Compactly supported real scalar potentials realizing the Hardy uncertainty endpoint for Schrödinger evolutions

@misc{fu2026compactly,
  title = {{Compactly supported real scalar potentials realizing the Hardy uncertainty endpoint for Schrödinger evolutions}},
  author = {Xiao-Ming Fu and Tianyang Sun},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20202},
  eprint = {2609.20202},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20202},
}

TY  - UNPB
TI  - Compactly supported real scalar potentials realizing the Hardy uncertainty endpoint for Schrödinger evolutions
AU  - Xiao-Ming Fu
AU  - Tianyang Sun
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.20202
UR  - https://arxiv.org/abs/2609.20202
AB  - At the critical Hardy Gaussian weight for the Schrödinger equation in one space dimension on [0,1], the known nonzero scalar example in the weighted L^2 class carries a complex-valued potential. Cassano and Fanelli observed that the existence of a real-valued scalar endpoint example was open, and produced examples with real electric and magnetic potentials only after introducing a magnetic potential. We construct a nonzero smooth solution of i∂_t u+∂_x^2 u=Vu on ℝ×[0,1] such that e^x^2/4u(·,0),e^x^2/4u(·,1)∈ L^2(ℝ) and the potential is bounded, smooth, real-valued, and purely scalar and is supported in one fixed compact spatial interval for all times. This compact-support endpoint example is the main result. We also record the explicit rational-tail realization underlying the construction. The main results of this paper were obtained by the multi-agent system Eureka and have subsequently been verified by the authors.
ER  -