On a necessary aspect for the Riesz basis property for indefinite Sturm‐Liouville problems
Copyright policy )On a necessary aspect for the Riesz basis property for indefinite Sturm‐Liouville problems
In 1996, H. Volkmer observed that the inequality urn:x-wiley:dummy:mana201300104:equation:mana201300104-math-0001is satisfied with some positive constant for a certain class of functions f on [ − 1, 1] if the eigenfunctions of the problem urn:x-wiley:dummy:mana201300104:equation:mana201300104-math-0003form a Riesz basis of the Hilbert space . Here the weight is assumed to satisfy a.e. on ( − 1, 1).We present two criteria in terms of Weyl–Titchmarsh m‐functions for the Volkmer inequality to be valid. Note that one of these criteria is new even for the classical HELP inequality. Using these results we improve the result of Volkmer by showing that this inequality is valid if the operator associated with the spectral problem satisfies the linear resolvent growth condition. In particular, we show that the Riesz basis property of eigenfunctions is equivalent to the linear resolvent growth if r is odd.
- University of Vienna Austria
LRG condition, 34B24, 26D10, 34L10, 47A10, 47A75, Indefinite Sturm-Liouville problem, Inequalities involving derivatives and differential and integral operators, 101002 Analysis, HELP inequality, Riesz basis, Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators, Weyl theory and its generalizations for ordinary differential equations, Mathematics - Spectral Theory, Sturm-Liouville theory, indefinite Sturm-Liouville problem, Mathematics - Classical Analysis and ODEs, Eigenvalue problems for linear operators, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Spectral Theory (math.SP)
LRG condition, 34B24, 26D10, 34L10, 47A10, 47A75, Indefinite Sturm-Liouville problem, Inequalities involving derivatives and differential and integral operators, 101002 Analysis, HELP inequality, Riesz basis, Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators, Weyl theory and its generalizations for ordinary differential equations, Mathematics - Spectral Theory, Sturm-Liouville theory, indefinite Sturm-Liouville problem, Mathematics - Classical Analysis and ODEs, Eigenvalue problems for linear operators, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Spectral Theory (math.SP)
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