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

TL;DR

This paper proves that, for most domains, the squares of Laplacian-Dirichlet eigenfunctions are linearly independent and the spectrum is non-resonant, with implications for control and decay rate optimization.

Contribution

It establishes generic linear independence of squared eigenfunctions and non-resonance of the spectrum for Laplacian-Dirichlet operators through domain perturbations.

Findings

Squares of eigenfunctions are generically linearly independent.

Spectrum is generically non-resonant.

Results enable applications in control and decay optimization.

Abstract

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.