Homogenization of point interactions
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866912977875107840 |
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| author | Cafiero, Domenico Correggi, Michele Fermi, Davide |
| author_facet | Cafiero, Domenico Correggi, Michele Fermi, Davide |
| contents | We consider a non-relativistic quantum particle in $\mathbb{R}^d$, $d=2$ or $d = 3$, interacting with singular zero-range potentials concentrated on a large collection of points. We analyze the homogenization regime where the intensities of the singular potentials and the distances between the points simultaneously go to zero as their number grows, while the total interaction strength remains finite. Assuming that the singular potentials have negative scattering lengths and are uniformly distributed, we prove the strong resolvent convergence as $N \to \infty$ of the family of operators to a Schrödinger operator with a regular electrostatic potential. The result is obtained via $Γ$-converge of the associated quadratic forms. Moreover, in presence of an external trapping potential, the convergence is lifted to uniform resolvent sense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21400 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Homogenization of point interactions Cafiero, Domenico Correggi, Michele Fermi, Davide Mathematical Physics Quantum Physics 35J10, 35P05, 47A07, 47B93, 81Q10, 35B27 We consider a non-relativistic quantum particle in $\mathbb{R}^d$, $d=2$ or $d = 3$, interacting with singular zero-range potentials concentrated on a large collection of points. We analyze the homogenization regime where the intensities of the singular potentials and the distances between the points simultaneously go to zero as their number grows, while the total interaction strength remains finite. Assuming that the singular potentials have negative scattering lengths and are uniformly distributed, we prove the strong resolvent convergence as $N \to \infty$ of the family of operators to a Schrödinger operator with a regular electrostatic potential. The result is obtained via $Γ$-converge of the associated quadratic forms. Moreover, in presence of an external trapping potential, the convergence is lifted to uniform resolvent sense. |
| title | Homogenization of point interactions |
| topic | Mathematical Physics Quantum Physics 35J10, 35P05, 47A07, 47B93, 81Q10, 35B27 |
| url | https://arxiv.org/abs/2603.21400 |