Geometric Approximation of Point Interactions in Three-Dimensional Domains
In this paper, we study a three-dimensional second-order elliptic operator with a point interaction in an arbitrary domain. The operator is supposed to be self-adjoint. We cut out a small cavity around the center of the interaction and consider an operator in such perforated domain with the Robin co...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/7/1031 |
| _version_ | 1797212275660029952 |
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| author | Denis Ivanovich Borisov |
| author_facet | Denis Ivanovich Borisov |
| author_sort | Denis Ivanovich Borisov |
| collection | DOAJ |
| description | In this paper, we study a three-dimensional second-order elliptic operator with a point interaction in an arbitrary domain. The operator is supposed to be self-adjoint. We cut out a small cavity around the center of the interaction and consider an operator in such perforated domain with the Robin condition on the boundary of the cavity. Our main result states that once the coefficient in this Robin condition is appropriately chosen, the operator in the perforated domain converges to that with the point interaction in the norm resolvent sense. We also succeed in establishing order-sharp estimates for the convergence rate. |
| first_indexed | 2024-04-24T10:39:48Z |
| format | Article |
| id | doaj.art-1978741d089c407087197a251d8316c0 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-24T10:39:48Z |
| publishDate | 2024-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-1978741d089c407087197a251d8316c02024-04-12T13:22:41ZengMDPI AGMathematics2227-73902024-03-01127103110.3390/math12071031Geometric Approximation of Point Interactions in Three-Dimensional DomainsDenis Ivanovich Borisov0Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, Ufa 450008, RussiaIn this paper, we study a three-dimensional second-order elliptic operator with a point interaction in an arbitrary domain. The operator is supposed to be self-adjoint. We cut out a small cavity around the center of the interaction and consider an operator in such perforated domain with the Robin condition on the boundary of the cavity. Our main result states that once the coefficient in this Robin condition is appropriately chosen, the operator in the perforated domain converges to that with the point interaction in the norm resolvent sense. We also succeed in establishing order-sharp estimates for the convergence rate.https://www.mdpi.com/2227-7390/12/7/1031point interactionsmall cavityRobin conditionnorm resolvent convergenceconvergence rate |
| spellingShingle | Denis Ivanovich Borisov Geometric Approximation of Point Interactions in Three-Dimensional Domains Mathematics point interaction small cavity Robin condition norm resolvent convergence convergence rate |
| title | Geometric Approximation of Point Interactions in Three-Dimensional Domains |
| title_full | Geometric Approximation of Point Interactions in Three-Dimensional Domains |
| title_fullStr | Geometric Approximation of Point Interactions in Three-Dimensional Domains |
| title_full_unstemmed | Geometric Approximation of Point Interactions in Three-Dimensional Domains |
| title_short | Geometric Approximation of Point Interactions in Three-Dimensional Domains |
| title_sort | geometric approximation of point interactions in three dimensional domains |
| topic | point interaction small cavity Robin condition norm resolvent convergence convergence rate |
| url | https://www.mdpi.com/2227-7390/12/7/1031 |
| work_keys_str_mv | AT denisivanovichborisov geometricapproximationofpointinteractionsinthreedimensionaldomains |