Take the FT of your first eq, and get K(E,p) = 1/(E-p*p) -- I'm dropping constants, except those that are really important.
Now get K(x,t) by inverting. So, K(x,t) == Integral dE dp{exp(-iEt +ipx)/(E-p*p)}. Do the E integration first, to handle the pole, push it below the E axis -- add...