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| Format: | Preprint |
| Published: |
2001
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/math/0105026 |
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| _version_ | 1866912655241904128 |
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| author | Lin, Kevin K. |
| author_facet | Lin, Kevin K. |
| contents | This paper presents numerical evidence that for quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy $E$ scales like $\hbar^{-\frac{D(K_E)+1}{2}}$ as $\hbar\to{0}$. Here, $K_E$ denotes the subset of the classical energy surface $\{H=E\}$ which stays bounded for all time under the flow generated by the Hamiltonian $H$ and $D(K_E)$ denotes its fractal dimension. Since the number of bound states in a quantum system with $n$ degrees of freedom scales like $\hbar^{-n}$, this suggests that the quantity $\frac{D(K_E)+1}{2}$ represents the effective number of degrees of freedom in scattering problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0105026 |
| institution | arXiv |
| publishDate | 2001 |
| record_format | arxiv |
| spellingShingle | Numerical Study of Quantum Resonances in Chaotic Scattering Lin, Kevin K. Spectral Theory Numerical Analysis This paper presents numerical evidence that for quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy $E$ scales like $\hbar^{-\frac{D(K_E)+1}{2}}$ as $\hbar\to{0}$. Here, $K_E$ denotes the subset of the classical energy surface $\{H=E\}$ which stays bounded for all time under the flow generated by the Hamiltonian $H$ and $D(K_E)$ denotes its fractal dimension. Since the number of bound states in a quantum system with $n$ degrees of freedom scales like $\hbar^{-n}$, this suggests that the quantity $\frac{D(K_E)+1}{2}$ represents the effective number of degrees of freedom in scattering problems. |
| title | Numerical Study of Quantum Resonances in Chaotic Scattering |
| topic | Spectral Theory Numerical Analysis |
| url | https://arxiv.org/abs/math/0105026 |