Theory of Probability and Mathematical Statistics
Schrodinger equation with Gaussian potential
Y. Hu
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Abstract: This paper studies the Schrodinger equation with fractional Gaussian noise potential of the form $\Delta u(x)= u(x)\diamond \dot W(x)$, $x\in \DDD$, $u(x)= \phi(x)$, $x\in \partial \DDD$, where $\Delta$ is the Laplacian on the $d$-dimensional Euclidean space $\RR^d$, $\DDD\subseteq \RR^d$ is a given domain with smooth boundary $\partial \DDD$, $\phi$ is a given nice function on the boundary $\partial \DDD$, and $\dot W$ is the fractional Gaussian noise of Hurst parameters $(H_1, \ldots, H_d)$ and $\diamond $ denotes the Wick product. We find a family of distribution spaces $(\WW_\la\,, \la>0)$, with the property $\WW_{\la}\subseteq \WW_\mu$ when $\la\le \mu$, such that under the condition $\sum_{i=1}^d H_i>d-2$, the solution exists uniquely in $\WW_{\la_0} $ when $\la_0$ is sufficiently big and the solution is not in $\WW_{\la_1}$ when $\la_1$ is sufficiently small.
Keywords: Fractional Brownian field, fractional Gaussian noise, Schrodinger equation, distribution spaces, chaos expansion, Poisson equation, multiplicative noise.
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