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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2604.01368 |
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| _version_ | 1866914439184252928 |
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| author | Betancor, Jorge J. Dalmasso, Estefanía Fariña, Juan C. Quijano, Pablo |
| author_facet | Betancor, Jorge J. Dalmasso, Estefanía Fariña, Juan C. Quijano, Pablo |
| contents | In this paper we consider the Schrödinger operator $\mathcal L_V= -Δ+ V$ in $\mathbb R^d$ with a non negative potential $V$, and $V\not\equiv 0$. We define the logarithmic Schrödinger operator $\log \mathcal L_V$ proving its main properties. We obtain a pointwise representation of $\log \mathcal L_V$ when $V$ satisfies a reverse Hölder inequality of exponent $q> \frac{d}{2}$ by using the semigroup of operators $\{T_t^V\}_{t>0}$ generated by $\mathcal L_V$. We consider the Lipschitz function space adapted to the Schrödinger setting to solve the initial value problem
\[ \begin{cases} \frac{\partial u}{\partial t}=-(\log \mathcal{L}_V)u, & \text{in } \mathbb{R}^n \times (0,\infty), \\ u(x,0)=f(x), & x \in \mathbb{R}^d \end{cases} \]
in terms of the fractional integral associated with $\mathcal L_V$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01368 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Logarithmic Schrödinger operators Betancor, Jorge J. Dalmasso, Estefanía Fariña, Juan C. Quijano, Pablo Analysis of PDEs In this paper we consider the Schrödinger operator $\mathcal L_V= -Δ+ V$ in $\mathbb R^d$ with a non negative potential $V$, and $V\not\equiv 0$. We define the logarithmic Schrödinger operator $\log \mathcal L_V$ proving its main properties. We obtain a pointwise representation of $\log \mathcal L_V$ when $V$ satisfies a reverse Hölder inequality of exponent $q> \frac{d}{2}$ by using the semigroup of operators $\{T_t^V\}_{t>0}$ generated by $\mathcal L_V$. We consider the Lipschitz function space adapted to the Schrödinger setting to solve the initial value problem \[ \begin{cases} \frac{\partial u}{\partial t}=-(\log \mathcal{L}_V)u, & \text{in } \mathbb{R}^n \times (0,\infty), \\ u(x,0)=f(x), & x \in \mathbb{R}^d \end{cases} \] in terms of the fractional integral associated with $\mathcal L_V$. |
| title | Logarithmic Schrödinger operators |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.01368 |