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

TL;DR

This work advances the spectral analysis of the fractional Schrödinger operator with Neumann boundary conditions by adapting Veliev's non-resonance/resonance framework to the fractional context ($\frac12<\ell<1$). It establishes that non-resonance eigenvalues of $(-\Delta_{\mathsf{NEU}})^{\ell}+q(x)$ asymptotically track those of $(-\Delta_{\mathsf{NEU}})^{\ell}$, and derives an explicit perturbative expansion in terms of the unperturbed eigenvalues $|\beta|^{2\ell}$ and recursively defined Fourier-data corrections $F_k$, valid within a non-resonant spectral region. The main result is an iterative, $F_k$-driven formula $\xi_N=|\beta|^{2\ell}+F_{k-1}+O(r^{-k\alpha_1(\ell)})$, clarifying how the potential $q$ perturbs high-frequency eigenvalues and highlighting connections to the fractional Laplacian's spectrum. This provides a rigorous framework for understanding spectral stability in nonlocal quantum models and informs potential applications where asymptotic spectral behavior governs dynamics.

Abstract

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.

Figures (5)

• Figure 1: Resonance domains for increasing values of $\ell$.
• Figure 2: The resonance domain for $\ell = 1$.
• Figure 3: Resonance domains for decreasing values of $\ell$.
• Figure 4: Intersection of resonance domains corresponding to different values of $\beta$.
• Figure 5: Resonance domains for increasing values of $r$.

Theorems & Definitions (18)

• Definition 3.1
• Lemma 3.2
• proof
• Remark 3.3
• Lemma 3.4
• proof
• Lemma 3.5
• proof
• Lemma 3.6
• proof