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RAYLEIGH-SCHRODINGER PERTURBATION THEORY:
PSEUDOINVERSE FORMULATION

by Brian J. McCartin


Preface

In Lord Rayleigh's investigation of vibrating strings with mild longitudinal density variation, a perturbation procedure was developed based upon the known analytical solution for a string of constant density. This technique was subsequently refined by Schrodinger and applied to problems in quantum mechanics and it has since become a mainstay of mathematical physics.

Mathematically, we have a discretized Laplacian-type operator embodied in a real symmetric matrix which is subjected to a small symmetric perturbation due to some physical inhomogeneity. The Rayleigh-Schrodinger procedure produces approximations to the eigenvalues and eigenvectors of the perturbed matrix by a sequence of successively higher order corrections to the eigenvalues and eigenvectors of the unperturbed matrix.

The difficulty with standard treatments of this procedure is that the eigenvector corrections are expressed in a form requiring the complete collection of eigenvectors of the unperturbed matrix. For large matrices this is clearly an undesirable state of affairs. Consideration of the thorny issue of multiple eigenvalues only serves to exacerbate this difficulty.

This malady can be remedied by expressing the Rayleigh-Schrodinger procedure in terms of the Moore-Penrose pseudoinverse. This permits these corrections to be computed knowing only the eigenvectors of the unperturbed matrix corresponding to the eigenvalues of interest. In point of fact, the pseudoinverse need not be explicitly calculated since only pseudoinverse-vector products are required. In turn, these may be efficiently calculated by a combination of matrix factorization, elmination/back substitution and orthogonal projection. However, the formalism of the pseudoinverse provides a concise formulation of the procedure and permits ready analysis of theoretical properties of the algorithm.

The present book provides a complete and self-contained treatment of the Rayleigh-Schrodinger perturbation theory based upon such a pseudoinverse formulation. The theory is built up gradually and many numerical examples are included. The intent of this spiral approach is to provide the reader with ready access to this important technique without being deluged by a torrent of formulae. Some redundancy has been intentionally incorporated into the presentation so as to make the chapters individually accessible.

Chapter 1 provides historical background relative to this technique and also includes several examples of how such perturbed eigenvalue problems arise in Applied Mathematics. Chapter 2 presents a self-contained summary of the most important facts about pseudoinverses needed in subsequent chapters. Chapter 3 treats the symmetric eigenvalue problem, first for linear perturbations and then for general analytic perturbations. The theory is then extended in Chapter 4 to the symmetric definite generalized eigenvalue problem.

Finally, Chapter 5 presents a detailed application of the previously developed theory to the technologically important problem of the analysis of inhomogeneous acoustic waveguides. Specifically, the walls of a duct (such as a muffler) are heated thereby producing a temperature gradient within the waveguide. The consequent perturbations to the propagating acoustic pressure waves are then calculated by applying the Rayleigh-Schrodinger pseudoinverse technique to the resulting generalized eigenvalue problem. Of particular interest is that this approach allows one to study the so-called degenerate modes of the waveguide. Enough background material is provided so as to be accessible to a wide scientific audience.

The target audience for this book includes practicing Engineers, Scientists and Applied Mathematicians. Particular emphasis has been placed upon including enough background material to also make the book accessible to graduate students in these same fields. The goal of the book has been not only to provide its readership with an understanding of the theory but also to give an appreciation for the context of this method within the corpus of Techniques of Applied Mathematics as well as to include sufficient examples and applications for them to apply the method in their own work. For those readers interested in the theoretical underpinnings of this technique, a generalized version of Rellich's Spectral Perturbation Theorem is presented and proved in the Appendix.


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