Exactly solvable Schrödinger operators related to the confluent equation

TL;DR

This work classifies and analyzes exactly solvable one-dimensional Schrödinger operators with complex potentials that admit resolvent expressions in terms of confluent and Bessel functions. It develops holomorphic basic families of closed operators on $L^2$ spaces, derives explicit resolvent kernels via Wronskian methods, and establishes transmutation identities that exchange spectral parameters with coupling constants across operator families, including Bessel, Whittaker, Morse, isotonic, and harmonic cases along with negative exponential potentials. The authors provide detailed boundary-condition analysis for singular endpoints and a concise review of the relevant special functions, along with limiting and mixed-boundary realizations. These solvable models serve as precise reference systems for spectral theory, perturbation, and scattering in quantum mechanics with complex potentials, and reveal deep interconnections among operator families through resolvent transmutations.

Abstract

Our paper investigates one-dimensional Schrödinger operators defined as closed operators on $L^2(\mathbb{R})$ or $L^2(\mathbb{R}_+)$ that are exactly solvable in terms of confluent functions (or, equivalently, Whittaker functions). We allow the potentials to be complex. They fall into three families: Whittaker operators (or radial Coulomb Hamiltonians), Schrödinger operators with Morse potentials and isotonic oscillators. For each of them, we discuss the corresponding basic holomorphic family of closed operators and the integral kernel of their resolvents. We also describe transmutation identities that relate these resolvents. These identities interchange spectral parameters with coupling constants across different operator families. A similar analysis is performed for one-dimensional Schrödinger operators solvable in terms of Bessel functions (which are reducible to special cases of Whittaker functions). They fall into two families: Bessel operators and Schrödinger operators with exponential potentials. To make our presentation self-contained, we include a short summary of the theory of closed one-dimensional Schrödinger operators with singular boundary conditions. We also provide a concise review of special functions that we use.

Figures (1)

• Figure 1: Domains of the parameters $k$ and $c=k^2$

Theorems & Definitions (40)

• Remark 1.1
• Remark 2.1
• Theorem 2.2
• Lemma 2.3
• Lemma 2.4: Special case of Schur's Criterion
• Lemma 2.5
• Lemma 2.6
• Theorem 3.1
• proof
• Theorem 3.2