  The aim of this paper is to show some examples of matrix-valued orthogonal
functions on the real line which are simultaneously eigenfunctions of a
second-order differential operator of Schr\"{o}dinger type and an integral
operator of Fourier type. As a consequence we derive integral representations
of these functions as well as other useful structural formulas. Some of these
functions are plotted to show the relationship with the Hermite or wave
functions.
