Is there an explicit formula for the Fourier transform of the generalized function of 2 variables $$\frac{1}{x+y^2+i0}?$$

Remark. **Equivalent question:** consider the Schroedinger equation one the line
$$i\frac{\partial}{\partial t}\Psi(x,t)=\frac{\partial^2}{\partial x^2}\Psi(x,t).$$

Find the kernel $K(x,t)$ such that for $t>0$ $$\Psi(x,t)=\int_{-\infty}^\infty K(x-y,t)\Psi(x,0)dy.$$