Asymptotic Behavior of the Non-resonance Eigenvalues of the Fractional Schrödinger Operator with Neumann Condition

We present an analytical investigation of the asymptotic behavior of non-resonance eigenvalues for the fractional Schrödinger operator under homogeneous Neumann boundary conditions. Our findings reveal an intriguing convergence: as the system evolves, the eigenvalues of the fractional Schrödinger 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.