Lazarev’s Results on the Algebraic Structure of the Set of Surface Potentials of a Linear Operator

Abstract

In Chap. 1 we proposed a construction of the potential for a given differential operator with density from the space of clear traces and we constructed the boundary equations for the density, which isolate the traces of solutions of the homogeneous differential equation. We saw that the choice of the Green operator used in the definition of the potential

$$ P_{\overline D \Gamma } \xi _\Gamma = \upsilon _{\overline D } - G_{\overline D D} L_{\overline D D} \upsilon _{\overline D } $$

with density ${{\xi }_{\Gamma }} = T{{r}_{\Gamma }}_{{\bar{D}}}{{\upsilon }_{{\bar{D}}}}$ as well as the choice of the space ΞГ of clear traces and that of the clear trace operator

$${\text{T}}{{r}_{{\Gamma \bar{D}}}}:V_{{\bar{D}}}^{ + } \to {{\Xi }_{\Gamma }}$$

is significantly nonunique. The structure of the set of all potentials was not studied in Chap. 1.