We are interested in the study of quantum chaos and scarring, especially in the semiclassical limit. In this regime (as Planck's constant goes to zero), we have shown that certain unstable periodic orbits permanently scar some quantum eigenfunctions.
We have also demonstrated the connection between scarring and tunneling in chaotic double-well potentials through the distribution of level splittings. We found the mean level splitting to have oscillations as a function of energy, as expected if scarring plays a role in determining the size of the splittings, and the spacing between peaks is observed to be periodic of period 2 pi hbar in action. Moreover, the size of the oscillations is directly correlated with the strength of scarring. (Phys. Rev. E 64, 016204, (2001), Phys. Rev. B 96, 094204 (2017)
Typical eigenfunctions for the double-well potential in the (a) near-integrable case without Gaussian perturbations and (b) chaotic case with Gaussian perturbations. The barrier is located at the right side.