Asymptotic Behavior of the Non-resonance Eigenvalues of the Fractional Schr\"odinger Operator with Neumann Condition

TL;DR

This paper analyzes the asymptotic behavior of non-resonance eigenvalues for the fractional Schr"odinger operator with Neumann boundary conditions, revealing their convergence to the eigenvalues of the fractional Laplace operator and providing new spectral insights.

Contribution

It derives a precise asymptotic formula for eigenvalues, showing their convergence and deepening understanding of spectral properties of fractional operators.

Findings

Eigenvalues of the fractional Schr"odinger operator converge to those of the fractional Laplace operator.

A new asymptotic formula for eigenvalues is established.

Spectral properties of fractional operators are better understood with potential applications in physics.

Abstract

We present an analytical investigation of the asymptotic behavior of non-resonance eigenvalues for the fractional Schr\"odinger operator under homogeneous Neumann boundary conditions. Our findings reveal an intriguing convergence: as the system evolves, the eigenvalues of the fractional Schr\"odinger operator increasingly resemble those of the fractional Laplace operator. By deriving a precise asymptotic formula, we provide new insights into the spectral properties of these operators, highlighting their deeper connections and potential applications in mathematical physics.