Higher order self-adjoint operators with polynomial coefficients
  Algebraic and analytic aspects of self-adjoint operators of order four or more with polynomial coefficients are investigated. As a consequence, a systematic way of constructing such operators is given. The procedure is applied to obtain many examples up to order 8; similar examples can be constructed for all even order operators. In particular, a complete classification of all order 4 operators is given. 
H. Azad, A. Laradji and M. T. Mustafa
math.CA