Semiclassical measures of eigenfunctions of the attractive Coulomb operator
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908425789636608 |
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| author | Lohr, Nicholas |
| author_facet | Lohr, Nicholas |
| contents | We characterize the set of semiclassical measures corresponding to sequences of eigenfunctions of the attractive Coulomb operator $\widehat{H}_{\hbar}:=-\frac{\hbar^2}{2}Δ_{\mathbb{R}^3}-\frac{1}{|x|}$. In particular, any Radon probability measure on the fixed negative energy hypersurface $Σ_E$ of the Kepler Hamiltonian $H$ in classical phase space that is invariant under the regularized Kepler flow is the semiclassical measure of a sequence of eigenfunctions of $\widehat{H}_{\hbar}$ with eigenvalue $E$ as $\hbar \to 0$. The main tool that we use is the celebrated Fock unitary conjugation map between eigenspaces of $\widehat{H}_{\hbar}$ and $-Δ_{\mathbb{S}^3}$. We first prove that for any Kepler orbit $γ$ on $Σ_E$, there is a sequence of eigenfunctions that converge in the sense of semiclassical measures to the delta measure supported on $γ$ as $\hbar \to 0$, and we finish using a density argument in the weak-* topology. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_09640 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Semiclassical measures of eigenfunctions of the attractive Coulomb operator Lohr, Nicholas Analysis of PDEs Mathematical Physics Spectral Theory We characterize the set of semiclassical measures corresponding to sequences of eigenfunctions of the attractive Coulomb operator $\widehat{H}_{\hbar}:=-\frac{\hbar^2}{2}Δ_{\mathbb{R}^3}-\frac{1}{|x|}$. In particular, any Radon probability measure on the fixed negative energy hypersurface $Σ_E$ of the Kepler Hamiltonian $H$ in classical phase space that is invariant under the regularized Kepler flow is the semiclassical measure of a sequence of eigenfunctions of $\widehat{H}_{\hbar}$ with eigenvalue $E$ as $\hbar \to 0$. The main tool that we use is the celebrated Fock unitary conjugation map between eigenspaces of $\widehat{H}_{\hbar}$ and $-Δ_{\mathbb{S}^3}$. We first prove that for any Kepler orbit $γ$ on $Σ_E$, there is a sequence of eigenfunctions that converge in the sense of semiclassical measures to the delta measure supported on $γ$ as $\hbar \to 0$, and we finish using a density argument in the weak-* topology. |
| title | Semiclassical measures of eigenfunctions of the attractive Coulomb operator |
| topic | Analysis of PDEs Mathematical Physics Spectral Theory |
| url | https://arxiv.org/abs/2303.09640 |