[Paper Review] On periodic boundary value problem for the Sturm-Liouville operator

A. S. Makin|arXiv (Cornell University)|Jan 18, 2006

Spectral Theory in Mathematical Physics3 citations

TL;DR

This paper investigates the Riesz basis property of eigenfunctions for the Sturm-Liouville operator with periodic or antiperiodic boundary conditions. It establishes sufficient conditions on the potential function $ q(x) \in L_1(0,1) $, based on the decay and ratio of its Fourier coefficients $ \alpha_n $ and $ \beta_n $, to determine whether the root system forms a Riesz basis in $ L_2(0,1) $. The key result is a dichotomy: under specific growth conditions on $ \alpha_n $ and $ \beta_n $, the system is either a Riesz basis (Theorem 1) or not a basis at all (Theorem 2).

ABSTRACT

We consider the Sturm-Liouville operator Lu=u''-q(x)u with periodic or antiperiodic boundary conditions. It is shown that depending of Fourier coefficients of the potential q(x) the system of root functions may have or may not have the basis property.

Motivation & Objective

• To determine necessary and sufficient conditions under which the eigenfunction and associated function system of the Sturm-Liouville operator forms a Riesz basis in $ L_2(0,1) $.
• To analyze the spectral properties of the Sturm-Liouville operator with periodic or antiperiodic boundary conditions when the potential $ q(x) $ is complex-valued and integrable.
• To characterize the set of potentials $ q(x) \in L_1(0,1) $ for which the root system is a Riesz basis, and to study the density of such sets in $ L_1(0,1) $.
• To establish a dichotomy between basis and non-basis properties based on the asymptotic behavior of Fourier coefficients $ \alpha_n $ and $ \beta_n $ of $ q(x) $.

Proposed method

• Define the Sturm-Liouville operator $ Lu = u'' - q(x)u $ with periodic or antiperiodic boundary conditions.
• Express the potential $ q(x) $ via its Fourier coefficients $ \alpha_n = \int_0^1 q(x) e^{2\pi i n x} dx $ and $ \beta_n = \int_0^1 q(x) e^{-2\pi i n x} dx $.
• Impose smoothness and periodicity conditions on $ q(x) $: $ q \in W_1^m[0,1] $ and $ q^{(j)}(0) = q^{(j)}(1) $ for $ j = 0, \dots, m-1 $.
• Analyze the asymptotic decay of $ \alpha_n $ and $ \beta_n $, particularly for even or odd $ n $, depending on the boundary condition case.
• Use the ratio $ |\alpha_n / \beta_n| $ and the decay rate $ |\alpha_n| > c_0 / n^{m+1} $ to classify the basis property of the eigenfunction system.
• Construct explicit examples of potentials satisfying the conditions of Theorem 2 to demonstrate non-basis behavior.

Experimental results

Research questions

• RQ1 Under what conditions on the Fourier coefficients $ \alpha_n $ and $ \beta_n $ of $ q(x) $ does the eigenfunction system of the Sturm-Liouville operator with periodic or antiperiodic boundary conditions form a Riesz basis in $ L_2(0,1) $?
• RQ2 Can the eigenfunction system fail to be a basis even when the potential $ q(x) $ is smooth and periodic?
• RQ3 How do the decay rates and relative magnitudes of $ \alpha_n $ and $ \beta_n $ influence the basis property of the root system?
• RQ4 Is the set of potentials for which the system is a Riesz basis dense in $ L_1(0,1) $, and what is the topological structure of the complement set?
• RQ5 Can explicit examples be constructed where the eigenfunction system is not a basis, despite the potential being smooth and periodic?

Key findings

• The eigenfunction and associated function system $ \{u_n(x)"} $ forms a Riesz basis in $ L_2(0,1) $ if, for all sufficiently large even (case 1, periodic) or odd (case 2, antiperiodic) $ n $, $ |\alpha_n| > c_0 / n^{m+1} $ and $ 0 < c_1 < |\alpha_n / \beta_n| < c_2 $, with $ c_0 > 0 $.
• If there exists a sequence of even (case 1) or odd (case 2) indices $ n_k $ such that $ |\alpha_{n_k}| > c_0 / n_k^{m+1} $, $ |\beta_{n_k}| > c_0 / n_k^{m+1} $, and $ |\alpha_{n_k}/\beta_{n_k}| + |\beta_{n_k}/\alpha_{n_k}| \to \infty $, then the system $ \{u_n(x)"} $ is not a basis in $ L_2(0,1) $.
• An explicit potential $ q(x) = \sum_{n=1}^\infty \gamma_n \left( \frac{e^{2\pi i n x}}{n^{\varepsilon_1}} + \frac{e^{-2\pi i n x}}{n^{\varepsilon_2}} \right) $ with $ 0 < \varepsilon_1 < \varepsilon_2 < 1 $, and $ \gamma_n = 1 $ only when $ n = 2^p $ (case 1) or $ n = 2^p + 1 $ (case 2), satisfies the conditions of Theorem 2 and thus yields a non-basis system.
• The set $ Q $ of potentials for which the system is a Riesz basis is dense in $ L_1(0,1) $, and so is its complement $ \bar{Q} $, indicating that both basis and non-basis potentials are topologically dense in $ L_1(0,1) $.
• The results establish a sharp dichotomy: the basis property depends critically on the relative decay and growth of the Fourier coefficients $ \alpha_n $ and $ \beta_n $, particularly their ratio.
• The theorems provide a complete characterization of the Riesz basis property in terms of spectral data (Fourier coefficients) of the potential, extending classical results on spectral expansions.