Zeroes of Eigenfunctions of Schrödinger Operators after Schwartzman
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916947210272768 |
|---|---|
| author | Wong, Willie Wai-Yeung |
| author_facet | Wong, Willie Wai-Yeung |
| contents | Consider a complete, connected, smooth, oriented Riemannian manifold $(M,g)$ with boundary, such that the first Betti number vanishes. Sol Schwartzman proved that for Schrödinger operators of the form $-Δ_g + V$ where $\Im(V)$ is signed, if $f: M\to\mathbb{C}$ is a non-vanishing element of its kernel, then $f$ has constant phase. The proof relied on dynamical systems methods applied to the gradient flow of the phase of $f$. In this manuscript we provide a more direct PDE argument that proves strengthened versions of the same facts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09739 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zeroes of Eigenfunctions of Schrödinger Operators after Schwartzman Wong, Willie Wai-Yeung Analysis of PDEs 2020 MSC-class: 35J10 (Primary) Consider a complete, connected, smooth, oriented Riemannian manifold $(M,g)$ with boundary, such that the first Betti number vanishes. Sol Schwartzman proved that for Schrödinger operators of the form $-Δ_g + V$ where $\Im(V)$ is signed, if $f: M\to\mathbb{C}$ is a non-vanishing element of its kernel, then $f$ has constant phase. The proof relied on dynamical systems methods applied to the gradient flow of the phase of $f$. In this manuscript we provide a more direct PDE argument that proves strengthened versions of the same facts. |
| title | Zeroes of Eigenfunctions of Schrödinger Operators after Schwartzman |
| topic | Analysis of PDEs 2020 MSC-class: 35J10 (Primary) |
| url | https://arxiv.org/abs/2509.09739 |