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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2509.01225 |
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| _version_ | 1866918388774731776 |
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| author | Kaminaga, Masahiro |
| author_facet | Kaminaga, Masahiro |
| contents | We study Stark Hamiltonians with a $δ$-interaction supported on a compact hypersurface in $\mathbb R^d$. Let $Σ$ be a compact Lipschitz hypersurface and let $α\in L^\infty(Σ;\mathbb R)$. We define the operator $H_{F,α}$ as a self--adjoint realization of the formal Hamiltonian $H_{F,0}+αδ_Σ$ by imposing transmission conditions across $Σ$. We then derive a boundary resolvent formula which expresses the resolvent of $H_{F,α}$ in terms of the free Stark resolvent and a boundary operator on $Σ$. This reduces the spectral problem to the boundary and shows that the interaction can be treated as a boundary perturbation at the resolvent level.
As an application, we prove that for every nonzero electric field the resolvent difference between $H_{F,α}$ and $H_{F,0}$ is compact on $L^2(\mathbb R^d)$. It follows that the essential spectrum of $H_{F,α}$ coincides with $\mathbb R$. The argument is based on trace mapping properties for compact Lipschitz hypersurfaces and does not rely on translation invariance of the background operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01225 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stark Hamiltonians with Hypersurface-Supported $δ$-Interactions: Self-Adjoint Realization and Boundary Resolvent Formula Kaminaga, Masahiro Mathematical Physics We study Stark Hamiltonians with a $δ$-interaction supported on a compact hypersurface in $\mathbb R^d$. Let $Σ$ be a compact Lipschitz hypersurface and let $α\in L^\infty(Σ;\mathbb R)$. We define the operator $H_{F,α}$ as a self--adjoint realization of the formal Hamiltonian $H_{F,0}+αδ_Σ$ by imposing transmission conditions across $Σ$. We then derive a boundary resolvent formula which expresses the resolvent of $H_{F,α}$ in terms of the free Stark resolvent and a boundary operator on $Σ$. This reduces the spectral problem to the boundary and shows that the interaction can be treated as a boundary perturbation at the resolvent level. As an application, we prove that for every nonzero electric field the resolvent difference between $H_{F,α}$ and $H_{F,0}$ is compact on $L^2(\mathbb R^d)$. It follows that the essential spectrum of $H_{F,α}$ coincides with $\mathbb R$. The argument is based on trace mapping properties for compact Lipschitz hypersurfaces and does not rely on translation invariance of the background operator. |
| title | Stark Hamiltonians with Hypersurface-Supported $δ$-Interactions: Self-Adjoint Realization and Boundary Resolvent Formula |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2509.01225 |