Strong scarring of logarithmic quasimodes

We consider a semiclassical pseudodifferential operator on a compact surface, such that the Hamiltonian flow generated by its principal symbol admits a hyperbolic periodic orbit at some energy. For an arbitrary small ε > 0 , we construct semiclassical families of quasimodes of this operator, with energy widths of order ε ℏ / | log ℏ | , and which feature a strong scar along that hyperbolic orbit. Our construction proceeds by controlling the evolution of Gaussian wavepackets up to the Ehrenfest time.