In this work, by stochastic analyses, we study stochastic representation, well-posedness, and regularity of generalized time fractional Schrödinger equation
$ \begin{equation*} \left\{\begin{aligned} \partial_t^wu(t,x)& = \mathcal{L} u(t,x)-\kappa(x)u(t,x),\; t\in(0,\infty),\; x\in \mathcal{X},\\ u(0,x)& = g(x),\; x\in \mathcal{X},\\ \end{aligned}\right. \end{equation*} $
where the potential $ \kappa $ is signed, $ \mathcal{X} $ is a Lusin space, $ \partial_t^w $ is a generalized time fractional derivative, and $ \mathcal{L} $ is infinitesimal generator in terms of semigroup induced by a symmetric Markov process $ X $. Our results are applicable to some typical physical models.
Citation: Rui Sun, Weihua Deng. A generalized time fractional Schrödinger equation with signed potential[J]. Communications in Analysis and Mechanics, 2024, 16(2): 262-277. doi: 10.3934/cam.2024012
In this work, by stochastic analyses, we study stochastic representation, well-posedness, and regularity of generalized time fractional Schrödinger equation
$ \begin{equation*} \left\{\begin{aligned} \partial_t^wu(t,x)& = \mathcal{L} u(t,x)-\kappa(x)u(t,x),\; t\in(0,\infty),\; x\in \mathcal{X},\\ u(0,x)& = g(x),\; x\in \mathcal{X},\\ \end{aligned}\right. \end{equation*} $
where the potential $ \kappa $ is signed, $ \mathcal{X} $ is a Lusin space, $ \partial_t^w $ is a generalized time fractional derivative, and $ \mathcal{L} $ is infinitesimal generator in terms of semigroup induced by a symmetric Markov process $ X $. Our results are applicable to some typical physical models.
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Rui Sun, Weihua Deng. A generalized time fractional Schrödinger equation with signed potential[J]. Communications in Analysis and Mechanics, 2024, 16(2): 262-277. doi: 10.3934/cam.2024012
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