Resonances with a background potential

Abstract

We consider a pair of operators H ₁ = −Δ+V and H ₂ = H ₁+W, where the “background potential” V is S _b-dilation analytic, V = O(‖x‖^−2−ε and W = O(e ^{−2α‖x‖} for ‖x‖ → ∞. The resolvent R ₁ (z) = (H ₁ - z ²)⁻¹ is shown to have an analytic continuation $\tilde R_1 (z)$from C⁺ to the region S _a,b = {z‖ ‖ arg z‖ < b, -a <

z < 0} as an operator from the exponentially weighted space L ^z,a(R ³) to L _z, − α(R ³). Resonances of (H ₁, H ₂) in S _a,b are then defined as poles of (1 + W R ₁(z)⁻¹ and identified with poles of the S-matrix of (H ₁, H ₂).