Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.18893 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918281816834048 |
|---|---|
| author | Ifa, Abdelwaheb Rouleux, Michel |
| author_facet | Ifa, Abdelwaheb Rouleux, Michel |
| contents | We revisit the well known Bohr-Sommerfeld quantization rule (BS) of order 2 for a self-adjoint 1-D h-Pseudo-differential operator within the algebraic and microlocal framework of Helffer and Sjoestrand; BS holds precisely when Gram matrix consisting of scalar products of some WKB solutions with respect to the ``flux norm'' (or microlocal Wronskian) is not invertible. We simplify somewhat our previous proof [A. Ifa H. Louati and M. Rouleux. Bohr-Sommerfeld Quantization Rules Revisited: the Method of Positive Commutators. J. Math. Sci. Univ. Tokyo, 25(2):2018] by working in spatial representation only, as in complex WKB theory for Schroedinger operator. We consider also the scattering problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18893 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The flux norm, Bohr-Sommerfeld Quantization Rules and the scattering problem for h $Ψ$DO's on the real line Ifa, Abdelwaheb Rouleux, Michel Mathematical Physics We revisit the well known Bohr-Sommerfeld quantization rule (BS) of order 2 for a self-adjoint 1-D h-Pseudo-differential operator within the algebraic and microlocal framework of Helffer and Sjoestrand; BS holds precisely when Gram matrix consisting of scalar products of some WKB solutions with respect to the ``flux norm'' (or microlocal Wronskian) is not invertible. We simplify somewhat our previous proof [A. Ifa H. Louati and M. Rouleux. Bohr-Sommerfeld Quantization Rules Revisited: the Method of Positive Commutators. J. Math. Sci. Univ. Tokyo, 25(2):2018] by working in spatial representation only, as in complex WKB theory for Schroedinger operator. We consider also the scattering problem. |
| title | The flux norm, Bohr-Sommerfeld Quantization Rules and the scattering problem for h $Ψ$DO's on the real line |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2407.18893 |