Eigenvalue asymptotics and uniform eigenfunction bounds for the fractional Laplacian in the interval

Technical Summary: Eigenvalue Asymptotics and Uniform Eigenfunction Bounds for the Fractional Laplacian in the Interval

Problem Statement
This paper addresses the spectral analysis of the fractional Laplacian operator $A = (-\Delta)^{\alpha/2}$ restricted to a bounded interval $I = (-1, 1)$ with zero exterior conditions. The operator is defined via the singular integral:
$$Af(x) = c_\alpha \text{ p.v.} \int_{\mathbb{R}} \frac{f(x) - f(y)}{|x-y|^{1+\alpha}} dy, \quad c_\alpha = \frac{\Gamma(1+\alpha)\sin(\pi\alpha/2)}{\pi}.$$
The study focuses on two primary objectives:

1. Eigenvalue Asymptotics: Deriving a precise three-term asymptotic expansion for the eigenvalues $\lambda_n(\alpha)$ as $n \to \infty$, improving upon existing two-term estimates and confirming numerical conjectures regarding the order of the remainder term.
2. Uniform Eigenfunction Bounds: Establishing that the $L^\infty$ norms of the normalized eigenfunctions $\phi_n$ are uniformly bounded with respect to both the eigenvalue index $n$ and the fractional order $\alpha \in (0, 2)$. This resolves a conjecture by Kwaśnicki regarding the behavior of eigenfunctions for small $\alpha$.

Methodology
The proofs rely on constructing accurate quasimodes (approximate eigenfunctions) by gluing generalized half-line eigenfunctions and analyzing the resulting residual errors.

• Quasimode Construction: The author constructs a quasimode $\tilde{\phi}_n$ by combining the generalized half-line eigenfunction $F_\alpha$ (defined on $(0, \infty)$) on both sides of the interval, separated by smooth cutoff functions $q(x)$ and $q(-x)$.

  • For the range $1 \le \alpha < 2$, the standard quasimode $\tilde{\phi}_n$ suffices. The residual $z_n = (A_I - \mu_n^\alpha)\tilde{\phi}_n$ is expanded to higher orders using Taylor expansions of the tail terms of the half-line eigenfunctions.
  • For the range $0 < \alpha < 1$, the standard construction yields an error of the same order as the desired third term. To overcome this, the author introduces a refined quasimode $w_n$ which removes the cutoffs from the correction terms $G_\alpha$, allowing for the use of exact quadratic-form identities on the interval $(0, L)$ where $L = 2\mu_n$.

• Asymptotic Analysis:

  • The eigenvalues are approximated using the Rayleigh quotient $E_{\alpha, I}[\tilde{\phi}_n] / \|\tilde{\phi}_n\|^2_{L^2}$.
  • The remainder terms are determined by evaluating inner products $\langle z_n, \tilde{\phi}_n \rangle$. This requires detailed asymptotic expansions of the correction term $G_\alpha(t)$ as $t \to \infty$ and its moments.
  • Specific attention is paid to the distinct behaviors of $G_\alpha$ for $0 < \alpha < 1$, $\alpha = 1$, and $1 < \alpha < 2$, particularly the appearance of logarithmic terms at $\alpha = 1$ and different power-law decays elsewhere.

• Uniform Bounds:

  • For $\alpha$ bounded away from zero, the bounds follow from heat-semigroup estimates and the proximity of $\phi_n$ to the quasimode.
  • For small $\alpha$, the author proves a uniform estimate $\|z_n\|_{L^2} = o(\alpha/n)$. This allows transferring an antiderivative bound from the quasimode to the true eigenfunction.
  • A truncated quadratic-form argument is employed. By testing the eigenvalue equation against the positive and negative parts of $(\phi_n - \Lambda)$, and utilizing the specific structure of the truncated kernel $k_\delta$, the author derives a contradiction if $\|\phi_n\|_{L^\infty}$ exceeds a specific constant, thereby proving uniform boundedness.

Key Contributions and Results

1. Three-Term Eigenvalue Asymptotics (Theorem 1)
The paper establishes the following three-term asymptotic formula for $\lambda_n(\alpha)$ as $n \to \infty$, where $\mu_n = \frac{n\pi}{2} - \frac{(2-\alpha)\pi}{8}$:
$$\lambda_n(\alpha) = \mu_n^\alpha + (-1)^n \kappa_\alpha \mu_n^{-2} + R_n(\alpha).$$
The remainder $R_n(\alpha)$ exhibits three distinct regimes:

• Case $0 < \alpha < 1$:
  $$R_n(\alpha) = \left( A_\alpha + (-1)^n B_\alpha \right) \mu_n^{-2-\alpha} + o_\alpha(\mu_n^{-2-\alpha}).$$
  Here, $A_\alpha$ and $B_\alpha$ are explicit constants involving Gamma functions and trigonometric terms. This confirms the numerical prediction of an $O(n^{-2-\alpha})$ remainder.
• Case $\alpha = 1$:
  $$R_n(1) = (-1)^{n+1} \frac{1}{2\pi^2} \mu_n^{-3} \log \mu_n + O(\mu_n^{-3}).$$
  This identifies a logarithmic correction term specific to the half-Laplacian.
• Case $1 < \alpha < 2$:
  $$R_n(\alpha) = (-1)^n \eta_\alpha \mu_n^{-3} + o_\alpha(\mu_n^{-3}).$$
  This confirms the $O(n^{-3})$ remainder predicted numerically.

2. Uniform Eigenfunction Bounds (Theorem 2)
The paper proves the existence of an absolute constant $C < \infty$ such that for all $n \ge 1$ and $0 < \alpha < 2$:
$$\|\phi_n\|_{L^\infty(I)} \le C.$$
This result settles the conjecture that the uniform boundedness observed for $\alpha \ge 1/2$ extends to the entire range $0 < \alpha < 2$. The proof provides a quantitative bound, noting that one can take $C = \exp(100)$.

Significance
The paper claims to significantly advance the understanding of the spectral properties of the fractional Laplacian in bounded domains:

• It improves upon the two-term asymptotic formulas previously established by Kulczycki–Kwaśnicki–Małecki–Stós and Kwaśnicki by providing a rigorous third term.
• It validates numerical simulations regarding the precise order of the remainder terms, particularly the $O(n^{-2})$ (or $O(n^{-2-\alpha})$) behavior.
• It resolves the open problem of uniform $L^\infty$ bounds for eigenfunctions across the full range of fractional orders, a property crucial for understanding the regularity and behavior of solutions to fractional PDEs near boundaries.

The author notes that while these results are specific to the one-dimensional interval, they provide a foundation for future investigations into higher-dimensional domains, where precise asymptotics for individual eigenvalues and sharp global $L^\infty$ estimates remain open problems.