Transmutation operators for Schrödinger equations with distributional potentials and the associated impedance equation

TL;DR

The paper develops a Polya-factorization–based framework for transmutation operators connecting the free operator $-\frac{d^2}{dx^2}$ to the distributional Schrödinger operator $-y''+q(x)y$ with $q\in W^{-1,2}(J)$. It constructs the integral transmutation operator $\mathbf{T}_f$ with kernel $K_f$ and derives SPPS and NSBF representations for the distributional solutions, while establishing the Darboux transform and extending the theory to the Sturm–Liouville equation in impedance form via the Liouville map. The work proves existence and construction results for nonvanishing regularizing functions $f$, demonstrates approximation and stability of the transmutation operators, and provides convergent NSBF series enabling efficient numeric treatment of direct and inverse spectral problems with distributional potentials. Together, these results offer a rigorous, unified approach for spectral analysis in settings involving measures and other distributional coefficients, with practical implications for computation and inverse problems in quantum and wave equations.

Abstract

We present the construction of an integral transmutation operator for the Schrödinger equation \[ -y'' + q(x)y = λy, \quad x \in J, \ λ\in \mathbb{C}, \] 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'' + λv = 0 $ into solutions of the Schrödinger equation. The construction of the integral transmutation operator relies on a new regularization of the distributional Schrödinger equation based on the Polya factorization in terms of a solution $f$ that does not vanish on the closure of $J$. Sufficient conditions for the existence of such a function $f$ are established, together with a method for its construction. 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 Schrödinger 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.

Figures (2)

• Figure 1: The domains $\mathcal{T}_{\ell}$ and $\mathcal{Q}_{\ell}$ and the transformation between them.
• Figure 2: The rectangle $C_{(\xi,\zeta)}$ contained within $\mathcal{Q}_{\ell}$.

Theorems & Definitions (52)

• Definition 1
• Proposition 2
• Definition 3
• Remark 4
• Theorem 5: Polya factorization
• Theorem 6
• Remark 7
• Remark 8
• Remark 9
• Proposition 10