Two-sided Lieb-Thirring bounds
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914948218617856 |
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| author | Bachmann, Sven Froese, Richard Schraven, Severin |
| author_facet | Bachmann, Sven Froese, Richard Schraven, Severin |
| contents | We prove upper and lower bounds for the number of eigenvalues of semi-bounded Schrödinger operators in all spatial dimensions. As a corollary, we obtain two-sided estimates for the sum of the negative eigenvalues of atomic Hamiltonians with Kato potentials. Instead of being in terms of the potential itself, as in the usual Lieb-Thirring result, the bounds are in terms of the landscape function, also known as the torsion function, which is a solution of $(-Δ+ V +M)u_M =1$ in $\mathbb{R}^d$; here $M\in\mathbb{R}$ is chosen so that the operator is positive. We further prove that the infimum of $(u_M^{-1} - M)$ is a lower bound for the ground state energy $E_0$ and derive a simple iteration scheme converging to $E_0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19023 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two-sided Lieb-Thirring bounds Bachmann, Sven Froese, Richard Schraven, Severin Mathematical Physics Spectral Theory 47B93, 81Q10, 35P20, 35J10 We prove upper and lower bounds for the number of eigenvalues of semi-bounded Schrödinger operators in all spatial dimensions. As a corollary, we obtain two-sided estimates for the sum of the negative eigenvalues of atomic Hamiltonians with Kato potentials. Instead of being in terms of the potential itself, as in the usual Lieb-Thirring result, the bounds are in terms of the landscape function, also known as the torsion function, which is a solution of $(-Δ+ V +M)u_M =1$ in $\mathbb{R}^d$; here $M\in\mathbb{R}$ is chosen so that the operator is positive. We further prove that the infimum of $(u_M^{-1} - M)$ is a lower bound for the ground state energy $E_0$ and derive a simple iteration scheme converging to $E_0$. |
| title | Two-sided Lieb-Thirring bounds |
| topic | Mathematical Physics Spectral Theory 47B93, 81Q10, 35P20, 35J10 |
| url | https://arxiv.org/abs/2403.19023 |