Approximation and Equidistribution of Phase Shifts: Spherical Symmetry
Consider a semiclassical Hamiltonian H[subscript V,h]:=h[superscript 2] Δ + V − E, where h > 0 is a semiclassical parameter, Δ is the positive Laplacian on R[superscript d],V is a smooth, compactly supported central potential function and E > 0 is an energy level. In this setting the scatter...
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| Format: | Article |
| Language: | English |
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Springer Berlin Heidelberg
2016
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| Online Access: | http://hdl.handle.net/1721.1/104921 |
| _version_ | 1826190215487684608 |
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| author | Datchev, Kiril Gell-Redman, Jesse Hassell, Andrew Humphries, Peter |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Datchev, Kiril Gell-Redman, Jesse Hassell, Andrew Humphries, Peter |
| author_sort | Datchev, Kiril |
| collection | MIT |
| description | Consider a semiclassical Hamiltonian H[subscript V,h]:=h[superscript 2] Δ + V − E, where h > 0 is a semiclassical parameter, Δ is the positive Laplacian on R[superscript d],V is a smooth, compactly supported central potential function and E > 0 is an energy level. In this setting the scattering matrix S[subscript h](E) is a unitary operator on L[superscript 2](S[superscript d−1]), hence with spectrum lying on the unit circle; moreover, the spectrum is discrete except at 1. We show under certain additional assumptions on the potential that the eigenvalues of S[subscript h](E) can be divided into two classes: a finite number ∼c[subscript d](R√E/h)[superscript d−1], as h→0, where B(0, R) is the convex hull of the support of the potential, that equidistribute around the unit circle, and the remainder that are all very close to 1. Semiclassically, these are related to the rays that meet the support of, and hence are scattered by, the potential, and those that do not meet the support of the potential, respectively. A similar property is shown for the obstacle problem in the case that the obstacle is the ball of radius R. |
| first_indexed | 2024-09-23T08:36:31Z |
| format | Article |
| id | mit-1721.1/104921 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T08:36:31Z |
| publishDate | 2016 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1049212022-09-30T09:57:53Z Approximation and Equidistribution of Phase Shifts: Spherical Symmetry Datchev, Kiril Gell-Redman, Jesse Hassell, Andrew Humphries, Peter Massachusetts Institute of Technology. Department of Mathematics Datchev, Kiril Consider a semiclassical Hamiltonian H[subscript V,h]:=h[superscript 2] Δ + V − E, where h > 0 is a semiclassical parameter, Δ is the positive Laplacian on R[superscript d],V is a smooth, compactly supported central potential function and E > 0 is an energy level. In this setting the scattering matrix S[subscript h](E) is a unitary operator on L[superscript 2](S[superscript d−1]), hence with spectrum lying on the unit circle; moreover, the spectrum is discrete except at 1. We show under certain additional assumptions on the potential that the eigenvalues of S[subscript h](E) can be divided into two classes: a finite number ∼c[subscript d](R√E/h)[superscript d−1], as h→0, where B(0, R) is the convex hull of the support of the potential, that equidistribute around the unit circle, and the remainder that are all very close to 1. Semiclassically, these are related to the rays that meet the support of, and hence are scattered by, the potential, and those that do not meet the support of the potential, respectively. A similar property is shown for the obstacle problem in the case that the obstacle is the ball of radius R. National Science Foundation (U.S.). (Postdoctoral Fellowship) 2016-10-21T19:27:54Z 2016-10-21T19:27:54Z 2013-11 2012-12 2016-08-18T15:24:04Z Article http://purl.org/eprint/type/JournalArticle 0010-3616 1432-0916 http://hdl.handle.net/1721.1/104921 Datchev, Kiril et al. “Approximation and Equidistribution of Phase Shifts: Spherical Symmetry.” Communications in Mathematical Physics 326.1 (2014): 209–236. en http://dx.doi.org/10.1007/s00220-013-1841-8 Communications in Mathematical Physics Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer-Verlag Berlin Heidelberg application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Datchev, Kiril Gell-Redman, Jesse Hassell, Andrew Humphries, Peter Approximation and Equidistribution of Phase Shifts: Spherical Symmetry |
| title | Approximation and Equidistribution of Phase Shifts: Spherical Symmetry |
| title_full | Approximation and Equidistribution of Phase Shifts: Spherical Symmetry |
| title_fullStr | Approximation and Equidistribution of Phase Shifts: Spherical Symmetry |
| title_full_unstemmed | Approximation and Equidistribution of Phase Shifts: Spherical Symmetry |
| title_short | Approximation and Equidistribution of Phase Shifts: Spherical Symmetry |
| title_sort | approximation and equidistribution of phase shifts spherical symmetry |
| url | http://hdl.handle.net/1721.1/104921 |
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