For slightly mass-supercritical NLS in any dimension, the low-energy unstable spectrum of the self-similar linearized operator consists exactly of the symmetry modes 0, -bi, and -2bi.
Spectra of lin- earized operators for nls solitary waves.SIAM Journal on Mathematical Analysis, 39(4):1070– 1111, 2008
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Mode stability for self-similar blowup of slightly supercritical NLS: I. low-energy spectrum
For slightly mass-supercritical NLS in any dimension, the low-energy unstable spectrum of the self-similar linearized operator consists exactly of the symmetry modes 0, -bi, and -2bi.