$p$-Hyperbolic Zolotarev Functions in Boundary Value Problems for a $p\,$th order Differential Operator

TL;DR

This paper constructs and analyzes a new class of hyperbolic Zolotarev functions related to boundary value problems for p-th order differential operators, providing explicit solutions and eigenfunction bases for the third derivative case.

Contribution

It introduces p-hyperbolic Zolotarev functions, describes boundary conditions, and explicitly solves the third derivative operator, expanding the understanding of spectral properties of higher-order differential operators.

Findings

Constructed fundamental solutions analogous to sines and cosines.

Described classes of self-adjoint boundary conditions.

Calculated the resolvent and eigenfunctions for the third derivative operator.

Abstract

For the self-adjoint operator of the $p$th derivative, a system of fundamental solutions is constructed. This system is analogues to the classical system of sines and cosines. The properties of such functions are studied. Classes of self-adjoint boundary conditions are described. For the operator of the third derivative, the resolvent is calculated and an orthonormal basis of eigenfunctions is given.