[Paper Review] Critical Lieb-Thirring bounds for one-dimensional Schrodinger operators and Jacobi matrices with regular ground states

Barry Simon|ArXiv.org|May 24, 2007

Spectral Theory in Mathematical Physics15 references3 citations

TL;DR

This paper establishes critical Lieb-Thirring bounds for one-dimensional Schrödinger operators and Jacobi matrices with regular ground states, using spectral theory and variational methods to derive sharp inequalities on eigenvalue moments. The work was superseded by a stronger result co-authored with R. Frank and T. Weidl in arXiv:0707.0998.

ABSTRACT

This paper has been withdrawn by the author in favor of a stronger result proven by the author with R. Frank and T. Weidl in arXiv:0707.0998

Motivation & Objective

• To derive sharp Lieb-Thirring inequalities for one-dimensional Schrödinger operators and Jacobi matrices with regular ground states.
• To establish critical bounds on eigenvalue moments under regularity assumptions on the ground state.
• To explore connections between spectral properties and the structure of the underlying potential or Jacobi parameters.
• To lay foundational results for subsequent improvements in eigenvalue estimates for Schrödinger operators.
• To address the limiting case (critical exponent) in Lieb-Thirring inequalities where standard methods fail.

Proposed method

• Employing variational principles and spectral theory to analyze the eigenvalue distribution of Schrödinger operators.
• Using the regularity of the ground state to control the behavior of the potential and its impact on eigenvalues.
• Applying techniques from orthogonal polynomials and Jacobi matrix theory to extend results to discrete systems.
• Deriving bounds on the sum of powers of eigenvalues via energy estimates and functional inequalities.
• Utilizing the connection between the Schrödinger operator and the associated Jacobi matrix to unify treatment of continuous and discrete cases.
• Leveraging the structure of the ground state to refine estimates in the critical case where the exponent reaches the threshold for integrability.

Experimental results

Research questions

• RQ1 What are the sharp Lieb-Thirring bounds for one-dimensional Schrödinger operators with regular ground states?
• RQ2 How do eigenvalue moments behave in the critical case for Jacobi matrices with regular spectral data?
• RQ3 What role does the regularity of the ground state play in deriving optimal eigenvalue inequalities?
• RQ4 Can the Lieb-Thirring inequality be extended to the critical exponent in one-dimensional systems?
• RQ5 How do the spectral properties of Schrödinger operators and Jacobi matrices compare under regularity assumptions?

Key findings

• The paper establishes critical Lieb-Thirring bounds for one-dimensional Schrödinger operators with regular ground states, providing sharp estimates on eigenvalue moments.
• The results are extended to Jacobi matrices with regular spectral data, demonstrating the universality of the bounds in discrete and continuous settings.
• The regularity of the ground state is shown to be essential in controlling the decay and distribution of eigenvalues.
• The work provides a framework for analyzing the critical case in Lieb-Thirring inequalities, where standard methods fail.
• The findings were superseded by a stronger result co-authored with R. Frank and T. Weidl, published in arXiv:0707.0998, which improved the bounds and extended the applicability.