Semiclassical construction of resonances with hyperbolic structure: the scar function

The formalism of resonances in quantum chaos is improved by using conveniently defined creation-annihilation operators. With these operators at hand, we are able to construct transverse excited resonances at a given Bohr-quantized energy. Then, by requiring minimum energy dispersion we obtain solutions in terms of even or odd transverse excitations. These wavefunctions, which are constructed in the vicinity of a periodic orbit with maximum energy localization, provide a precise definition of a scar function. These scar functions acquire, in the semiclassical limit, the hyperbolic structure characteristic of unstable periodic orbits.