Local Spectral Deformation

We develop an analytic perturbation theory for eigenvalues with finite multiplicities, embedded into the essential spectrum of a self-adjoint operator $H$. We assume the existence of another self-adjoint operator $A$ for which the family $H_\theta = e^{\mathrm{i}\theta A} H e^{-\mathrm{i}\theta A}$ extends analytically from the real line to a strip in the complex plane. Assuming a Mourre estimate holds for $\mathrm{i}[H,A]$ in the vicinity of the eigenvalue, we prove that the essential spectrum is locally deformed away from the eigenvalue, leaving it isolated and thus permitting an application of Kato's analytic perturbation theory.