Bounds for ratios of eigenvalues of the Dirichlet Laplacian

Ashbaugh, Mark S.; Benguria, Rafael D.

doi:10.1090/S0002-9939-1994-1186125-1

Bounds for ratios of eigenvalues of the Dirichlet Laplacian
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by Mark S. Ashbaugh and Rafael D. Benguria PDF

Proc. Amer. Math. Soc. 121 (1994), 145-150 Request permission

Abstract:

We use a doubling scheme to derive a bound for the ratio of the ${2^k}$th eigenvalue to the first for the Dirichlet Laplacian on a bounded domain $\Omega \subset {\mathbb {R}^n}$. The explicit bounds we obtain derive from the optimal bound ${({\lambda _2}/{\lambda _1})_\Omega } \leq {({\lambda _2}/{\lambda _1})_{n -{\text {dimensional ball}}}}$ (the Payne-Pólya-Weinberger conjecture) recently proved by us.

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