2.4. Explicit representation of the solution operator🔗

Every Fourier multiplier M_f can be represented as a convolution operator with kernel K = 𝓕⁻ f. Therefore, the explicit representation of the solution operator follows from the calculation of the Fourier transform of the complex Gaussian as outlined in the previous chapter.

Once the Fourier transform of the Gaussian has been established, it only remains to show that the Gaussian indeed is a fundamental solution to the Schrödinger equation. The fact that it solves the equation follows from the calculation of the Fourier transform and the fact that for t = 0 it is the identity follows from calculating the limit t → 0 for the Fourier transformed Gaussian and the fact that the Fourier transform of the delta distribution is the constant 1 function.