Counterexample to the second eigenfunction having one zero for a non-local Schrodinger operator
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912501006860288 |
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| author | Andrews, Ben Chen, Sophie |
| author_facet | Andrews, Ben Chen, Sophie |
| contents | We demonstrate that the second eigenfunction of a perturbed fractional Laplace operator on a bounded interval can exhibit two sign changes, in stark contrast with the classical expectation that it should have exactly one zero. Our construction employs the Kato-Rellich regular perturbation theory to analyse an infinite potential well eigenvalue problem, and then uses an energy-minimisation argument to extend this counterexample to finite potential wells. Although our detailed analysis focuses on the case where s takes value 1/2 (the Cauchy process), our approach strongly suggests that similar phenomena occur for other rational values of s in (0, 1). At the time of writing, this result provides one of the first rigorous insights into the qualitative behaviour of eigenfunctions for perturbed nonlocal Schrodinger operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19016 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counterexample to the second eigenfunction having one zero for a non-local Schrodinger operator Andrews, Ben Chen, Sophie Analysis of PDEs We demonstrate that the second eigenfunction of a perturbed fractional Laplace operator on a bounded interval can exhibit two sign changes, in stark contrast with the classical expectation that it should have exactly one zero. Our construction employs the Kato-Rellich regular perturbation theory to analyse an infinite potential well eigenvalue problem, and then uses an energy-minimisation argument to extend this counterexample to finite potential wells. Although our detailed analysis focuses on the case where s takes value 1/2 (the Cauchy process), our approach strongly suggests that similar phenomena occur for other rational values of s in (0, 1). At the time of writing, this result provides one of the first rigorous insights into the qualitative behaviour of eigenfunctions for perturbed nonlocal Schrodinger operators. |
| title | Counterexample to the second eigenfunction having one zero for a non-local Schrodinger operator |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.19016 |