Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

The notion of approximate eigenvalues applied to an integral equation of laser theory


Author: H. J. Landau
Journal: Quart. Appl. Math. 35 (1977), 165-172
MSC: Primary 78.45
DOI: https://doi.org/10.1090/qam/446101
MathSciNet review: 446101
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Abstract: The integral operator with kernel $ {\left( {i\eta /\pi } \right)^{1/2}}\exp \left[ { - i\eta {{\left( {x - y} \right)}^2}} \right]$ on the interval $ \left\vert x \right\vert$, $ \left\vert y \right\vert \le 1$ serves to model the behavior of a class of lasers. Although the kernel is simple, it is not Hermitian; this presents a major obstacle to a theoretical understanding of the equation--indeed, even the existence of eigenvalues is difficult to prove. We here introduce a definition of approximate eigenvalues and eigenfunctions, and argue that these will model the physical problem equally well. We then show that, for $ \eta $ sufficiently large, each point $ \lambda $ with $ \left\vert \lambda \right\vert = 1$ is an approximate eigenvalue, and that the number of mutually orthogonal approximate eigenfunctions corresponding to $ \lambda $ grows faster than any constant multiple of $ \sqrt \eta $ This confirms a conjecture of J. A. Cochran and E. W. Hinds, supported by numerical evidence. In physical terms, it shows that for large Fresnel number the laser cannot be expected to settle to a single mode.


References [Enhancements On Off] (What's this?)

  • [1] J. A. Cochran and E. W. Hinds, Eigensystems associated with the complex-symmetric kernels of laser theory, SIAM J. Appl. Math. 26, 776-786 (1974) MR 0366064
  • [2] F. Riesz and B. Sz.-Nagy, Functional analysis, Ungar, N. Y., 1971
  • [3] H. J. Landau, On Szegö's cigenvalue distribution theorem and non-Hermitian kernels, J. d'Analyse Math. 28, 335-357 (1975) MR 0487600
  • [4] M. Kac, W. L. Murdock and G. Szegö, On the eigenvalues of certain Hermitian forms, J. Rat. Mech. Anal. 2, 767-800 (1953) MR 0059482
  • [5] E. G. Titchmarsh, Introduction to the theory of Fourier integrals, Oxford Univ. Press, 1948

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DOI: https://doi.org/10.1090/qam/446101
Article copyright: © Copyright 1977 American Mathematical Society

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