Exactly solvable Schr\"odinger operators related to the confluent equation

TL;DR

This paper classifies and analyzes exactly solvable one-dimensional Schr"odinger operators using confluent and Bessel functions, exploring their resolvents, spectral properties, and interrelations through transmutation identities.

Contribution

It introduces a comprehensive framework for exactly solvable Schr"odinger operators with complex potentials, detailing their resolvent kernels and interrelations via transmutation identities.

Findings

Explicit resolvent kernels for Whittaker, Morse, and isotonic oscillator operators.

Transmutation identities linking spectral parameters and coupling constants.

Extension to operators solvable via Bessel functions.

Abstract

Our paper investigates one-dimensional Schr\"odinger 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), Schr\"odinger 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 Schr\"odinger operators solvable in terms of Bessel functions (which are reducible to special cases of Whittaker functions). They fall into two families: Bessel operators and Schr\"odinger operators with exponential potentials. To make our presentation self-contained, we include a short summary of the theory of closed one-dimensional Schr\"odinger operators with singular boundary conditions. We also provide a concise review of special functions that we use.