Band invariants for perturbations of the harmonic oscillator
We study the direct and inverse spectral problems for semiclassical operators of the form S = S[subscript 0] + ℏ[superscript 2]V, where S[subscript 0] = 1/2(−ℏ[superscript 2]Δ[subscript Rn] + |x|[superscript 2]) is the harmonic oscillator and V:R[superscript n] → R is a tempered smooth function. We...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
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Elsevier
2015
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| Online Access: | http://hdl.handle.net/1721.1/99441 https://orcid.org/0000-0003-2641-1097 |
| _version_ | 1826210812119744512 |
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| author | Uribe, A. Wang, Z. Guillemin, Victor W. |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Uribe, A. Wang, Z. Guillemin, Victor W. |
| author_sort | Uribe, A. |
| collection | MIT |
| description | We study the direct and inverse spectral problems for semiclassical operators of the form S = S[subscript 0] + ℏ[superscript 2]V, where S[subscript 0] = 1/2(−ℏ[superscript 2]Δ[subscript Rn] + |x|[superscript 2]) is the harmonic oscillator and V:R[superscript n] → R is a tempered smooth function. We show that the spectrum of S forms eigenvalue clusters as ℏ tends to zero, and compute the first two associated “band invariants”. We derive several inverse spectral results for V, under various assumptions. In particular we prove that, in two dimensions, generic analytic potentials that are even with respect to each variable are spectrally determined (up to a rotation). |
| first_indexed | 2024-09-23T14:55:47Z |
| format | Article |
| id | mit-1721.1/99441 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T14:55:47Z |
| publishDate | 2015 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | mit-1721.1/994412022-09-29T11:28:16Z Band invariants for perturbations of the harmonic oscillator Uribe, A. Wang, Z. Guillemin, Victor W. Massachusetts Institute of Technology. Department of Mathematics Guillemin, Victor W. We study the direct and inverse spectral problems for semiclassical operators of the form S = S[subscript 0] + ℏ[superscript 2]V, where S[subscript 0] = 1/2(−ℏ[superscript 2]Δ[subscript Rn] + |x|[superscript 2]) is the harmonic oscillator and V:R[superscript n] → R is a tempered smooth function. We show that the spectrum of S forms eigenvalue clusters as ℏ tends to zero, and compute the first two associated “band invariants”. We derive several inverse spectral results for V, under various assumptions. In particular we prove that, in two dimensions, generic analytic potentials that are even with respect to each variable are spectrally determined (up to a rotation). 2015-10-23T18:13:40Z 2015-10-23T18:13:40Z 2012-06 2011-11 Article http://purl.org/eprint/type/JournalArticle 00221236 1096-0783 http://hdl.handle.net/1721.1/99441 Guillemin, V., A. Uribe, and Z. Wang. “Band Invariants for Perturbations of the Harmonic Oscillator.” Journal of Functional Analysis 263, no. 5 (September 2012): 1435–1467. https://orcid.org/0000-0003-2641-1097 en_US http://dx.doi.org/10.1016/j.jfa.2012.05.022 Journal of Functional Analysis Creative Commons Attribution-Noncommercial-NoDerivatives http://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf Elsevier Arxiv |
| spellingShingle | Uribe, A. Wang, Z. Guillemin, Victor W. Band invariants for perturbations of the harmonic oscillator |
| title | Band invariants for perturbations of the harmonic oscillator |
| title_full | Band invariants for perturbations of the harmonic oscillator |
| title_fullStr | Band invariants for perturbations of the harmonic oscillator |
| title_full_unstemmed | Band invariants for perturbations of the harmonic oscillator |
| title_short | Band invariants for perturbations of the harmonic oscillator |
| title_sort | band invariants for perturbations of the harmonic oscillator |
| url | http://hdl.handle.net/1721.1/99441 https://orcid.org/0000-0003-2641-1097 |
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