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Abstract  

For vibrating strings with symmetric single-well densities, it is known that the ratio λ211 is maximized when the density is constant. In this note, we extend this result to a class of symmetric densities.

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Abstract  

We establish several comparison results on the eigenvalue gap for vibrating strings with symmetric single-well densities or symmetric double-well densities.

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Abstract

We consider the Dirichlet operator H t=−d 2/dx 2+q(x) on L 2([t,∞)), where q is a convex potential with q(x)→∞ as x→∞. We show that the eigenvalue gap Γ(t) of H t 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.

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