= Countability of elastic eigenfrequencies
Distinct squared traction-free <eigenfrequencies> in a bounded lossless elastic body have <eigenfunctions> <orthogonal> in $L^2(\mathcal D,\rho\,dV)^3$, by the <Betti identity for elastodynamic fields>. Normalize one field from each <eigenspace>. This is an <orthonormal set> in a <separable Hilbert space>, so it is countable: disjoint balls of radius less than $1/\sqrt2$ around its unit vectors must contain distinct points of a countable dense set. Positive and negative <frequency> partners add at most a factor of two. This argument proves countability without requiring a separate compact-resolvent proof.
Back to article page
