On the relation between quantum lifetimes and classical stability for the systems with a saddle-type potential

Relations between quantum-mechanical and classical properties of open systems with a saddle-type potential, for which at a given energy only one unstable periodic orbit exists, are studied. By considering the convergence of the Gutzwiller trace formula [J. Math. Phys. 12, 343 (1971)] it is confirmed that both for homogeneous and inhomogeneous potentials the poles of the formula are located below the real energy axis, i.e., these kind of potentials do not support bound states, in general. Within the harmonic approximation the widths of resonant (transition) states are proportional to the values of Lyapunov exponent of the single periodic orbit calculated at the energies which are equal to the resonance positions. The accuracy of the semiclassical relation is discussed and demonstrated for several examples.