RESONANCES FOR THIN BARRIERS ON THE CIRCLE

Abstract. We study high energy resonances for the operator − ∆ + V ⊗ δ∂Ω when V has strong frequency dependence. The operator − ∆ + V ⊗ δ∂Ω is a model Hamiltonian for both quantum corrals [1] [2] and leaky quantum graphs [6]. We study a model for the case where the potential behaves like a quantum point interaction. In particular, we give sharp bounds on the resonance free region when V ≡ h−γV0> 0, γ ≤ 1, Ω is the unit disk, and h−1 ∼ λ is the frequency. We also give a lower bound on the number of resonances in logarithmic size strips. 1