Quantum Harmonic Oscillator Complete System

[YAHA]

Over which interval do the wave functions of a harmonic oscillator form a complete and orthogonal system? Is it (-inf,+inf)? The case with particle in a box is rather clear(system is complete and orthogonal only for the interval of the well), however the harmonic oscillator is a bit less intuitive.

 

[Matterwave]

http://en.wikipedia.org/wiki/Hermite_polynomials

The Hermite polynomials form an orthogonal, complete, basis for the Hilbert space.