In this paper, we consider the Fourier-type functionals on Wiener space. We then establish the analytic Feynman integrals involving the ${diamond}$-convolutions. Further, we give an approach to solution of the Schr$ddot{o}$dinger equation via Fourier-type functionals. Finally, we use this approach to obtain solutions of the Schr$ddot{o}$dinger equations for harmonic oscillator and double-well potential. The Schr$ddot{o}$dinger equations for harmonic oscillator and double-well potential are meaningful subjects in quantum mechanics.