Spectra of Linearized Operators for NLS Solitary Waves
  Nonlinear Schr\"odinger (NLS) equations with focusing power nonlinearities have solitary wave solutions. The spectra of the linearized operators around these solitary waves are intimately connected to stability properties of the solitary waves, and to the long-time dynamics of solutions of (NLS). We study these spectra in detail, both analytically and numerically. 
Shu-Ming Chang, Stephen Gustafson, Kenji Nakanishi, Tai-Peng Tsai