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2026
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| Online Access: | https://doi.org/10.5281/zenodo.19259784 |
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| author | Trees, Nala |
| author_facet | Trees, Nala |
| contents | <p>We study one-dimensional radial Schrodinger-type operators on a bounded interval (0, R*) with a singular or steep boundary wall near r = R*. We separate two regimes. First, an ideal inverse-square wall V(r) >= C(R* - r)^{-2} with C > 3/4 is limit-point at the terminal radius. Second, a family of steep but finite collar barriers W_eta supported in [R* - eta, R*) with height m_eta satisfying m_eta eta -> infinity converges, in the Mosco / strong-resolvent sense, to the Dirichlet realisation on the bounded host interval. This yields convergence of each fixed low-lying eigenvalue to the Dirichlet value as the wall steepens. Finally, if the interior potential V_0 is bounded below by more than -pi^2/(R*)^2, then the Dirichlet operator has compact resolvent and strictly positive lowest eigenvalue. The ideal-wall endpoint theorem and the steep-collar reduction are logically separate results: the latter does not require the former. DOI: 10.5281/zenodo.19259784</p> <p>Keywords:<br>Schrodinger operators; singular potentials; limit-point theory; Dirichlet reduction; Mosco convergence; spectral convergence</p> <p> </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19259784 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Soft-Wall Barriers and Dirichlet Reduction Trees, Nala <p>We study one-dimensional radial Schrodinger-type operators on a bounded interval (0, R*) with a singular or steep boundary wall near r = R*. We separate two regimes. First, an ideal inverse-square wall V(r) >= C(R* - r)^{-2} with C > 3/4 is limit-point at the terminal radius. Second, a family of steep but finite collar barriers W_eta supported in [R* - eta, R*) with height m_eta satisfying m_eta eta -> infinity converges, in the Mosco / strong-resolvent sense, to the Dirichlet realisation on the bounded host interval. This yields convergence of each fixed low-lying eigenvalue to the Dirichlet value as the wall steepens. Finally, if the interior potential V_0 is bounded below by more than -pi^2/(R*)^2, then the Dirichlet operator has compact resolvent and strictly positive lowest eigenvalue. The ideal-wall endpoint theorem and the steep-collar reduction are logically separate results: the latter does not require the former. DOI: 10.5281/zenodo.19259784</p> <p>Keywords:<br>Schrodinger operators; singular potentials; limit-point theory; Dirichlet reduction; Mosco convergence; spectral convergence</p> <p> </p> |
| title | Soft-Wall Barriers and Dirichlet Reduction |
| url | https://doi.org/10.5281/zenodo.19259784 |