Quantum ring of singularity $X^{p}+XY^{q}$

In this paper, we will prove that the quantum ring of the quasi-homogeneous polynomial $X^{p}+XY^{q}(p\ge 2,q>1)$ with some admissible symmetry group $G$ defined by Fan-Jarvis-Ruan-Witten theory is isomorphic to the Milnor ring of its mirror dual polynomial $X^{p}Y+Y^{q}$. We will construct an concrete isomorphism between them. The construction is a little bit different in case $(p-1,q)=1$ and case $(p-1,q)=d>1$. Some other problems including the correspondence between the pairings of both Frobenius algebras has also been discussed.