Exponential solutions of $y\sp"+(r-q)y=0$ and the least eigenvalue of Hill's equation

Thomas T. Read
1975 Proceedings of the American Mathematical Society  
It is shown that if q is a nonnegative continuous function on [O, °°) such that for some positive constants A and L, lim inf fX+ ql/2(t)dt > AL, V_.r^ -7 X X-'OO 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|| < YE/(AL + 1)] . A refinement of the proof is used to show that for all sufficiently large values of k the least eigenvalue X(k) of the two parameter Hill equation
more » ... meter Hill equation y + (X -kp)y = 0 satisfies an inequality of the form X(k) > Pk + B Ak\" where P = min p if k > 0, P = max p if k < 0, and ß is a constant between 0 and 1 that depends on the periodic function p.
doi:10.1090/s0002-9939-1975-0377184-2 fatcat:eggz6wfoanaxrpijuu2k2ziifa