[Paper Review] Properties of the l=1 radial part of the Laplace operator in a special scalar product

T. A. Bolokhov|arXiv (Cornell University)|Oct 27, 2015

Spectral Theory in Mathematical Physics3 references4 citations

TL;DR

This paper investigates self-adjoint extensions of the $ l=1 $ radial part of the Laplace operator in a non-standard scalar product arising from the transverse vector field decomposition in $ \mathbb{R}^3 $. It derives spectral properties, resolvent kernels, and spectral decompositions for both the operator and its inverse, revealing a continuous spectrum on $[0, \infty)$ and discrete eigenvalues at $ -\kappa^2 $ and $ -2^{2/3}\kappa^{-4} $, with explicit eigenfunctions and integral kernels provided.

ABSTRACT

We develop self-adjoint extensions of the l=1 radial part of the Laplace operator in a special scalar product. The product arises as the transfer of the plain product from R^3 into the set of functions parametrizing one of the two components of the transverse vector field. The similar extensions are treated for the square of inverse operator of the radial part in question.

Motivation & Objective

• To develop self-adjoint extensions of the $ l=1 $ radial part of the Laplace operator in a scalar product derived from the transverse vector field structure in $ \mathbb{R}^3 $.
• To analyze the spectral properties of the operator $ T_1 $ and its inverse in this non-standard scalar product, which differs from the standard $ L^2 $-based setting.
• To construct explicit resolvent kernels and spectral decompositions for both the operator and its inverse, including eigenfunctions for discrete eigenvalues.
• To clarify the physical relevance of these extensions in the context of the electromagnetic field Hamiltonian in Coulomb gauge, particularly for the kinetic and potential energy terms.

Proposed method

• The scalar product $ \langle u, v \rangle_l = \int_0^\infty \left( \overline{u'}v' + \frac{l(l+1)}{r^2}\overline{u}v \right) dr $ is used, which arises from the $ L^2 $ inner product on transverse vector fields in $ \mathbb{R}^3 $.
• The radial operator $ T_l = -\frac{d^2}{dr^2} + \frac{l(l+1)}{r^2} $ is studied for $ l=1 $, with self-adjoint extensions characterized by boundary conditions involving a parameter $ \kappa $.
• The resolvent kernel $ \tilde{R}(r,s;\lambda) $ is constructed explicitly using solutions to the differential equation $ (T_1 - \lambda^2)u = 0 $, with complex $ \lambda $.
• Spectral decomposition is derived via the kernel $ \hat{p}_\lambda(r) = \frac{i}{\sqrt{2\pi}\lambda^2} D\left( \sqrt{\frac{\overline{d}}{d}}e^{i\lambda r} - \sqrt{\frac{d}{\overline{d}}}e^{-i\lambda r} + \frac{2i\lambda^3}{|d|}e^{-\lambda r} \right) $, yielding a continuous spectral measure.
• For the inverse operator $ T_1^{-1} $, a second family of extensions is constructed using a different domain condition involving $ \varkappa $, leading to a modified quadratic form and spectral kernel.
• The spectral decomposition of the inverse operator is expressed as $ Q_{\varkappa}^{-1}(u) = \iint Q_{\varkappa}^{-1}(r,s) u(r) \overline{u(s)} \, dr \, ds $, with $ Q_{\varkappa}^{-1}(r,s) $ involving eigenfunctions and a discrete term proportional to $ \varkappa^{-4} $.

Experimental results

Research questions

• RQ1 What are the self-adjoint extensions of the $ l=1 $ radial Laplacian in the scalar product derived from transverse vector fields in $ \mathbb{R}^3 $?
• RQ2 How do the spectral properties of $ T_1 $ and its inverse differ in this non-standard scalar product compared to the standard $ L^2 $ setting?
• RQ3 What is the explicit form of the resolvent kernel and spectral decomposition for the extended operator $ T_1 $?
• RQ4 How do the discrete eigenvalues and eigenfunctions emerge in the extended operator framework, and what is their dependence on the extension parameter $ \kappa $?
• RQ5 Can the quadratic form associated with $ T_1^{-1} $ be extended meaningfully in this scalar product, and if so, how is it represented?

Key findings

• The self-adjoint extensions of $ T_1 $ in the scalar product $ \langle u, v \rangle_1 $ possess a continuous spectrum on $[0, \infty)$, with spectral density given by the function $ \hat{p}_\lambda(r) $ in equation (59).
• For negative values of the extension parameter $ \kappa $, the operator $ T_1 $ has a simple discrete eigenvalue at $ -\kappa^2 $, with corresponding eigenfunction given by equation (32).
• The resolvent kernel $ \tilde{R}(r,s;\lambda) $ is explicitly constructed as a combination of exponential and modified Bessel-type terms, with a closed-form expression involving $ d $, $ \overline{d} $, and $ \alpha_{\pm}(\lambda) $.
• The spectral decomposition of $ T_1 $ is given by equation (29), expressing the identity operator as an integral over the continuous spectrum with kernel $ \hat{p}_\lambda(r) $.
• For the inverse operator $ T_1^{-1} $, a second family of self-adjoint extensions is constructed using a domain condition involving $ \varkappa $, leading to a discrete eigenvalue at $ -2^{2/3}\kappa^{-4} $.
• The spectral decomposition of the inverse operator is expressed via a kernel $ Q_{\varkappa}^{-1}(r,s) $ that includes both a continuous part and a discrete term proportional to $ \varkappa^{-4} $, as shown in equation (59).