Analyticity and hp discontinuous Galerkin approximation of nonlinear Schr\"odinger eigenproblems

TL;DR

This paper investigates nonlinear Schrödinger eigenproblems with singular potentials, demonstrating that eigenfunctions are analytic in weighted spaces and that an hp discontinuous Galerkin method achieves exponential convergence, validated by numerical tests.

Contribution

It establishes the analyticity of eigenfunctions for singular potentials and nonlinearities, and proves exponential convergence of an hp dG method for these problems.

Findings

Eigenfunctions are analytic in weighted Sobolev spaces.

The hp dG method converges exponentially with mesh refinement.

Numerical tests confirm theoretical error estimates.

Abstract

We study a class of nonlinear eigenvalue problems of Schr\"{o}dinger type, where the potential is singular on a set of points. Such problems are widely present in physics and chemistry, and their analysis is of both theoretical and practical interest. In particular, we study the regularity of the eigenfunctions of the operators considered, and we propose and analyze the approximation of the solution via an isotropically refined $hp$ discontinuous Galerkin (dG) method. We show that, for weighted analytic potentials and for up-to-quartic polynomial nonlinearities, the eigenfunctions belong to analytic-type non homogeneous weighted Sobolev spaces. We also prove quasi optimal a priori estimates on the error of the dG finite element method; when using an isotropically refined $hp$ space the numerical solution is shown to converge with exponential rate towards the exact eigenfunction. We conclude with a series of numerical tests to validate the theoretical results.