Engineering Electrodynamics: A collection of theorems, principles and field representations by Ramakrishna Janaswamy

Author:Ramakrishna Janaswamy
Language: eng
Format: mobi, epub
ISBN: 9780750317160
Publisher: IOP Publishing
Published: 2020-11-29T16:00:00+00:00

The range or closure of the range of is a proper subset of . In this case λ is said to belong to the residual spectrum of L.

The spectrum of L consists of all values of λ which belong to either the resolvent set, the continuous spectrum (singular spectrum), or the residual spectrum7.

12.3.2 Spectral theory of second order differential operators

Consider the self adjoint differential operator

under the inner product

where the weight function is real and in . The weight function has been added in the definition of the operator and in the corresponding definition of the inner product to accommodate problems encountered in various coordinates in electrodynamics. We are interested in the solution of the related problem , where λ is a given complex number. Let Hs be the Hilbert space consisting of all functions for which . The coefficients and q are real-valued functions in and are continuous in . If the interval is finite and all assumptions on the coefficients hold for the closed interval , the problem is said to be regular; otherwise it is singular. In the singular problem, the solutions of need not lie in Hs. The Green’s function is taken to satisfy the equation or equivalently, the equation subject to appropriate boundary conditions of the form . For it may also be written formally in terms of the resolvent operator Rλ as

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.