The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent

Details The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent

2008-09-17

arxiv.org/abs/0809.2889v3

Mathematics

Optimization and Control

Scientific paper

The paper deals with the genericity of domain-dependent spectral properties of the Laplacian-Dirichlet operator. In particular we prove that, generically, the squares of the eigenfunctions form a free family. We also show that the spectrum is generically non-resonant. The results are obtained by applying global perturbations of the domains and exploiting analytic perturbation properties. The work is motivated by two applications: an existence result for the problem of maximizing the rate of exponential decay of a damped membrane and an approximate controllability result for the bilinear Schr\"odinger equation.

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