LAUSR

Let L=−Δ+V$L=-\Delta+V$ be a Schrödinger operator on Rn$\mathbb{R}^{n}$, where n≥3$n\ge3$ and the nonnegative potential V belongs to the reverse Hölder class RHq1$RH_{q_{1}}$ for some q1>n/2$q_{1} > n/2$. Let b belong to a new Campanato space Λνθ(ρ)$\Lambda_{\nu}^{\theta}(\rho)$ and IβL$\mathcal {I}_{\beta}^{L}$ be the fractional integral operator associated with L. In this paper, we study the boundedness of the commutators [b,IβL]$[b,\mathcal {I}_{\beta}^{L}]$ with b∈Λνθ(ρ)$b \in\Lambda_{\nu}^{\theta}(\rho)$ on local generalized Morrey spaces LMp,φα,V,{x0}$LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}$, generalized Morrey spaces Mp,φα,V$M_{p,\varphi}^{\alpha,V}$ and vanishing generalized Morrey spaces VMp,φα,V$VM_{p,\varphi}^{\alpha,V}$ associated with Schrödinger operator, respectively. When b belongs to Λνθ(ρ)$\Lambda_{\nu}^{\theta}(\rho)$ with θ>0$\theta >0$, 0<ν<1$0<\nu<1$ and (φ1,φ2)$(\varphi_{1},\varphi_{2})$ satisfies some conditions, we show that the commutator operator [b,IβL]$[b,\mathcal {I}_{\beta}^{L}]$ are bounded from LMp,φ1α,V,{x0}$LM_{p,\varphi_{1}}^{\alpha,V,\{ x_{0}\}}$ to LMq,φ2α,V,{x0}$LM_{q,\varphi_{2}}^{\alpha,V,\{x_{0}\}}$, from Mp,φ1α,V$M_{p,\varphi_{1}}^{\alpha,V}$ to Mq,φ2α,V$M_{q,\varphi_{2}}^{\alpha,V}$ and from VMp,φ1α,V$VM_{p,\varphi_{1}}^{\alpha,V}$ to VMq,φ2α,V$VM_{q,\varphi_{2}}^{\alpha ,V}$, 1/p−1/q=(β+ν)/n$1/p-1/q=(\beta+\nu)/n$.