Failure of analytic hypoellipticity in a class of PDOs

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For the hypoelliptic differential operators P=∂² x+(xᵏ∂ y -xˡ∂ₜ)² introduced by T. Hoshiro, generalizing a class of M. Christ, in the cases of k and l left open in the analysis, the operators P also fail to be analytic hypoelliptic (except for (k,l)=(0,1)), in accordance with Treves' conjecture. The proof is constructive, suitable for generalization, and relies on evaluating a family of eigenvalues of a non-self-adjoint operator.