eigenvalue $\lambda (k)$ of the two parameter Hill equation $y'' + (\lambda - kp)y = 0$ satisfies an inequality of the form $\lambda (k) \geqslant Pk + {B_\beta }|k{|^\beta }$ where $P = \min p$ if $k > 0$, $P = \max p$ if $k < 0$, and $\beta$ is a constant between 0 and 1 that depends on the periodic function $p$.