A failing eigenfunction expansion associated with an indefinite Sturm-Liouville problem

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• 作者：Andreas Fleige ^{andreas-fleige@versanet.de}
• 关键词：Indefinite Sturm&ndash ; Liouville problem ; Riesz basis ; Non-semibounded sesquilinear form ; Eigenfunction expansion
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2012
• 出版时间：15 May 2012
• 年：2012
• 卷：389
• 期：2
• 页码：932-949
• 全文大小：261 K

文摘

We consider the indefinite Sturm-Liouville problem , where satisfies . Conditions are presented such that the (normed) eigenfunctions form a Riesz basis of the Hilbert space (using known results for a modified problem). The main focus is on the non-Riesz basis case: We construct a function having no eigenfunction expansion . Furthermore, a sequence is constructed such that the ¡°Fourier series?does not converge in . These problems are closely related to the regularity property of the closed non-semibounded symmetric sesquilinear form with Dirichlet boundary conditions in where . For the associated operator we construct elements in the difference between and the domain of the associated regular closed form, i.e. .