You're almost remembering correctly. Usually you have a differential operator L (often the Laplacian) and a potential term V(x) (where x may be a vector) giving

You're almost remembering correctly. Usually you have a differential operator L (often the Laplacian) and a potential term V(x) (where x may be a vector) giving the equation L{y} = V(x) or in quantum mechanics L{y} = V(x)y (now L is i times nabla) Then you expand the potential V(x) as a power series to the quadratic term, which gives you a harmonic oscillator.