Transmutation operators and complete systems of solutions for the radial Schrodinger equation

A transmutation operator T transmuting harmonic functions into solutions of the radial Schrodinger equation Su := (Delta(d) - q(r)) u = 0 is studied. The potential q is assumed to be continuously differentiable, and the Schrodinger equation is considered in a star-shaped domain Omega subset of R-d (with d >= 2). Several new properties of the transmutation operator are established including the operator relation r(2) (Delta(d) - q(r)) T = Tr-2 Delta(d), valid on C-2 functions and its boundedness on the Bergman space. A Fourier-Jacobi series expansion of the integral transmutation kernel is derived, and with its aid, an infinite system of solutions of the radial Schrodinger equation is obtained, which is shown to be complete with respect to the uniform norm. Explicit construction of the system is derived. In the case of Omega being an open ball centered in the origin, the system of solutions represents an orthogonal basis of the corresponding Bergman space.