Criticality transition for positive powers of the discrete Laplacian on the half line
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910911213600768 |
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| author | Gerhat, Borbala Krejcirik, David Stampach, Frantisek |
| author_facet | Gerhat, Borbala Krejcirik, David Stampach, Frantisek |
| contents | We study the criticality and subcriticality of powers $(-Δ)^α$ with $α>0$ of the discrete Laplacian $-Δ$ acting on $\ell^2(\mathbb{N})$. We prove that these positive powers of the Laplacian are critical if and only if $α\ge 3/2$. We complement our analysis with Hardy type inequalities for $(-Δ)^α$ in the subcritical regimes $α\in (0,3/2)$. As an illustration of the critical case, we describe the negative eigenvalues emerging by coupling the discrete bilaplacian with an arbitrarily small localized potential. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_09919 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Criticality transition for positive powers of the discrete Laplacian on the half line Gerhat, Borbala Krejcirik, David Stampach, Frantisek Spectral Theory Mathematical Physics Classical Analysis and ODEs Functional Analysis Probability 26D15, 26A33 We study the criticality and subcriticality of powers $(-Δ)^α$ with $α>0$ of the discrete Laplacian $-Δ$ acting on $\ell^2(\mathbb{N})$. We prove that these positive powers of the Laplacian are critical if and only if $α\ge 3/2$. We complement our analysis with Hardy type inequalities for $(-Δ)^α$ in the subcritical regimes $α\in (0,3/2)$. As an illustration of the critical case, we describe the negative eigenvalues emerging by coupling the discrete bilaplacian with an arbitrarily small localized potential. |
| title | Criticality transition for positive powers of the discrete Laplacian on the half line |
| topic | Spectral Theory Mathematical Physics Classical Analysis and ODEs Functional Analysis Probability 26D15, 26A33 |
| url | https://arxiv.org/abs/2307.09919 |