Phase rigidity and avoided level crossings in the complex energy plane

We consider the effective Hamiltonian of an open quantum system, its biorthogonal eigenfunctions φ_λ, and define the value ${r}_{{λ}}=({{φ}}_{{λ}}{}{{φ}}_{{λ}})∕⟨{{φ}}_{{λ}}{}{{φ}}_{{λ}}⟩$ that characterizes the phase rigidity of the eigenfunctions ${{φ}}_{{λ}}$. In the scenario with avoided level crossings, ${r}_{{λ}}$ varies between 1 and 0 due to the mutual influence of neighboring resonances. The variation of ${r}_{{λ}}$ is an internal property of an open quantum system. In the literature, the phase rigidity ${ρ}$ of the scattering wave function ${{Ψ}}CE$ is considered. Since ${{Ψ}}CE$ can be represented in the interior of the system by the ${{φ}}_{{λ}}$, the phase rigidity ${ρ}$ of the ${{Ψ}}CE$ is related to the ${r}_{{λ}}$ and therefore also to the mutual influence of neighboring resonances. As a consequence, the reduction of the phase rigidity ${ρ}$ to values smaller than 1 should be considered, at least partly, as an internal property of an open quantum system in the overlapping regime. The relation to measurable values such as the transmission through a quantum dot, follows from the fact that the transmission is, in any case, resonant at energies that are determined by the real part of the eigenvalues of the effective Hamiltonian. We illustrate the relation between phase rigidity ρ and transmission numerically for small open cavities.