[Paper Review] Estimates for Periodic Eigenvalues of the Differential Operator $\mathbf{(-1)^{m}d^{2m}/dx^{2m}+V}$ with V -- Distribution

Volodymyr Molyboga|arXiv (Cornell University)|Mar 11, 2014

Spectral Theory in Mathematical Physics2 references3 citations

TL;DR

This paper establishes asymptotic estimates for the periodic eigenvalues of the differential operator $(-1)^m d^{2m}/dx^{2m} + V$, where $V$ is a complex-valued distribution in the Sobolev space $H^{-m\alpha}_{\text{per}}[-1,1]$ with $0 \leq \alpha < 1$. Using spectral analysis in Fourier space and operator-theoretic techniques, it derives precise decay rates for the eigenvalue gaps and shifts, showing that the spectrum's behavior reflects the regularity of $V$ through its Fourier coefficients.

ABSTRACT

The periodic eigenvalue problem for the differential operator $(-1)^{m}d^{2m}/dx^{2m}+V$ is studied for complex-valued distribution V in the Sobolev space $H^{-mα}_{per}[-1,1]\;(m\in\mathbb{N},\; 0\leqα&lt;1)$. The following result is shown: The periodic spectrum consists of a sequence $(λ_{k})_{k\geq0}$ of complex eigenvalues satisfying the asymptotics (for any $\varepsilon&gt;0$) $$ λ_{2n-1},λ_{2n}=n^{2m}π^{2m}+\hat{V}(0)\pm \sqrt{\hat{V}(-2n)\hat{V}(2n)}+o(n^{m(2α-1+\varepsilon)}), $$ where $\hat{V}(k)$ denote the Fourier coefficients of V.

Motivation & Objective

• To analyze the spectral behavior of the differential operator $(-1)^m d^{2m}/dx^{2m} + V$ under periodic boundary conditions when $V$ is a complex distribution in $H^{-m\alpha}_{\text{per}}[-1,1]$.
• To determine how the regularity of $V$, measured by $\alpha$, influences the asymptotic distribution of the periodic eigenvalues.
• To establish sharp estimates for the eigenvalue gaps $\gamma_{mn} = \lambda_{2n} - \lambda_{2n-1}$ and the midpoints $\tau_{mn} = (\lambda_{2n} + \lambda_{2n-1})/2$ in terms of the Fourier coefficients of $V$.
• To characterize the regularity of real-valued, 1-periodic distributions $V$ via the decay of the eigenvalue gaps $\gamma_{mn}$, linking $V \in H^{-m\beta}_{\text{per}}$ to $\gamma_{mn} \in h^{-m\beta}$.

Proposed method

• The problem is transformed into a spectral analysis of the Fourier multiplier operator $\hat{L}_m = D_m + B$, where $D_m(k,j) = k^{2m}\pi^{2m}\delta_{kj}$ and $B(k,j) = \hat{V}(k-j)$.
• The spectrum of $L_m$ is studied via the resolvent identity $\lambda - D_m - B = D^{1/2}_{m\lambda}(I_{m\lambda} - S_{m\lambda})D^{1/2}_{m\lambda}$, enabling perturbation analysis.
• The eigenvalue asymptotics are derived using the convolution lemma, which ensures boundedness of the operator $B = v * \cdot$ from $h^{m(2-\alpha)}$ to $h^{-m\alpha}$.
• The analysis exploits the invariance of the operator under parity decomposition: $h_+^{-m\alpha}$ (even Fourier modes) and $h_-^{-m\alpha}$ (odd Fourier modes), allowing separate treatment of even and odd eigenvalue pairs.
• The key estimates are obtained by comparing $\gamma_{mn}$ to $2|\hat{V}(2n)|$, with error terms controlled by the decay of $\hat{V}(k)$ in $h^{-m\alpha}$, leading to the stated asymptotic bounds.
• The proof uses iterative refinement: assuming $\gamma_{mn} \in h^{-m\beta}$ and $V \in H^{-m\alpha}_{\text{per}}$, the method iteratively improves the regularity of $V$ until $V \in H^{-m\beta}_{\text{per}}$ is confirmed.

Experimental results

Research questions

• RQ1 How do the periodic eigenvalues of $(-1)^m d^{2m}/dx^{2m} + V$ behave asymptotically when $V$ is a distribution in $H^{-m\alpha}_{\text{per}}$?
• RQ2 What is the precise decay rate of the eigenvalue gaps $\gamma_{mn} = \lambda_{2n} - \lambda_{2n-1}$ in terms of the Fourier coefficients $\hat{V}(k)$?
• RQ3 How does the regularity of $V$, measured by $\alpha$, affect the spectral asymptotics of the operator?
• RQ4 Can the $L^2$-regularity of $V$ be characterized by the decay of the eigenvalue gaps $\gamma_{mn}$ for real-valued, 1-periodic distributions?
• RQ5 Under what conditions does $\gamma_{mn} \in h^{-m\beta}$ imply $V \in H^{-m\beta}_{\text{per}}$?

Key findings

• For any $\varepsilon > 0$, the midpoints $\tau_{mn}$ satisfy $\tau_{mn} - n^{2m}\pi^{2m} - \hat{V}(0) \in h^{m(1 - 2\alpha - \varepsilon)}$, showing that the deviation of the spectral center from the free eigenvalue is controlled by the regularity of $V$.
• The eigenvalue gap $\gamma_{mn}$ satisfies $\min_{\pm} |\gamma_{mn} \pm 2\sqrt{\hat{V}(-2n)\hat{V}(2n)}| \in h^{m(1/2 - \alpha)}$ if $0 \leq \alpha < 1/2$, and $h^{m(1 - 2\alpha - \varepsilon)}$ if $1/2 \leq \alpha < 1$, indicating a phase transition at $\alpha = 1/2$.
• For real-valued, 1-periodic $V$, the condition $V \in H^{-m\beta}_{\text{per}}$ is equivalent to $\gamma_{mn} \in h^{-m\beta}$, establishing a sharp spectral characterization of distributional regularity.
• The asymptotic expansion $\lambda_{2n-1}, \lambda_{2n} = n^{2m}\pi^{2m} + \hat{V}(0) \pm \sqrt{\hat{V}(-2n)\hat{V}(2n)} + o(n^{m(2\alpha - 1 + \varepsilon)})$ holds uniformly for bounded sets of $V$ in $H^{-m\alpha}_{\text{per}}$.
• The spectrum of $L_m$ is independent of the choice of $\alpha$ in $[\beta, 1)$ when $V \in H^{-m\beta}_{\text{per}}$, implying that the spectral data is stable under regularity improvements.
• The proof relies on iterative refinement: if $\gamma_{mn} \in h^{-m\beta}$ and $V \in H^{-m\alpha}_{\text{per}}$, then $\hat{V}(k) \in h^{m(-\alpha + \delta)}$ for some $\delta > 0$, and after finitely many steps, $\hat{V}(k) \in h^{-m\beta}$, so $V \in H^{-m\beta}_{\text{per}}$.