In this paper, our aim is to prove the existence of normalized ground state for the following Schrödinger systems with potentials \(\begin{aligned}{\left\{ \begin{array}{ll} -\Delta u_1+V_1(x)u_1+\lambda _1 u_1=\partial _1 G(u_1,u_2)\;\quad & \hbox {in}\;{\mathbb {R}}^N,\\ -\Delta u_2+V_2(x)u_2+\lambda _2 u_2=\partial _2G(u_1,u_2)\;\quad & \hbox {in}\;{\mathbb {R}}^N,\\ 0<u_1,u_2\in H^1({\mathbb {R}}^N), N\ge 1,\\ \int _{{\mathbb {R}}^N}u_1^2 \textrm{d} x=a_1, \int _{{\mathbb {R}}^N}u_2^2 \textrm{d} x=a_2. \end{array}\right. }\end{aligned}\) The potentials \(V_1(x),V_2(x)\) are general such that \(\inf \text {ess}~\sigma (-\Delta +V_\iota )>-\infty \) , which are allowed to be singular at some points. And the nonlinearities \(G(u_1,u_2)\) are considered of the form \(\begin{aligned} {\left\{ \begin{array}{ll} G(u_1, u_2):=\sum _{i=1}^{\ell }\frac{\mu _i}{p_i}|u_1|^{p_i}+\sum _{j=1}^{m}\frac{\nu _j}{q_j}|u_2|^{q_j}+\sum _{k=1}^{n}\beta _k |u_1|^{r_{1,k}}|u_2|^{r_{2,k}},~~\ell ,m,n\in {\mathbb {N}}^+_0,\\ \mu _i, \nu _j,\beta _k>0, ~2<r_{1,k}+r_{2,k}, p_i, q_j<2+\frac{4}{N}, ~r_{1,k}, r_{2,k}>1, \\ i=1,2,\cdots , \ell ; j=1,2,\cdots , m; k=1,2,\cdots , n. \end{array}\right. } \end{aligned}\) Under the mass sub-critical assumption, the normalized ground states are obtained as the minimum of the functional J on the manifold \(S_{a_1,a_2}\) . Since the functional is not weak lower semi-continuous, to prove the minimizing problem is achievable, the key step is establishing the strict sub-additive inequality. Among its main ingredients is the study of the sharp decay of the positive solutions and the interaction estimates.
