Comments on Rayleigh’s Quotient and Perturbation Theory for the Eigenvalue Problem

1988 ◽  
Vol 55 (4) ◽  
pp. 986-988 ◽  
Author(s):  
Christophe Pierre
Keyword(s):  
Perturbation Theory ◽  
Eigenvalue Problem ◽  
Rayleigh’S Quotient

The connection between Rayleigh’s quotient and perturbation theory for the eigenvalue problem is studied. Equivalence of these techniques is proven under certain conditions.

scholarly journals Pseudoinverse formulation of Rayleigh-Schrödinger perturbation theory for the symmetric matrix eigenvalue problem

2003 ◽  
Vol 2003 (9) ◽  
pp. 459-485
Author(s):  
Brian J. McCartin
Keyword(s):  
Perturbation Theory ◽  
Eigenvalue Problem ◽  
Symmetric Matrix ◽  
Comprehensive Treatment ◽  
Matrix Eigenvalue Problem ◽  
Degenerate Problem ◽  
Matrix Eigenvalue ◽  
Penrose Pseudoinverse ◽  
Schrödinger Perturbation

A comprehensive treatment of Rayleigh-Schrödinger perturbation theory for the symmetric matrix eigenvalue problem is furnished with emphasis on the degenerate problem. The treatment is simply based upon the Moore-Penrose pseudoinverse thus distinguishing it from alternative approaches in the literature. In addition to providing a concise matrix-theoretic formulation of this procedure, it also provides for the explicit determination of that stage of the algorithm where each higher-order eigenvector correction becomes fully determined. The theory is built up gradually with each successive stage appended with an illustrative example.

scholarly journals Perturbation Theory of Relativistic Eigenvalue Problem

1953 ◽  
Vol 10 (1) ◽  
pp. 108-109
Author(s):  
Nobumichi Mugibayashi ◽  
Mikio Namiki
Keyword(s):  
Perturbation Theory ◽  
Eigenvalue Problem

scholarly journals Closed form bound-state perturbation theory

1980 ◽  
Vol 3 (2) ◽  
pp. 351-368 ◽  
Author(s):  
Ollie J. Rose ◽  
Carl G. Adler
Keyword(s):  
Perturbation Theory ◽  
Eigenvalue Problem ◽  
Closed Form ◽  
Bound States ◽  
Integral Form ◽  
Approximate Solutions ◽  
Perturbation Parameter ◽  
Bound State ◽  
The Given ◽  
Natural Way

The perturbed Schrödinger eigenvalue problem for bound states is cast into integral form using Green's Functions. A systematic algorithm is developed and applied to the resulting equation giving rise to approximate solutions expressed as functions of the given perturbation parameter. As a by-product, convergence radii for the traditional Rayleigh-Schrödinger and Brillouin-Wigner perturbation theories emerge in a natural way.

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