Dispersion for the Schr{ö}dinger equation on the line with short-range array of delta potentials
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| Format: | Preprint |
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2026
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| _version_ | 1866908920024399872 |
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| author | Duboscq, Romain Durand-Simonnet, Élio Coz, Stefan Le |
| author_facet | Duboscq, Romain Durand-Simonnet, Élio Coz, Stefan Le |
| contents | We study dispersive properties of the one-dimensional Schr{ö}dinger equation with a short-range array of delta interactions. More precisely, we consider the self-adjoint operator obtained by perturbing the free Laplacian on the line with a real-valued sequence of Dirac delta potentials and belonging to weighted ${\ell}$^1(Z) spaces. Under suitable decay assumptions on the coupling constants and in the absence of a zero-energy resonance, we establish the L^1 (R) $\rightarrow$ L^$\infty$ (R) dispersive estimate with decay rate |t|^{-1/2} for the associated Schr{ö}dinger group. The proof relies on a limiting absorption principle in weighted spaces, explicit representation of the resolvent kernel in terms of Jost solutions and Born series expansion of the Friedrichs extension of the perturbed operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05098 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dispersion for the Schr{ö}dinger equation on the line with short-range array of delta potentials Duboscq, Romain Durand-Simonnet, Élio Coz, Stefan Le Analysis of PDEs We study dispersive properties of the one-dimensional Schr{ö}dinger equation with a short-range array of delta interactions. More precisely, we consider the self-adjoint operator obtained by perturbing the free Laplacian on the line with a real-valued sequence of Dirac delta potentials and belonging to weighted ${\ell}$^1(Z) spaces. Under suitable decay assumptions on the coupling constants and in the absence of a zero-energy resonance, we establish the L^1 (R) $\rightarrow$ L^$\infty$ (R) dispersive estimate with decay rate |t|^{-1/2} for the associated Schr{ö}dinger group. The proof relies on a limiting absorption principle in weighted spaces, explicit representation of the resolvent kernel in terms of Jost solutions and Born series expansion of the Friedrichs extension of the perturbed operator. |
| title | Dispersion for the Schr{ö}dinger equation on the line with short-range array of delta potentials |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.05098 |