[Paper Review] On the norming constants of the Sturm-Liouville problem

Tigran Harutyunyan, Avetik Pahlevanyan|arXiv (Cornell University)|Aug 19, 2016

Spectral Theory in Mathematical Physics5 references3 citations

TL;DR

This paper derives new asymptotic formulae for the norming constants of Sturm-Liouville problems with summable potentials, refining previous results by incorporating smooth dependence on boundary conditions. The key contribution is a precise asymptotic expansion for norming constants $ a_n $ and $ b_n $, including $ O(1/n^2) $ remainder terms, and proving the absolute continuity of associated spectral functions, which establishes a novel regularity property of the spectral data under variation of boundary conditions.

ABSTRACT

We derive new asymptotic formulae for the norming constants of Sturm-Liouville problem with summable potentials, which generalize and make more precise previously known formulae. Moreover, our formulae take into account the smooth dependence of norming constants on boundary conditions. We also find some new properties of the remainder terms of asymptotics.

Motivation & Objective

• To derive improved asymptotic expansions for norming constants of Sturm-Liouville problems with summable potentials.
• To analyze the smooth dependence of norming constants on boundary conditions $ \alpha $ and $ \beta $, which had not been systematically studied before.
• To establish new regularity properties of the spectral data, particularly the absolute continuity of functions derived from spectral coefficients.
• To refine and generalize previously known asymptotic formulae by including higher-order terms and precise remainder estimates.

Proposed method

• Derivation of asymptotic expansions for eigenfunctions and norming constants using the solution behavior of the Sturm-Liouville equation with $ q \in L^1_{\mathbb{R}}[0,\pi] $.
• Introduction of the quantity $ \ae_n(q,\alpha,\beta) = -\frac{1}{2}\int_0^\pi (\pi - t)q(t)\sin[2(n + \delta_n)\tau]dt $ as a key term in the norming constant expansion.
• Use of Fourier series techniques to analyze the spectral function $ k(x) = \sum_{n=2}^\infty \frac{\ae_n}{n + \delta_n} \cos[(n + \delta_n)x] $, showing its absolute continuity on $ (0, 2\pi) $.
• Application of asymptotic expansions for $ \delta_n(\alpha,\beta) $, including $ \delta_n = \frac{\cot\beta - \cot\alpha}{\pi n} + O(1/n^2) $, to control the dependence on boundary conditions.
• Use of $ L^2 $-based estimates and $ O(1/n^2) $ bounds on remainder terms $ r_n, \tilde{r}_n, p_n, \tilde{p}_n $, ensuring uniform convergence in $ \alpha, \beta $, and $ q $ from bounded $ L^1 $ sets.
• Proof of absolute continuity of $ k(x) $ via decomposition into $ l_1, l_2, l_3 $, each shown to be in $ AC[0,2\pi] $ using decay estimates on Fourier coefficients of $ \tilde{\sigma} \in AC[0,2\pi] $.

Experimental results

Research questions

• RQ1 How do the norming constants of the Sturm-Liouville problem depend smoothly on the boundary conditions $ \alpha $ and $ \beta $, especially when $ q \in L^1_{\mathbb{R}}[0,\pi] $?
• RQ2 What is the precise asymptotic behavior of the $ L^2 $-norms of eigenfunctions $ \varphi_n $ and $ \psi_n $, and can higher-order terms be explicitly captured?
• RQ3 Can the spectral function $ k(x) = \sum_{n=2}^\infty \frac{\ae_n}{n + \delta_n} \cos[(n + \delta_n)x] $ be shown to be absolutely continuous on $ (0, 2\pi) $?
• RQ4 What is the optimal rate of decay for the remainder terms in the asymptotic expansion of norming constants, and can it be uniformly bounded in $ \alpha, \beta $, and $ q $?

Key findings

• The asymptotic formula for the norming constant $ a_n(q,\alpha,\beta) $ includes a leading term $ \frac{\pi}{2} $, a correction term proportional to $ \frac{\ae_n}{n + \delta_n} $, and a remainder $ r_n = O(1/n^2) $, uniformly in $ \alpha, \beta \in [0,\pi] $ and $ q \in BL^1_{\mathbb{R}}[0,\pi] $.
• Similarly, $ b_n(q,\alpha,\beta) $ admits an asymptotic expansion with the same $ O(1/n^2) $ remainder, capturing the dependence on $ \sin^2\beta $ and $ \cos^2\beta $, respectively.
• The function $ k(x) = \sum_{n=2}^\infty \frac{\ae_n}{n + \delta_n} \cos[(n + \delta_n)x] $ is absolutely continuous on $ (0, 2\pi) $, a new regularity result for spectral data under boundary variation.
• For $ \alpha, \beta \in (0,\pi) $, the decomposition of $ k_2(x) $ into $ l_1, l_2, l_3 $ shows that all components are in $ AC[0,2\pi] $, relying on $ \tilde{\sigma} \in AC[0,2\pi] $ and $ O(1/n) $ decay of Fourier coefficients.
• In the special case $ \alpha = \pi, \beta = 0 $, the function $ k_2(x) $ reduces to a Fourier cosine series of an absolutely continuous function, confirming $ k \in AC(0,2\pi) $.
• The remainder terms $ r_n, \tilde{r}_n, p_n, \tilde{p}_n $ are all bounded by $ O(1/n^2) $, improving upon earlier estimates and ensuring uniform convergence across boundary conditions and potentials.