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Feynman path formula for the time fractional Schrödinger equation
Discrete and Continuous Dynamical Systems-Series S ( IF 1.3 ) Pub Date : 2020-01-16 , DOI: 10.3934/dcdss.2020246
Hassan Emamirad , , Arnaud Rougirel

In this paper, we define $ E_ \alpha(t^ \alpha A) $, where $ A $ is the generator of an uniformly bounded ($ C_0 $) semigroup and $ E_ \alpha(z) $ the Mittag-Leffler function. Since the mapping $ t\mapsto E_ \alpha(t^ \alpha A) $ has not the semigroup property, we cannot use the Trotter formula for representing the Feynman operator calculus. Thus for the Hamiltonian $ H_ \alpha = -\frac{{\hbar_ \alpha2}}{{2m}}\Delta +V(x) $, we express $ E_ \alpha(t^ \alpha H_ \alpha ) $ by subordination principle of the Feynman path integral and we retrieve the corresponding Green function.

中文翻译:

时间分数薛定ding方程的费曼路径公式

在本文中,我们定义$ E_ \ alpha(t ^ \ alpha A)$,其中$ A $是一致有界($ C_0 $)半群的生成器,而$ E_ \ alpha(z)$ Mittag-Leffler函数。由于映射$ t \ mapsto E_ \ alpha(t ^ \ alpha A)$不具有semigroup属性,因此我们无法使用Trotter公式来表示Feynman运算符演算。因此,对于哈密顿量$ H_ \ alpha =-\ frac {{\\ hbar_ \ alpha2}} {{2m}} \ Delta + V(x)$,我们表示$ E_ \ alpha(t ^ \ alpha H_ \ alpha)$根据费曼路径积分的从属原理,我们检索到相应的格林函数。
更新日期:2020-01-16
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