On Threshold Eigenvalues and Resonances for the Linearized NLS Equation

Abstract

Abstract. We prove the instability of threshold resonances and eigenvalues of the linearized NLS operator. We compute the asymptotic approximations of the eigenvalues appearing from the endpoint singularities in terms of the perturbations applied to the original NLS equation. Our method involves such techniques as the Birman-Schwinger principle and the Feshbach map.

On Orbital Instability of Spectrally Stable Vortices of the NLS in the Plane

“…Hence the assumption of absence of embedded eigenvalues seems reasonable. Similarly, the edge of the essential spectrum is shown in [19,78] generically to be neither an eigenvalue nor resonance. Again, we did not check whether taking V generic makes the L ω associated to the cubic-quintic NLS generic in that sense, although this is even more likely than for the issue of embedded eigenvalues.…”

A degenerate edge bifurcation in the 1D linearized nonlinear Schrödinger equation

“…The generic bifurcation of resonances and eigenvalues from the edge of the essential spectrum was studied by [8] and [23] in three dimensions. Edge bifurcations have also been studied in one dimensional systems using the Evans function in [18] and [19].…”

“…Our work is distinct from [8,23] due to the unique challenges of working in one dimension, in particular the strong singularity of the free resolvent at zero energy, which among other things necessitates a double Lyapunov-Schmidt reduction procedure.…”

Solvability Conditions for Some Systems of Nonlinear Non-Fredholm Elliptic Equations

“…[5]), as distinct from the standard situation (see e.g. [1], [7]). We substitute (1.2) into (1.1) and easily obtain the following system of nonlocal elliptic equations with…”