A remark on the Dixmier Conjecture

The Dixmier Conjecture says that every endomorphism of the (first) Weyl algebra A1 (over a field of characteristic zero) is an automorphism, i.e., if PQ−QP=1 for some P,Q∈A1 then A1=K⟨P,Q⟩. The Weyl algebra A1 is a Z-graded algebra. We prove that the Dixmier Conjecture holds if the elements P and Q are sums of no more than two homogeneous elements of A (there is no restriction on the total degrees of P and Q).