On one dimensional Quantum Zakharov system

In this paper, we discuss the properties of one dimensional quantum Zakharov system which describes the nonlinear interaction between the quantum Langmuir and quantum ion-acoustic waves. The system with initial data $(E(0),n(0),{\mathrm{\partial }}_{t}n(0))\in H^k⨁H^l⨁H^{l-2}$ is local well posedness in low regularity spaces. Especially, the low regularity result for $k$ satisfies $-3\mathrm{/}4<k\le -1\mathrm{/}4$ is obtained by using the key observation that the convoluted phase function is convex and careful bilinear analysis. The result can not be obtained by using only Strichartz inequalities for “Schr"{o}dinger” waves.