##
D. Pelinovsky and A. Sakovich

#
Internal modes of discrete solitons near the
anti-continuum limit of the dNLS equation

###
Physica D 240, 265-281 (2011)

**Abstract:**

Discrete solitons of the discrete nonlinear Schrödinger (dNLS) equation become compactly
supported in the anti-continuum limit of the zero coupling between lattice sites. Eigenvalues
of the linearization of the dNLS equation at the discrete soliton determine its spectral and
linearized stability. All unstable eigenvalues of the discrete solitons near the anti-continuum
limit were characterized earlier for this model. Here we analyze the resolvent operator and
prove that it is uniformly bounded in the neighborhood of the continuous spectrum if the
discrete soliton is simply connected in the anti-continuum limit. This result rules out existence
of internal modes (neutrally stable eigenvalues of the discrete spectrum) of such discrete
solitons near the anti-continuum limit.

**Keywords:**

Discrete nonlinear Schrodinger equations, discrete solitons, anti-continuum limit,
internal modes, resolvent operators