On a differential operator containing large complex parameter

Abstract

Consider a differential operator depending on the parameter . This operator arises in the process of constructing quasi-exponential solutions for the Schrödinger and conductivity equations. Since thesesolutions constitute a widely used and powerful tool for proving uniqueness in inverse problems, properties of the operator P are of interest. We obtain an explicit integral representation for a tempered fundamental solution E of P known in the literature as Faddeev Green function. Given a compactly supported distribution ƒ we investigate asymptotic behavior of the convolution E * ƒ for large |x||ζ|. An explicit asymptotic expansion obtained leads to construction of aparametrix for P which decays rapidly with respect to both |x|and|ζ|. We also consider the Dirichlet problem for the Schrodinger type operator P + q with a compactly supported potential. We obtain sufficient conditions for existence of asymptotic boundary layer type solutions which decay rapidly away from the boundary with respect to |ζ|.

Keywords:

• Differential operator
• Fadeev-Green function
• Schroedinger operator
• boundary layer

AMS:

• 35J25
• 35J05
• 35J10