Discrete determinants and the Gel'fand-Yaglom formula

I present a partly pedagogic discussion of the Gel'fand-Yaglom formula for the functional determinant of a linear, one-dimensional, second-order difference operator, in the simplest settings. The formula is a textbook one in discrete Sturm-Liouville theory and orthogonal polynomials. A 2 × 2 ma...

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Bibliographic Details
Published in:Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2012-06, Vol.45 (21), p.215203-22
Main Author: Dowker, J S
Format: Article
Language:English
Subjects:
Chebyshev
Continuums
Deltas
Determinants
Eigenvalues
Exact sciences and technology
Mathematical analysis
Mathematical models
Physics
Polynomials
Sturm-Liouville
Sturm-Liouville theory
Textbooks
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Summary:I present a partly pedagogic discussion of the Gel'fand-Yaglom formula for the functional determinant of a linear, one-dimensional, second-order difference operator, in the simplest settings. The formula is a textbook one in discrete Sturm-Liouville theory and orthogonal polynomials. A 2 × 2 matrix approach is developed and applied to Robin boundary conditions. Euler-Rayleigh sums of eigenvalues are computed. A delta potential is introduced as a simple, non-trivial example and extended, in an appendix, to the general case. The continuum limit is considered in a non-rigorous way and a rough comparison with zeta regularized values is made. Vacuum energies are also considered in the free case. Chebyshev polynomials act as free propagators and their properties are developed using the two-matrix formulation, which appears to be novel and has some advantages. A trace formula, rather than the more usual determinant one, is derived for the Gel'fand-Yaglom function.
ISSN:1751-8113
1751-8121
DOI:10.1088/1751-8113/45/21/215203