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 on the interval , 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 sufficiently large, each point with is an approximate eigenvalue, and that the number of mutually orthogonal approximate eigenfunctions corresponding to grows faster than any constant multiple of 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.

**[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