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We examine the conditions under which the solution of the radial stationarySchr\"odinger equation for the sextic anharmonic oscillator can be expanded interms of Hermite functions. We find that this is possible for an infinitehierarchy of potentials discriminated by the parameter setting the strength ofthe centrifugal barrier. The $N$'th member of the hierarchy involves $N$solutions for $N$ generally different values of the energy. For a particularmember of the hierarchy, there exist infinitely many bound states with squareintegrable wave functions, written in terms of the Hermite functions, whichvanish at the origin and at infinity. These bound states correspond to distinctvalues of the parameter setting the strength of the harmonic term. We alsoinvestigate connection with the polynomial solutions of the sextic oscillatorobtained from the formalism of quasi-exactly solvable potentials.

Hermite function solutions of the Schr\"odinger equation for the sextic oscillator