The automorphism groups of the vertex operator algebras $V_L^{+}$: general case

In this article, we give a method of calculating the automorphism groups of the vertex operator algebras $V_L^{+}$ associated with even lattices $L$. For example, by using this method we determine the automorphism groups of $V_L^{+}$ for even lattices of rank one, two and three, and even unimodular lattices.