First algorithms and solvability-complexity classifications for computing Lebesgue measure, capacity, and fractal dimensions of spectra of infinite-dimensional linear operators.
Computing the Sound of the Sea in a Seashell
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abstract
The question of whether there exists an approximation procedure to compute the resonances of any Helmholtz resonator, regardless of its particular shape, is addressed. A positive answer is given, and it is shown that all that one has to assume is that the resonator chamber is bounded and that its boundary is $\mathcal C^2$. The proof is constructive, providing a universal algorithm which only needs to access the values of the characteristic function of the chamber at any requested point.
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On the computation of geometric features of spectra of linear operators on Hilbert spaces
First algorithms and solvability-complexity classifications for computing Lebesgue measure, capacity, and fractal dimensions of spectra of infinite-dimensional linear operators.