Spectral-Edge Continuation, Dimensional Crossover, and Endpoint Stability for Partially Confined Nonlinear Schrödinger Equations
Abstract
We study the global geometry and endpoint dynamics of positive solutions of −Δu+|y|2u+λu=g(u), (y,z)∈ℝd×ℝm, m=N−d. Despite the loss of coercivity at λ=−d, we construct a connected component of centered positive solutions with frequency projection (−d,∞), assuming subcritical asymptotics of g at zero and infinity, a zero-mass Liouville condition, and a strict Hamiltonian sign condition. At the lower edge, λ+d→ 0+, anisotropic rescaling selects the lowest confined harmonic mode coupled to an m-dimensional scalar ground state. As λ→∞, isotropic blow-up yields the N-dimensional scalar ground state, revealing a dimensional crossover. The endpoint profiles and mass laws hold for every sequence of centered positive solutions, not merely along this component. The proof combines Gaussian--exponential localization on compact frequency strips, Lyapunov--Schmidt reduction, compact spectral perturbation, and Leray--Schauder continuation. At both ends, we establish uniqueness and differentiability of the centered branch, identify the Morse indices and symmetry kernels of the real and imaginary linearizations in the full energy space, and derive sharp mass-slope asymptotics. Under standard Hamiltonian-flow hypotheses, the resulting slope criterion classifies orbital stability and instability away from the mass-critical powers. The mass map on this component, combined with endpoint control on the full centered solution set, yields existence, nonexistence, and multiplicity for normalized standing waves, revealing the roles of both critical exponents 2+4/m and 2+4/N.
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BibTeX
@misc{zhong2026spectraledge,
title = {{Spectral-Edge Continuation, Dimensional Crossover, and Endpoint Stability for Partially Confined Nonlinear Schrödinger Equations}},
author = {Xuexiu Zhong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14585},
eprint = {2610.14585},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14585},
}
RIS
TY - UNPB
TI - Spectral-Edge Continuation, Dimensional Crossover, and Endpoint Stability for Partially Confined Nonlinear Schrödinger Equations
AU - Xuexiu Zhong
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14585
UR - https://arxiv.org/abs/2610.14585
AB - We study the global geometry and endpoint dynamics of positive solutions of −Δu+|y|^2u+λu=g(u), (y,z)∈ℝ^d×ℝ^m, m=N−d. Despite the loss of coercivity at λ=−d, we construct a connected component of centered positive solutions with frequency projection (−d,∞), assuming subcritical asymptotics of g at zero and infinity, a zero-mass Liouville condition, and a strict Hamiltonian sign condition. At the lower edge, λ+d→ 0^+, anisotropic rescaling selects the lowest confined harmonic mode coupled to an m-dimensional scalar ground state. As λ→∞, isotropic blow-up yields the N-dimensional scalar ground state, revealing a dimensional crossover. The endpoint profiles and mass laws hold for every sequence of centered positive solutions, not merely along this component. The proof combines Gaussian--exponential localization on compact frequency strips, Lyapunov--Schmidt reduction, compact spectral perturbation, and Leray--Schauder continuation. At both ends, we establish uniqueness and differentiability of the centered branch, identify the Morse indices and symmetry kernels of the real and imaginary linearizations in the full energy space, and derive sharp mass-slope asymptotics. Under standard Hamiltonian-flow hypotheses, the resulting slope criterion classifies orbital stability and instability away from the mass-critical powers. The mass map on this component, combined with endpoint control on the full centered solution set, yields existence, nonexistence, and multiplicity for normalized standing waves, revealing the roles of both critical exponents 2+4/m and 2+4/N.
ER -