English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

We set out to build a framework for self-adjoint extension theory for powersof the Jacobi differential operator that does not make use of classicaldeficiency elements. Instead, we rely on simpler functions that capture theimpact of these elements on extensions but are defined by boundary asymptotics.This new perspective makes calculations much more accessible and allows for amore nuanced analysis of the associated domains.  The maximal domain for $n$-th composition of the Jacobi operator ischaracterized in terms of a smoothness condition for each derivative, and theendpoint behavior of functions in the underlying Hilbert space can then beclassified, for $j\in\mathbb{N}_0$, by $(1-x)^j$, $(1+x)^j$,$(1-x)^{-\alpha+j}$ and $(1+x)^{\beta+j}$. Most of these behaviors can onlyoccur when functions are in the associated minimal domain, and this leads to aformulation of the defect spaces with a convenient basis. Self-adjointextensions, including the important left-definite domain, are then given interms of the new basis functions for the defect spaces using GKN theory.Comments are made for the Laguerre operator as well.

Properties and Decompositions of Domains for Powers of the Jacobi Differential Operator