Isospectral Convex Domains in Euclidean Space
Isospectral Convex Domains in Euclidean Space
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DOI：
10.4310/mrl.1994.v1.n5.a2
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发表时间：
1994
影响因子：
1
通讯作者：
Carolyn S. Gordon;David L. Webb
中科院分区：
文献类型：
--
作者：
Carolyn S. Gordon;David L. Webb
Mark Kac’s question “Can one hear the shape of a drum?” [K] asks whether the eigenvalue spectrum of the Laplace operator on a domain in the Euclidean plane determines the domain up to congruence. Urakawa [U] constructed examples of isospectral domains in R for n ≥ 4. Kac’s original question was answered negatively in [GWW]; see [BCDS] for many other examples. Thus far, all examples of noncongruent isospectral domains in Euclidean space are nonconvex. Thus the following question has remained open: are there pairs of convex domains in Euclidean space which are isospectral but not congruent? In this note, we show that one can modify slightly Urakawa’s construction to obtain a pair of convex domains in Euclidean n-space for n ≥ 4 which are noncongruent yet isospectral for both Dirichlet and Neumann boundary conditions. The proof of Neumann isospectrality uses very recent results of Hubert Pesce; we wish to thank him for discussing his work with us, for reading and commenting on a preliminary version, and for furnishing references.
