A Deep Learning Method for Computing Eigenvalues of the Fractional Schr\"odinger Operator

TL;DR

This paper introduces a deep learning approach that efficiently computes multiple eigenvalues of the fractional Schrödinger operator, even in high dimensions and on irregular domains, advancing numerical methods for quantum mechanics problems.

Contribution

The paper presents a novel neural network architecture with a specialized loss function tailored for fractional Schrödinger eigenvalue problems, capable of handling high-dimensional and irregular domain cases.

Findings

Successfully computed up to 30 eigenvalues for various operators

Demonstrated effectiveness on high-dimensional and irregular domains

Proposed a conjecture for fractional order isospectral problems

Abstract

We present a novel deep learning method for computing eigenvalues of the fractional Schr\"odinger operator. Our approach combines a newly developed loss function with an innovative neural network architecture that incorporates prior knowledge of the problem. These improvements enable our method to handle both high-dimensional problems and problems posed on irregular bounded domains. We successfully compute up to the first 30 eigenvalues for various fractional Schr\"odinger operators. As an application, we share a conjecture to the fractional order isospectral problem that has not yet been studied.