Perturbation of Schrödinger Operators by Complex-Valued Rapidly Oscillating Potentials

抄録

Consider any essentially self-adjoint Schrödinger operator S₀ = −Δ + q₁(x) + q₂(x), Dom(S₀) = C₀^∞(R^N) where q₁(x) ∈ L_loc² satisfies q₁(x) ≥ 0 and q₂(x) ∈ L_loc² is a (−Δ)-bounded real-valued multiplication operator with bound less than 1. Let now q₃(x) be a rapidly oscillating potential, e.g., q₃(x) = |x|³ sin |x|⁵ or (1 + |x|²)⁻¹e^|x| cos(e^|x|) with a singularity near |x| = ∞. In this paper, it is guaranteed that the perturbation T₀ = S₀ + q₃(x) is also essentially self-adjoint. Moreover, their Friedrichs extensions T and S have the same essential spectrum, i.e., σ_ess(T) = σ_ess(S). In fact, we study these problems more generally, i.e., for complex-valued potentials.