In the semiclassical limit, the transition ratio for optimal cuboid shapes in Robin Riesz means differs from the ratio where the second term in the eigenvalue asymptotics changes sign.
Ivrii,The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary
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Optimizing Riesz means of Robin Laplace operators on cuboids in a semiclassical limit
In the semiclassical limit, the transition ratio for optimal cuboid shapes in Robin Riesz means differs from the ratio where the second term in the eigenvalue asymptotics changes sign.