Bounds for Ratios of Eigenvalues of the Dirichlet Laplacian
Mark S. Ashbaugh, Rafael D. Benguria · Proceedings of the American Mathematical Society · 1994
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.