Special potentials for relativistic Laplacians I: Fractional Rollnik-class
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911305504391168 |
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| author | Ascione, Giacomo Ishida, Atsuhide Lőrinczi, József |
| author_facet | Ascione, Giacomo Ishida, Atsuhide Lőrinczi, József |
| contents | We propose a counterpart of the classical Rollnik-class of potentials for fractional and massive relativistic Laplacians, and describe this space in terms of appropriate Riesz potentials. These definitions rely on precise resolvent estimates. We show that Coulomb-type potentials are elements of fractional Rollnik-class up to but not including the critical singularity of the Hardy potential. For the operators with fractional exponent $α= 1$ there exists no fractional Rollnik potential, however, in low dimensions we make sense of these classes as limiting cases by using $Γ$-convergence. In a second part of the paper we derive detailed results on the self-adjointness and spectral properties of relativistic Schrödinger operators obtained under perturbations by fractional Rollnik potentials. We also define an extended fractional Rollnik-class which is the maximal space for the Hilbert-Schmidt property of the related Birman-Schwinger operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08805 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Special potentials for relativistic Laplacians I: Fractional Rollnik-class Ascione, Giacomo Ishida, Atsuhide Lőrinczi, József Functional Analysis Mathematical Physics We propose a counterpart of the classical Rollnik-class of potentials for fractional and massive relativistic Laplacians, and describe this space in terms of appropriate Riesz potentials. These definitions rely on precise resolvent estimates. We show that Coulomb-type potentials are elements of fractional Rollnik-class up to but not including the critical singularity of the Hardy potential. For the operators with fractional exponent $α= 1$ there exists no fractional Rollnik potential, however, in low dimensions we make sense of these classes as limiting cases by using $Γ$-convergence. In a second part of the paper we derive detailed results on the self-adjointness and spectral properties of relativistic Schrödinger operators obtained under perturbations by fractional Rollnik potentials. We also define an extended fractional Rollnik-class which is the maximal space for the Hilbert-Schmidt property of the related Birman-Schwinger operators. |
| title | Special potentials for relativistic Laplacians I: Fractional Rollnik-class |
| topic | Functional Analysis Mathematical Physics |
| url | https://arxiv.org/abs/2405.08805 |