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We de...ne and study some properties of operators formally given by e A = a 1 2 x + i@y; 1 2 y i@x : In particular we show that when e A is essentially self-adjoint and the symbol a belongs to a certain Shubin class then the spectrum of e A is discrete and coincides with that of the usual Weyl operator A = a w (x; i@x). In addition, the eigenvalues of e A are in this case in...nitely degenerate and the corresponding eigenfunctions can be obtained from those of A using a family of isometries L 2doi:10.1080/03605300802501434 fatcat:ybz2rqgwvbcpvbha2oapwazxhe