Spectral properties and self-adjoint extensions of the third power of the radial Laplace operator

TL;DR

This paper investigates the spectral properties and self-adjoint extensions of the third power of the radial Laplace operator, analyzing boundary conditions that influence the operator's spectrum and physical implications.

Contribution

It provides explicit calculations of resolvents and spectral decompositions for specific boundary conditions of the third power of the radial Laplace operator.

Findings

Derived spectral decompositions for different boundary conditions.

Identified boundary conditions leading to long-range interactions.

Showed emergence of long-range action from boundary conditions.

Abstract

We consider self-adjoint extensions of differential operators of the type $ (-\frac{d^2}{dr^2} + \frac{l(l+1)}{r^2})^3 $ on the real semi-axis for l=1,2 with two kinds of boundary conditions: first that nullify the value of a function and its first derivative and second that nullify the 4th (l=1) or the 3rd (l=2) derivative. We calculate the expressions for the correponding resolvents and derive spectral decompositions. These types of boundary conditions are interesting from the physical point of view, especially the second ones, which give an example of emergence of long-range action in exchange for a singularity at the origin.