POLE STRINGS OF EXPONENTIALLY DAMPED PERIODIC POTENTIALS

We have calculated resonances for a particular class of exponentially damped (oscillatory) potentials, which exhibit multiple strings of eig en-values ( poles) in the complex plane. The aim of this contribution is to give evidence to the observation that each single pole string is connected with only a certain part (subunit) of the potential. In oth er words the complete spectrum of the Hamiltonian can be cleanly decom posed into spectra of ''relevant'' Hamiltonians, each of which exhibit only a single pole string. The ''relevant'' Hamiltonians contain as p otential only part of the complete potential, namely one well and the minimal environment, i.e. the adjacent barriers. Quite contrary to exp ectation we found no delocalisation of the particle trapped in such a potential. In addition to this remarkable behaviour, we found localisa tion even for very low barrier heights and extremely shallow wells giv ing resonances very near the threshold.