Lieb-Thirring Inequalities for Jacobi Matrices

Barry Simon; Dirk Hundertmark

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Lieb-Thirring Inequalities for Jacobi Matrices

Dirk Hundertmark, Barry Simon

Mathematical Physics · Submitted 13 Dec 2001

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim For a Jacobi matrix J on l^2(Z_+) with Ju(n)=a_{n-1} u(n-1) + b_n u(n) + a_n u(n+1), we prove that \sum_{|E|>2} (E^2 -4)^{1/2} \leq \sum_n |b_n| + 4\sum_n |a_n -1|.

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For a Jacobi matrix J on l^2(Z_+) with Ju(n)=a_{n-1} u(n-1) + b_n u(n) + a_n u(n+1), we prove that \sum_{|E|>2} (E^2 -4)^{1/2} \leq \sum_n |b_n| + 4\sum_n |a_n -1|. We also prove bounds on higher moments and some related results in higher dimension.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	For a Jacobi matrix J on l^2(Z_+) with Ju(n)=a_{n-1} u(n-1) + b_n u(n) + a_n u(n+1), we prove that \sum_{\|E\|>2} (E^2 -4)^{1/2} \leq \sum_n \|b_n\| + 4\sum_n \|a_n -1\|.	result 0.98	0.96	0.93	asserted 1.00
s1	We also prove bounds on higher moments and some related results in higher dimension.	contribution 0.69	0.94	0.05	asserted 1.00

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Original abstract

For a Jacobi matrix J on l^2(Z_+) with Ju(n)=a_{n-1} u(n-1) + b_n u(n) + a_n u(n+1), we prove that \sum_{|E|>2} (E^2 -4)^{1/2} \leq \sum_n |b_n| + 4\sum_n |a_n -1|. We also prove bounds on higher moments and some related results in higher dimension.

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With a different proof (not using the convolution property) the sharp inequalities for the Schrödinger operator and the Jacobi operator were first obtained in [ref] and in [ref] , respectively.

A Sharp Lieb-Thirring Inequality for Functional Difference Operators · Introduction · 0.81

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