Avoided level crossings in open quantum systems

At high level density, two states avoid usually crossing at the critical value $a_{\mathrm{c}\mathrm{r}}$ of the parameter $a$ by which the system is controlled. The wavefunctions of the two states are mixed in a finite parameter range around $a_{\mathrm{c}\mathrm{r}}$. This holds true for discrete states as well as for narrow resonance states which are coupled via the environment of scattering wavefunctions. We study the influence of avoided level crossings onto four overlapping complex eigenvalues of a symmetric non-Hermitian operator. The mixing of the two wavefunctions around $a_{\mathrm{c}\mathrm{r}}$ is simulated, in each case, by assuming a Gaussian distribution around $a_{\mathrm{c}\mathrm{r}}$. At high level density, the Gaussian distributions related to avoided crossings of different levels may overlap. Here, new effects arise, especially from the imaginary part of the coupling term via the environment. The results show, moreover, the influence of symmetries onto the multi-level avoided crossing phenomenon.