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...

Let A be a pseudodifferential operator on ${ℝ}^N$ whose Weyl symbol a is a strictly positive smooth function on $W={ℝ}^N\times {ℝ}^N$ such that $|{\partial }^{\alpha }a|\le C_{\alpha }{a}^{1-\varrho }$ for some ϱ>0 and all |α|>0, ${\partial }^{\alpha }a$ is bounded for large |α|, and $li{m}_{w\to \infty }a\left(w\right)=\infty$. Such an operator A is essentially selfadjoint, bounded from below, and its spectrum is discrete. The remainder term in the Weyl asymptotic formula for the distribution of the eigenvalues of A is estimated. This is done by applying the method of approximate spectral projectors of Tulovskiĭ and Shubin.