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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.13977 |
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| _version_ | 1866910377164406784 |
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| author | Han, Dan Molchanov, Stanislav Vainberg, Boris |
| author_facet | Han, Dan Molchanov, Stanislav Vainberg, Boris |
| contents | The research explores a high irregularity, commonly referred to as intermittency, of the solution to the non-stationary parabolic Anderson problem: \begin{equation*}
\frac{\partial u}{\partial t} = \varkappa \mathcal{L}u(t,x) + ξ_{t}(x)u(t,x) \end{equation*} with the initial condition \(u(0,x) \equiv 1\), where \((t,x) \in [0,\infty)\times \mathbb{Z}^d\). Here, \(\varkappa \mathcal{L}\) denotes a non-local Laplacian, and \(ξ_{t}(x)\) is a correlated white noise potential. The observed irregularity is intricately linked to the upper part of the spectrum of the multiparticle Schrödinger equations for the moment functions \(m_p(t,x_1,x_2,\cdots,x_p) = \langle u(t,x_1)u(t,x_2)\cdots u(t,x_p)\rangle\).
In the first half of the paper, a weak form of intermittency is expressed through moment functions of order $p\geq 3$ and established for a wide class of operators $\varkappa \mathcal{L}$ with a positive-definite correlator $B=B(x))$ of the white noise. In the second half of the paper, the strong intermittency is studied. It relates to the existence of a positive eigenvalue for the lattice Schrödinger type operator with the potential $B$. This operator is associated with the second moment $m_2$. Now $B$ is not necessarily positive-definite, but $\sum B(x)\geq 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13977 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral Analysis of Lattice Schrödinger-Type Operators Associated with the Nonstationary Anderson Model and Intermittency Han, Dan Molchanov, Stanislav Vainberg, Boris Mathematical Physics Analysis of PDEs Probability Spectral Theory 60H25, 60H15, 81Q10, 37H15, 35B40 The research explores a high irregularity, commonly referred to as intermittency, of the solution to the non-stationary parabolic Anderson problem: \begin{equation*} \frac{\partial u}{\partial t} = \varkappa \mathcal{L}u(t,x) + ξ_{t}(x)u(t,x) \end{equation*} with the initial condition \(u(0,x) \equiv 1\), where \((t,x) \in [0,\infty)\times \mathbb{Z}^d\). Here, \(\varkappa \mathcal{L}\) denotes a non-local Laplacian, and \(ξ_{t}(x)\) is a correlated white noise potential. The observed irregularity is intricately linked to the upper part of the spectrum of the multiparticle Schrödinger equations for the moment functions \(m_p(t,x_1,x_2,\cdots,x_p) = \langle u(t,x_1)u(t,x_2)\cdots u(t,x_p)\rangle\). In the first half of the paper, a weak form of intermittency is expressed through moment functions of order $p\geq 3$ and established for a wide class of operators $\varkappa \mathcal{L}$ with a positive-definite correlator $B=B(x))$ of the white noise. In the second half of the paper, the strong intermittency is studied. It relates to the existence of a positive eigenvalue for the lattice Schrödinger type operator with the potential $B$. This operator is associated with the second moment $m_2$. Now $B$ is not necessarily positive-definite, but $\sum B(x)\geq 0$. |
| title | Spectral Analysis of Lattice Schrödinger-Type Operators Associated with the Nonstationary Anderson Model and Intermittency |
| topic | Mathematical Physics Analysis of PDEs Probability Spectral Theory 60H25, 60H15, 81Q10, 37H15, 35B40 |
| url | https://arxiv.org/abs/2403.13977 |