Hydrogen-like Schrödinger Operators at Low Energies
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2022
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866912231616151552 |
|---|---|
| author | Sussman, Ethan |
| author_facet | Sussman, Ethan |
| contents | Consider a Schrödinger operator on an asymptotically Euclidean manifold $X$ of dimension at least two, and suppose that the potential is of attractive Coulomb-like type. Using Vasy's second 2nd-microlocal approach, "the Lagrangian approach," we analyze -- uniformly, all the way down to $E=0$ -- the output of the limiting resolvent $R(E\pm i 0) = \lim_{ε\to 0^+} R(E\pm i ε)$. The Coulomb potential causes the output of the low-energy resolvent to possess oscillatory asymptotics which differ substantially from the sorts of asymptotics observed in the short-range case by Guillarmou, Hassell, Sikora, and (more recently) Hintz and Vasy. Specifically, the compound asymptotics at low energy and large spatial scales are more delicate, and the resolvent output is smooth all the way down to $E=0$. In fact, we will construct a compactification of $(0,1]_E\times X$ on which the resolvent output is given by a specified (and relatively complicated) function that oscillates as $r\to\infty$ times something polyhomogeneous. As a corollary, we get complete and compatible asymptotic expansions for solutions to the scattering problem as functions of both position and energy, with a transitional regime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_08355 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hydrogen-like Schrödinger Operators at Low Energies Sussman, Ethan Analysis of PDEs Mathematical Physics Classical Analysis and ODEs Differential Geometry Spectral Theory Primary 35P25, Secondary 58J40, 58J47, 58J50 Consider a Schrödinger operator on an asymptotically Euclidean manifold $X$ of dimension at least two, and suppose that the potential is of attractive Coulomb-like type. Using Vasy's second 2nd-microlocal approach, "the Lagrangian approach," we analyze -- uniformly, all the way down to $E=0$ -- the output of the limiting resolvent $R(E\pm i 0) = \lim_{ε\to 0^+} R(E\pm i ε)$. The Coulomb potential causes the output of the low-energy resolvent to possess oscillatory asymptotics which differ substantially from the sorts of asymptotics observed in the short-range case by Guillarmou, Hassell, Sikora, and (more recently) Hintz and Vasy. Specifically, the compound asymptotics at low energy and large spatial scales are more delicate, and the resolvent output is smooth all the way down to $E=0$. In fact, we will construct a compactification of $(0,1]_E\times X$ on which the resolvent output is given by a specified (and relatively complicated) function that oscillates as $r\to\infty$ times something polyhomogeneous. As a corollary, we get complete and compatible asymptotic expansions for solutions to the scattering problem as functions of both position and energy, with a transitional regime. |
| title | Hydrogen-like Schrödinger Operators at Low Energies |
| topic | Analysis of PDEs Mathematical Physics Classical Analysis and ODEs Differential Geometry Spectral Theory Primary 35P25, Secondary 58J40, 58J47, 58J50 |
| url | https://arxiv.org/abs/2204.08355 |