[Paper Review] Exceptional points of radial potentials at positive energies

Samuli Siltanen, Janne P. Tamminen|arXiv (Cornell University)|Jul 8, 2013

Quantum Mechanics and Non-Hermitian Physics3 citations

TL;DR

This paper investigates exceptional points in radial potentials for positive energy Schrödinger equations using a novel numerical method for the Faddeev Green's function. It confirms the theoretical absence of exceptional points for small potentials and large spectral parameters, while numerically identifying such points for certain compactly supported radial potentials.

ABSTRACT

A numerical investigation is conducted for the exceptional points of the positive energy problem $(-\Delta+q_0)u = Eu$, where $E>0$ and $q_0$ is a real-valued, radially symmetric and compactly supported potential. In this case the scattering transform for positive energy is shown to be real-valued and radially symmetric. A new computational method for the Faddeev Green's function for positive energy is introduced. The numerical results show exceptional points for some potentials and supports the only known theoretical result in our case: the absence of exceptional points for small enough potentials and for large enough spectral parameters.

Motivation & Objective

• To investigate the existence of exceptional points in the positive energy scattering problem for radially symmetric, compactly supported potentials.
• To develop a new computational method for the Faddeev Green's function tailored to positive energy scattering.
• To numerically verify the theoretical prediction that exceptional points do not exist for small potentials or large spectral parameters.
• To explore whether exceptional points emerge for specific radial potentials under positive energy conditions.

Proposed method

• A numerical scheme is developed to compute the Faddeev Green's function for positive energy scattering in radial potentials.
• The scattering transform is computed under the assumption of radial symmetry and real-valuedness, simplifying the problem structure.
• The method leverages the radial symmetry of the potential $ q_0 $ and the resulting real-valuedness of the scattering transform.
• The algorithm solves the integral equation for the Green's function using quadrature and iterative techniques suitable for compactly supported potentials.
• The approach is validated by reproducing known theoretical results on the absence of exceptional points for small potentials and large $ E $.
• Numerical continuation techniques are used to detect exceptional points by monitoring spectral parameter bifurcations.

Experimental results

Research questions

• RQ1 Do exceptional points exist for positive energy scattering in radial, compactly supported potentials?
• RQ2 Can a new numerical method accurately compute the Faddeev Green's function for positive energy problems?
• RQ3 Is the theoretical absence of exceptional points for small potentials and large spectral parameters confirmed numerically in this setting?
• RQ4 What types of radial potentials give rise to exceptional points at positive energies?

Key findings

• The scattering transform for positive energy is real-valued and radially symmetric when the potential $ q_0 $ is real, radial, and compactly supported.
• A new numerical method for the Faddeev Green's function is successfully implemented and validated for positive energy problems.
• Numerical results confirm the absence of exceptional points for small potentials and large spectral parameters, consistent with theoretical predictions.
• Exceptional points are numerically identified for certain radial potentials, indicating their existence under specific conditions.