Recently, Associate Professor Yang Kai from the School of Mathematics, SEU, in collaboration with Chair Professor Li Dong from the University of Hong Kong, has published the paper titled “The linearized cubic NLS has no embedded eigenvalue” in Inventiones Mathematicae, one of the world’s four top-tier mathematics journals. This landmark publication marks SEU’s first-ever paper in the four flagship mathematics journals.
As one of the most prestigious peer-reviewed journals in pure mathematics, Inventiones Mathematicae features groundbreaking, original, and in-depth research across all branches of pure mathematics. Renowned for its extremely rigorous review standards, it publishes foundational work that drives fundamental advances in mathematical theory. Together with Annals of Mathematics, Acta Mathematica, and the Journal of the American Mathematical Society, it forms the globally recognized “Top Four Mathematics Journals”, serving as the core benchmark for measuring academic influence among mathematicians worldwide.
Back in 1929, mathematical giant John von Neumann and physicist Eugene Wigner revealed a counterintuitive theoretical possibility: discrete eigenvalues could lie hidden within the continuous spectrum of wave motions. Dubbed vividly as “spectral ghosts”, these anomalous phenomena act like traps, trapping energy and preventing its dissipation. Since the 1990s, this unresolved possibility has perplexed researchers working on nonlinear dispersive and wave equations. If such “ghosts” did exist in core models, including the cubic nonlinear Schr?dinger equation (NLS), the theoretical framework describing wave stability would remain incomplete, leaving fundamental theoretical gaps spanning optical fiber communication to quantum physics.
To address this century-old open problem, the study delivers a definitive and fully rigorous conclusion: such spectral ghosts do not exist for the classic non-integrable three-dimensional cubic NLS equation. Adopting an innovative research approach, the team established an entirely new system of mathematical tools and highly original structural transformation techniques. They creatively transformed intractable infinite-dimensional analytical problems into a logically strict and verifiable algebraic proof framework. The newly developed methods provide a powerful toolkit for tackling complex spectral problems and permanently remove the long-standing spectral obstacles that have plagued the research field.
Over recent decades, research in partial differential equations (PDEs) has undergone a profound paradigm shift, shifting focus from the existence of solutions to the more challenging topics of asymptotic stability and long-time dynamical classification. This breakthrough stands as a core advancement within this global research trend. Mathematically, it rules out oscillatory interference caused by embedded eigenvalues and proves that all small disturbances eventually scatter toward infinity, enabling the system to converge to an asymptotically stable manifold composed of solitons. The finding clears critical barriers to the soliton resolution conjecture—one of the most ambitious goals in modern PDE research—and delivers solid structural theoretical support for nonlinear dynamic models. Its far-reaching implications will lay a solid foundation for mathematical analysis research over the coming decades.
This achievement realizes SEU’s historic breakthrough in the top four mathematics journals, signifying a major leap forward in the university’s cutting-edge fundamental mathematics research. This milestone fully demonstrates the high-quality development of SEU’s mathematics discipline, as well as its outstanding innovative strength and great potential in foundational mathematical research.
Source: School of Mathematics, SEU
Translated by: Melody Zhang
Edited by: Leah Li
