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

Abstract

A transmutation operator transmuting harmonic functions into solutions of the radial Schrödinger equation is studied. The potential is assumed to be continuously differentiable, and the Schrödinger equation is considered in a star‐shaped domain (with ). Several new properties of the transmutation operator are established including the operator relation valid on 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 Schrödinger 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 being an open ball centered in the origin, the system of solutions represents an orthogonal basis of the corresponding Bergman space.