## WKB analysis to global solvability and hypoellipticity.(English)Zbl 0842.35021

In the paper under consideration global regularity and solvability of differential and pseudodifferential operators on the two-dimensional torus $$T^2$$ are investigated. Their properties are studied and explained by using the methods of the WKB analysis. The WKB formal solutions, i.e. the formal power series with respect to a large parameter whose coefficients may have poles of arbitrary large orders, enable the authors to give sharp results on global hypoellipticity and solvability.
More precisely, a necessary and sufficient condition is given for the global hypoellipticity on $$T^2$$ of an $$m \times m$$ system. Siegel conditions play an important role here, and it is shown that they can be invariantly defined under realizations and changes of variables for the corresponding WKB solutions. A single equation with characteristics of variable multiplicity is studied, and under the assumption that a smooth formal solution of Riccati equation exists, a sharp result on hypoellipticity is proved.

### MSC:

 35H10 Hypoelliptic equations 34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations

### Keywords:

WKB analysis; global hypoellipticity
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