On the multilevel internal structure of the asymptotic distribution of resonances

Abstract We prove that the set of resonances Σ ( H ) has a multilevel asymptotic structure for the following classes of Hamiltonians H : Schrodinger operators with point interactions y j ∈ R 3 , quantum graphs, and 1-D photonic crystals. In the case of N ≥ 2 point interactions, Σ ( H ) consists of a finite number of sequences with logarithmic asymptotics. Leading parameters μ n of these sequences are connected with the geometry of the family Y = { y j } j = 1 N . The minimal parameter μ min corresponds to sequences with ‘more narrow’ and so more observable resonances. The asymptotic density of narrow resonances is expressed via the multiplicity of μ min and is connected with symmetries of Y . In the cases of quantum graphs and 1-D photonic crystals, the decomposition of Σ ( H ) into asymptotic sequences is proved under commensurability conditions. To address the general quantum graph, we introduce asymptotic density functions for two classes of complex strips. The obtained results and effects are compared with obstacle scattering.