In this paper using a transform defined by the translation operator we introduce the concept of spectrum of sequences that are bounded by a given polynomial. We apply this spectral theory to study the asymptotic behavior of solutions of fractional linear difference equations. One of the obtained results is an extension of a famous Katznelson-Tzafriri Theorem, saying that the α-resolvent operator that is associated with the fractional equation, satisfies an asymptotic estimate of Katznelson-Tzafriri type, provided that it is bounded by the polynomial, and the spectrum of the fractional equation on the unit circle is either empty or consists of only one element 1. Three concrete examples are also included to illustrate the obtained results.

We consider the energy-critical stochastic cubic nonlinear Schrödinger equation on R⁴ with additive noise, and with the non-vanishing boundary conditions at spatial infinity. By viewing this equation as a perturbation to the energy-critical cubic nonlinear Schrödinger equation on R⁴, we prove global well-posedness in the energy space. Moreover, we establish unconditional uniqueness of solutions in the energy space.

We define a nonlinear q-difference system _N,(M-,M+) as a connection preserving deformation of a certain q-difference linear equation. We also study its relation to a series _N,M defined as a certain generalization of q-hypergeometric functions.

We prove low regularity a priori estimates for the derivative nonlinear Schrödinger equation in Besov spaces with positive regularity index conditional upon small L²-norm. This covers the full subcritical range. We use the power series expansion of the perturbation determinant introduced by Killip-Vişan-Zhang for completely integrable PDE. This makes it possible to derive low regularity conservation laws from the perturbation determinant.