WKB for semiclassical operators: How to fly over caustics (and more)
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918411204820992 |
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| author | Ngoc, San Vũ |
| author_facet | Ngoc, San Vũ |
| contents | The method initiated by Wentzel, Kramers, and Brillouin to find approximate solutions to the Schrödinger equation lies at the origin of the spectacular development of microlocal and semiclassical analysis. When used naively, the approach appears to break down at caustics, but Maslov showed how a simple generalization could overcome this difficulty. In this paper, after a partial historical review, we take advantage of more recent advances in microlocal analysis to present a unified treatment of this generalized Maslov-WKB method, using a microlocal sheaf-theoretic approach. This framework provides a rigorous proof of the Bohr Sommerfeld Einstein Brillouin Keller quantization conditions for the eigenvalues of general semiclassical operators (pseudodifferential and Berezin Toeplitz) in one degree of freedom. We also review some applications and extensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25601 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | WKB for semiclassical operators: How to fly over caustics (and more) Ngoc, San Vũ Spectral Theory Mathematical Physics Analysis of PDEs 81Q20 (Primary) 58J50, 58K50, 34E20, 35S30, 35A27, 53D12 The method initiated by Wentzel, Kramers, and Brillouin to find approximate solutions to the Schrödinger equation lies at the origin of the spectacular development of microlocal and semiclassical analysis. When used naively, the approach appears to break down at caustics, but Maslov showed how a simple generalization could overcome this difficulty. In this paper, after a partial historical review, we take advantage of more recent advances in microlocal analysis to present a unified treatment of this generalized Maslov-WKB method, using a microlocal sheaf-theoretic approach. This framework provides a rigorous proof of the Bohr Sommerfeld Einstein Brillouin Keller quantization conditions for the eigenvalues of general semiclassical operators (pseudodifferential and Berezin Toeplitz) in one degree of freedom. We also review some applications and extensions. |
| title | WKB for semiclassical operators: How to fly over caustics (and more) |
| topic | Spectral Theory Mathematical Physics Analysis of PDEs 81Q20 (Primary) 58J50, 58K50, 34E20, 35S30, 35A27, 53D12 |
| url | https://arxiv.org/abs/2603.25601 |