Two-parameter localization and related phase transition for a Schrödinger operator in balls and spherical shells
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2019
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911922284134400 |
|---|---|
| author | Jia, Chen Zhang, Zhimin Zhao, Lewei |
| author_facet | Jia, Chen Zhang, Zhimin Zhao, Lewei |
| contents | Here we investigate the two-parameter high-frequency localization for the eigenfunctions of a Schrödinger operator with a singular inverse square potential in high-dimensional balls and spherical shells as the azimuthal quantum number $l$ and the principal quantum number $k$ tend to infinity simultaneously, while keeping their ratio as a constant, generalizing the classical one-parameter localization for Laplacian eigenfunctions [SIAM J. Appl. Math. 73:780-803, 2013]. We prove that the eigenfunctions in balls are localized around an intermediate sphere whose radius is increasing with respect to the $l$-$k$ ratio. The eigenfunctions decay exponentially inside the localized sphere and decay polynomially outside. Furthermore, we discover a novel second-order phase transition for the eigenfunctions in spherical shells as the $l$-$k$ ratio crosses a critical value. In the supercritical case, the eigenfunctions are localized around a sphere between the inner and outer boundaries of the spherical shell. In the critical case, the eigenfunctions are localized around the inner boundary. In the subcritical case, no localization could be observed, giving rise to localization breaking. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1909_09878 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Two-parameter localization and related phase transition for a Schrödinger operator in balls and spherical shells Jia, Chen Zhang, Zhimin Zhao, Lewei Mathematical Physics Analysis of PDEs Spectral Theory Quantum Physics 35J05, 35J10, 35P99, 33C10, 35Q40 Here we investigate the two-parameter high-frequency localization for the eigenfunctions of a Schrödinger operator with a singular inverse square potential in high-dimensional balls and spherical shells as the azimuthal quantum number $l$ and the principal quantum number $k$ tend to infinity simultaneously, while keeping their ratio as a constant, generalizing the classical one-parameter localization for Laplacian eigenfunctions [SIAM J. Appl. Math. 73:780-803, 2013]. We prove that the eigenfunctions in balls are localized around an intermediate sphere whose radius is increasing with respect to the $l$-$k$ ratio. The eigenfunctions decay exponentially inside the localized sphere and decay polynomially outside. Furthermore, we discover a novel second-order phase transition for the eigenfunctions in spherical shells as the $l$-$k$ ratio crosses a critical value. In the supercritical case, the eigenfunctions are localized around a sphere between the inner and outer boundaries of the spherical shell. In the critical case, the eigenfunctions are localized around the inner boundary. In the subcritical case, no localization could be observed, giving rise to localization breaking. |
| title | Two-parameter localization and related phase transition for a Schrödinger operator in balls and spherical shells |
| topic | Mathematical Physics Analysis of PDEs Spectral Theory Quantum Physics 35J05, 35J10, 35P99, 33C10, 35Q40 |
| url | https://arxiv.org/abs/1909.09878 |