On the local well-posedness for the nonlinear Schrödinger equation with spatial variable coefficient

We study a new type of nonlinear Schrödinger equation where the coefficient of Laplacian depends on spatial variable. Based on a modified Hankel transform and delicate frequency estimates, we establish the local well-posedness of the NLS with spatial variable coefficient in the weighted Lebesgue space for n≥2n≥2, and extend the Strichartz estimates to the non-radially Schrödinger equation with spatial variable coefficient in 2D Euclidian space.