Resonances and viscosity limit for the Wigner-von Neumann type Hamiltonian

Abstract

The resonances for the Wigner-von Neumann type Hamiltonian are defined by the periodic complex distortion in the Fourier space. Also, following Zworski, we characterize resonances as the limit points of discrete eigenvalues of the Hamiltonian with a quadratic complex absorbing potential in the viscosity type limit.

Resonances as viscosity limits for black box perturbations

“…Zworski [Zw18] showed that scattering resonances of −∆ + V , V ∈ L ∞ comp , are limits of eigenvalues of −∆ + V − iεx 2 as ε → 0+ . The situation is very different for potentials of the Wigner-von Neumann type, in which case Kameoka and Nakamura [KaNa20] showed that the corresponding limits exist away from a discrete set of thresholds. Using an approach closer to [KaNa20] than [Zw18], the author extended Zworski's result to potentials which are exponentially decaying [Xi20].…”

“…The situation is very different for potentials of the Wigner-von Neumann type, in which case Kameoka and Nakamura [KaNa20] showed that the corresponding limits exist away from a discrete set of thresholds. Using an approach closer to [KaNa20] than [Zw18], the author extended Zworski's result to potentials which are exponentially decaying [Xi20]. In this paper we show that the CAP method is also valid for an abstractly defined class of black box perturbations of the Laplacian in R n which can be analytically extended from R n to a conic neighborhood in C n near infinity.…”