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An L-basis associated to a linear second-order ordinary differential operatorL is an infinite sequence of functions {\phi_k}_{k=0}^{\infty} such thatL\phi_k=0 for k=0,1, L\phi_k=k(k-1)\phi_{k-2}, for k=2,3,... and all \phi_ksatisfy certain prescribed initial conditions. We study the transmutationoperators related to L in terms of the transformation of powers of theindependent variable {(x-x_{0})^k}_{k=0}^{\infty} to the elements of theL-basis and establish a precise form of the transmutation operator realizingthis transformation. We use this transmutation operator to establish acompleteness of an infinite system of solutions of the stationary Schr\"odingerequation from a certain class. The system of solutions is obtained as anapplication of the theory of bicomplex pseudoanalytic functions and itscompleteness was a long sought result. Its use for constructing reproducingkernels and solving boundary and eigenvalue problems has been considered evenwithout the required completeness justification. The obtained result on thecompleteness opens the way for further development and application of the toolsof pseudoanalytic function theory.

Transmutations, L-bases and complete families of solutions of the stationary Schr\"odinger equation in the plane