On the eigenvector algebra of the product of elements with commutator one in the first Weyl algebra

Abstract: Let $A_1=K < X, Y | [Y,X]=1>$ be the (first) Weyl algebra over a field $K$ of characteristic zero. It is known that the set of eigenvalues of the inner derivation $\ad (YX)$ of $A_1$ is $\Z$. Let $ A_1\ra A_1$, $X\mapsto x$, $Y\mapsto y$, be a $K$-algebra homomorphism, i.e. $[y,x]=1$. It is proved that the set of eigenvalues of the inner derivation $\ad (yx)$ of the Weyl algebra $A_1$ is $\Z$ and the eigenvector algebra of $\ad (yx)$ is $K< x,y> $ (this would be an easy corollary of the Problem/Conjecture of Dixmier of 1968 [still open]: {\em is an algebra endomorphism of $A_1$ an automorphism?}).