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Integral Equations
Published in Ronald B. Guenther, John W. Lee, Sturm-Liouville Problems, 2018
Ronald B. Guenther, John W. Lee
Many eigenvalue problems that arise in applications are equivalent to eigenvalue problems for integral operators that are self-adjoint. This is fortunate because such eigenvalue problems behave in a way strictly analogous to eigenvalue problems for self-adjoint matrices. The eigenvalues for self-adjoint matrices are real and there is a corresponding set of orthonormal eigenvectors that are a basis for the underlying real or complex Euclidean space. Such a basis is strictly analogous to standard basis i, j, and k in three space and is equally useful for theoretical and computational purposes. Virtually all of the properties in the matrix setting carry over to the infinite dimensional, integral equations setting. (See Section 2.3 for further discussion of the matrix case.)
Boundary Value Problems
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
where L[x,D]=-Dp(x)D+q(x) $ L[x, D] = - Dp(x)D + q(x) $ with D = d/dx. We consider self‐adjoint differential operators of the second order for two reasons. First, all eigenvalues of a self‐adjoint operator are real numbers and eigenfunctions corresponding to distinct eigenvalues are orthogonal. Second, a nonself‐adjoint differential equation can be reduced (see §4.1.3) to a self‐adjoint counterpart. Moreover, if the boundary value problem generated by the differential operator L[x,D] $ L[x, D] $ and the boundary conditions B0, Bℓ is nonself‐adjoint, then the corresponding Green’s function is not symmetric and its eigenvectors are not orthogonal.
Linear Algebra
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
One of the most important theorems in linear algebra is the Spectral Theorem. This theorem tells us when a matrix can be diagonalized. In fact, it goes beyond matrices to the diagonalization of linear operators. We learn in linear algebra that linear operators can be represented by matrices once we pick a particular representation basis. Diagonalization is simplest for finite dimensional vector spaces and requires some generalization for infinite dimensional vectors spaces. Examples of operators to which the spectral theorem applies are self-adjoint operators (more generally normal operators on Hilbert spaces). We will explore some of these ideas later in the course. The spectral theorem provides a canonical decomposition, called the spectral decomposition, or eigendecomposition, of the underlying vector space on which it acts.
On the Sturm–Liouville problem describing an ocean waveguide covered by pack ice
Published in Applicable Analysis, 2022
Boris P. Belinskiy, Don B. Hinton, Lakmali Weerasena, Mohammad M. Khan
We finally note that the modern literature contains study of diverse wave propagation problems in the ocean waveguides, with ice-covered or partially ice-covered surface, including waveguides of varying depth (see, e.g. [2, 5–9, 11, 17, 18, 31]). Further, our study is based on the theory of self-adjoint operators. Yet, the similar study of viscoelastic-type models (see, e.g. [36]) would require using more complex methods of non-self-adjoint operators. We believe that further study of wave propagation in the layered ocean covered by ice may require a spectral analysis similar to the analysis in our paper. The question appears whether Propositions 3.3, 3.5, 3.6, Remarks 3.4, 4.5, 5.4, 5.5, 5.6, Theorems 4.2, 4.3,5.1, numerical analysis in Section 6, and asymptotic analysis in Section 7 hold (or may be generalized) in the case of the boundary condition more complex than (12), e.g. (80). We hope to study these issues later.