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In this paper, by using Picard-Fuchs equations and Chebyshev criterion, westudy the bifurcate of limit cycles for quadratic Hamilton system $S^{(2)}$ and$S^{(3)}$: $\dot{x}= y+2axy+by^2$, $\dot{y}=-x+x^2-ay^2$ with$a\in(-\frac{1}{2},1)$, $b=(1-a)(1+2a)^{1/2}$ and $a=1$, $b=0$ respectively,under perturbations of piecewise smooth polynomials with degree $n$. Thediscontinuity is the line $y=0$. We bound the number of zeros of first orderMelnikov function which controls the number of limit cycles bifurcating fromthe center. It is proved that the upper bounds of the number of limit cyclesfor $S^{(2)}$ and $S^{(3)}$ are respectively $25n+161$ $(n\geq3)$ and $24n+126$$(n\geq3)$ (taking into account the multiplicity).

On the number of limit cycles for polycycles $S^{(2)}$ and $S^{(3)}$ in  quadratic Hamilton systems under perturbations of piecewise smooth  polynomials