Transmutation operators for Schr\"odinger equations with distributional potentials and the associated impedance equation

Abstract

We present the construction of an integral transmutation operator for the Schr\"odinger equation \[ -y'' + q(x)y = \lambda y, \quad x \in J, \ \lambda \in \mathbb{C}, \] in the case where $q$ is the distributional derivative of an $L^2$ function on a bounded interval $J \subset \mathbb{R}$. Such a transmutation operator transforms solutions of $ v'' + \lambda v = 0 $ into solutions of the Schr\"odinger equation. The construction of the integral transmutation operator relies on a new regularization of the distributional Schr\"odinger equation based on the Polya factorization in terms of a solution $f$ that does not vanish on the closure of $J$. The existence of such a function $f$ is established, together with a constructive method for its computation. As a consequence of the Polya factorization, we obtain an integro-differential transmutation operator for the associated Sturm--Liouville operator in impedance form related to $f$, along with smoothness conditions for the transmutation kernel. Furthermore, we introduce the Darboux transform for both the Schr\"odinger and impedance operators and describe their relationships with the corresponding transmutation operators. Finally, we develop several series representations for the solutions, including the spectral parameter power series and the Neumann series of spherical Bessel functions.