PreprintarXiv licencearXiv
Absence of non-threshold embedded eigenvalues for nonlinear Schrödinger equations
Abstract
We consider the focusing nonlinear Schrödinger equation on ℝd i∂tu+Δu+|u|p−1u=0. We prove the absence of non-threshold embedded eigenvalues for the linearized operator around the positive ground state when d=1,2 and p>1, or when d≥3 and 1<p≤1+4/(d−2). Starting from the fluid variables of NLS, we derive a biharmonic wave equation for the phase gradient after linearization around the ground state. Seeking a virial identity for this equation, we identify a mechanism for establishing a repulsive structure, which guides the subsequent ODE analysis to exclude embedded eigenvalues.
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BibTeX
@misc{kim2026absence,
title = {{Absence of non-threshold embedded eigenvalues for nonlinear Schrödinger equations}},
author = {Taegyu Kim},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26111},
eprint = {2610.26111},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26111},
}
RIS
TY - UNPB
TI - Absence of non-threshold embedded eigenvalues for nonlinear Schrödinger equations
AU - Taegyu Kim
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26111
UR - https://arxiv.org/abs/2610.26111
AB - We consider the focusing nonlinear Schrödinger equation on ℝ^d i∂_tu+Δu+|u|^p−1u=0. We prove the absence of non-threshold embedded eigenvalues for the linearized operator around the positive ground state when d=1,2 and p>1, or when d≥3 and 1<p≤1+4/(d−2). Starting from the fluid variables of NLS, we derive a biharmonic wave equation for the phase gradient after linearization around the ground state. Seeking a virial identity for this equation, we identify a mechanism for establishing a repulsive structure, which guides the subsequent ODE analysis to exclude embedded eigenvalues.
ER -