Construction and application of Bergman-type reproducing kernels for boundary and eigenvalue problems in the plane

摘要

We show how the Bergman-type reproducing kernels for the elliptic operator D = div p grad + q with variable coefficients defined in a bounded domain in the plane can be constructed using pseudoanalytic function theory and in particular pseudoanalytic formal powers. Under certain conditions on the coefficients p and q and with the aid of pseudoanalytic function theory a complete system of null solutions of the operator can be obtained following a simple algorithm consisting in recursive integration. Then the complete system of solutions is used for constructing the corresponding reproducing kernel. We study theoretical and numerical aspects of the method and apply it to solve boundary value and eigenvalue problems for the stationary Schr?dinger operator in bounded domains.