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We consider a discrete non-linear Schrödinger equation on Z and show that, after adding a small potential localized in the time-frequency space, one can construct a three-parametric family of non-decaying spacetime quasiperiodic solutions to this equation. The proof is based on the Craig–Wayne–Bourgain method combined with recent techniques of dealing with Anderson localization for two-dimensional quasiperiodic operators with degenerate frequencies.

Let , where , , are pairwise coprime. Let be the discrete Schrödinger operator, where Δ is the discrete Laplacian on and the potential is Γ‐periodic. We prove three rigidity theorems for discrete periodic Schrödinger operators in any dimension :

If at some energy level, Fermi varieties of two real‐valued Γ‐periodic potentialsVandYare the same (this feature is referred to asFermi isospectralityofVandY), andYis a separable function, thenVis separable;

We show that the sublinear bound of the bad Green’s functions implies explicit logarithmic bounds of moments for long range operators in arbitrary dimension.

We discover that the distribution of (frequency and phase) resonances plays a role in determining the spectral type of supercritical quasi-periodic Schrödinger operators. In particular, we disprove the 2nd spectral transition line conjecture of Jitomirskaya in the early 1990s.