= Spectrum of a flat torus
{title2=$\lambda_w=4\pi^2|w|^2$}
With the nonnegative <Laplace-Beltrami operator>, a frequency $w$ in the <dual lattice> gives the <eigenfunction> $e^{2\pi i\langle w,x\rangle}$ and <eigenvalue> $4\pi^2|w|^2$. Its multiplicity is the number of dual vectors with that norm. <Fourier series> prove completeness. In dimension two the shortest vector, shortest independent vector and covolume determine a reduced <Gram matrix>, proving <spectral rigidity>.
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