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Solutions of a particle with fractional $δ$-potential in a fractional dimensional space

A Fourier transformation in a fractional dimensional space of order $\la$ ($0<\la\leq 1$) is defined to solve the Schrödinger equation with Riesz fractional derivatives of order $\a$. This new method is applied for a particle in a fractional $δ$-potential well defined by $V(x) =- γδ^{\la}(x)$, where $γ>0$ and $δ^{\la}(x)$ is the fractional Dirac delta function. A complete solutions for the energy values and the wave functions are obtained in terms of the Fox H-functions. It is demonstrated that the eigen solutions are exist if $0< \la<\a$. The results for $\la= 1$ and $\a=2$ are in exact agreement with those presented in the standard quantum mechanics.