For vibrating strings with symmetric single-well densities, it is known that the ratio λ2/λ11 is maximized when the density is constant. In this note, we extend this result to a class of symmetric densities.
We consider the Dirichlet operator Ht=−d2/dx2+q(x) on L2([t,∞)), where q is a convex potential with q(x)→∞ as x→∞. We show that the eigenvalue gap Γ(t) of Ht is monotone increasing as t increases from −∞ to ∞. We also show that Γ(t) is strictly increasing if q is not linear at infinity. An asymptotic estimate of Γ(t) for quadratic potentials is obtained.