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We provide complementary semiclassical bounds for the Riesz means $R_1(z)$ ofthe eigenvalues of various biharmonic operators, with a second term in theexpected power of $z$. The method we discuss makes use of the averagedvariational principle (AVP), and yields two-sided bounds for individualeigenvalues, which are semiclassically sharp. The AVP also yields comparisonswith Riesz means of different operators, in particular Laplacians.

Semiclassical bounds for spectra of biharmonic operators