Flat tori with large Laplacian eigenvalues in dimensions up to eight
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We consider the optimization problem of maximizing the k-th Laplacian eigenvalue, λ_k, over flat d-dimensional tori of fixed volume. For k=1, this problem is equivalent to the densest lattice sphere packing problem. For larger k, this is equivalent to the NP-hard problem of finding the d-dimensional (dual) lattice with longest k-th shortest lattice vector. As a result of extensive computations, for d ≤ 8, we obtain a sequence of flat tori, T_k,d, each of volume one, such that the k-th Laplacian eigenvalue of T_k,d is very large; for each (finite) k the k-th eigenvalue exceeds the value in (the k→∞ asymptotic) Weyl's law by a factor between 1.54 and 2.01, depending on the dimension. Stationarity conditions are derived and numerically verified for T_k,d and we describe the degeneration of the tori as k →∞.
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