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

Abstract

A transmutation operator boldT transmuting harmonic functions into solutions of the radial Schrödinger equation boldSu:=()△d−qfalse(rfalse)u=0 is studied. The potential q is assumed to be continuously differentiable, and the Schrödinger equation is considered in a star‐shaped domain normalΩ⊂double-struckRd (with d⩾2). Several new properties of the transmutation operator are established including the operator relation r2△d−q(r)T=Tr2△d, valid on C2 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 normalΩ being an open ball centered in the origin, the system of solutions represents an orthogonal basis of the corresponding Bergman space.

Transmutation Operator Method for Solving Heat Conduction Problem

“…Further, the fourth conjugation condition is proved similarly. The conjugation conditions for the functions 12 21 22 , u u u .…”

“…In works [3,4,5,7,14,16,18,20] the theoretical foundations of transmutation operator's method are created. Papers [8,[10][11][12][13]15,17,19] are devoted to various applications of the method, including the solving of partial differential equations. The most important of the applications of the developed method is the solution of boundary value problems in areas with axial symmetry [3,11,12].…”

“…Papers [8,[10][11][12][13]15,17,19] are devoted to various applications of the method, including the solving of partial differential equations. The most important of the applications of the developed method is the solution of boundary value problems in areas with axial symmetry [3,11,12]. The transmutation operator method allows problems with axial symmetry to be reduced to boundary value problems in domains with plane symmetry.…”