Krein extension of a differential operator of even order

TL;DR

This paper characterizes the Krein extension of a high-order differential operator on a finite interval, detailing boundary conditions and classifying all non-negative and finite negative square extensions.

Contribution

It provides a comprehensive description of the Krein extension and all non-negative and finite negative square extensions for the differential operator of even order.

Findings

Explicit boundary conditions for the Krein extension

Classification of all non-negative extensions

Description of extensions with finite negative squares

Abstract

We describe the Krein extension of minimal operator associated with the expression A:=(-1)^n*d^(2n)/dx^(2n) on a finite interval (a,b) in terms of boundary conditions. All non-negative extensions of the operator A as well as extensions with a finite number of negative squares are described.