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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2405.01903 |
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| _version_ | 1866914783106695168 |
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| author | Breteaux, Sébastien Faupin, Jérémy Grasselli, Viviana |
| author_facet | Breteaux, Sébastien Faupin, Jérémy Grasselli, Viviana |
| contents | We study the number $N_{<0}(H_s)$ of negative eigenvalues, counting multiplicities, of the fractional Schrödinger operator $H_s=(-Δ)^s-V(x)$ on $L^2(\mathbb{R}^d)$, for any $d\ge1$ and $s\ge d/2$. We prove a bound on $N_{<0}(H_s)$ which depends on $s-d/2$ being either an integer or not, the critical case $s=d/2$ requiring a further analysis. Our proof relies on a splitting of the Birman-Schwinger operator associated to this spectral problem into low- and high-energies parts, a projection of the low-energies part onto a suitable subspace, and, in the critical case $s=d/2$, a Cwikel-type estimate in the weak trace ideal $\mathcal{L}^{2,\infty}$ to handle the high-energies part. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01903 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the number of bound states for fractional Schr{ö}dinger operators with critical and super-critical exponent Breteaux, Sébastien Faupin, Jérémy Grasselli, Viviana Analysis of PDEs We study the number $N_{<0}(H_s)$ of negative eigenvalues, counting multiplicities, of the fractional Schrödinger operator $H_s=(-Δ)^s-V(x)$ on $L^2(\mathbb{R}^d)$, for any $d\ge1$ and $s\ge d/2$. We prove a bound on $N_{<0}(H_s)$ which depends on $s-d/2$ being either an integer or not, the critical case $s=d/2$ requiring a further analysis. Our proof relies on a splitting of the Birman-Schwinger operator associated to this spectral problem into low- and high-energies parts, a projection of the low-energies part onto a suitable subspace, and, in the critical case $s=d/2$, a Cwikel-type estimate in the weak trace ideal $\mathcal{L}^{2,\infty}$ to handle the high-energies part. |
| title | On the number of bound states for fractional Schr{ö}dinger operators with critical and super-critical exponent |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.01903 |