Large eigenvalues of a Sturm-Liouville problem

Extract

Consider the eigenvalue problem

where q(x) is a (bounded) function in the class L(a, b). We may suppose without loss of generality that 0 ≤ α < π and 0 ≤ β < π. Then as is well known there are infinitely many eigenvalues λ_r (r = 0, 1, 2, …) and λ_n ∽ n²π² (b − a)⁻² as n → ∞.