A generalized time fractional Schrödinger equation with signed potential

In this work, by stochastic analyses, we study stochastic representation, well-posedness, and regularity of generalized time fractional Schrödinger equation

\begin{document}$ \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*} $\end{document}

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.