Higher order self-adjoint operators with polynomial coefficients

TL;DR

This paper explores the algebraic and analytic properties of high-order self-adjoint operators with polynomial coefficients, providing a systematic construction method and classifying all order 4 cases, with examples up to order 8.

Contribution

It introduces a systematic approach to construct high-order self-adjoint operators with polynomial coefficients and classifies all order 4 cases.

Findings

Complete classification of order 4 operators

Construction of examples up to order 8

Method applicable to all even orders

Abstract

Algebraic and analytic aspects of self-adjoint operators of order four or more with polynomial coefficients are investigated. As a consequence, a systematic way of constructing such operators is given. The procedure is applied to obtain many examples up to order 8; similar examples can be constructed for all even order operators. In particular, a complete classification of all order 4 operators is given.