Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



On the eigentheory of operators which exhibit a slow variation

Authors: Lawrence Sirovich and B. W. Knight
Journal: Quart. Appl. Math. 38 (1981), 469-488
MSC: Primary 47A70; Secondary 35B40, 45C05, 92A09
DOI: https://doi.org/10.1090/qam/614554
MathSciNet review: 614554
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Abstract: A class of linear operators which exhibit slow variation is considered. If the kernel of the operator is $ K\left( {x - y,\frac{1}{2}\varepsilon \left( {x + y} \right)} \right)$, $ \varepsilon $ the parameter of slowness, then its Wigner transform is defined to be $ \tilde K\left( {p, q} \right) = \smallint K\left( {u, p} \right)\exp \left( { - iup} \right)du$. The eigenvalues of such operators are shown to follow an area rule: if the curve $ {\lambda _n} = \tilde K\left( {p, q} \right)$ contains the area $ A\left( \lambda \right) = \left( {2n + 1} \right)\pi \varepsilon $ then $ {\lambda _n}$ is an eigenvalue. Forms for the corresponding eigenfunctions are also obtained. Classical WKB theory is shown to be a special case and other examples are given.

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

  • [1] F. Ratliff, Mach bands: quantitative studies on neural networks in the retina, Holden-Day, San Francisco, 1967
  • [2] T. N. Cornsweet, Visual perception, Academic Press, New York, 1970
  • [3] H. R. Wilson and S. C. Giese, Threshold visibility of frequency gradient patterns, Vision Res. 17, 1177-1190 (1977)
  • [4] J. O. Limb and C. B. Rubenstein, A model of threshold vision incorporating inhomogeneity of the visual field, Vision Res. 17, 571-584 (1977)
  • [5] A. J. Doorn, J. J. Koendrink, and M. A. Bouman, The influence of the retinal inhomogeneity on the perception of spatial patterns, Kybernetic 10, 223-230 (1972)
  • [6] J. E. Robson and N. Graham, Probability summation and regional variation in sensitivity across the visual field, Vision Res. (in press)
  • [7] Lawrence Sirovich, Boundary effects in neural networks, SIAM J. Appl. Math. 39 (1980), no. 1, 142–160. MR 585834, https://doi.org/10.1137/0139012
  • [8] L. Sirovich, S. Brodie, and B. W. Knight, The effect of boundaries on the response of a neural network, Biophys. J. 28, 423-446 (1979)
  • [9] E. P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40, 749-759 (1932)
  • [10] H. J. Groenewald, On the principles of quantum mechanics, Physica 12, 405-460 (1946)
  • [11] J. E. Moyal, Quantum mechanics as a statistical theory, Proc. Cambridge Philos. Soc. 45 (1949), 99–124. MR 0029330
  • [12] J. Bruer, The classical limit of quantum theory, Ph.D. thesis, The Rockefeller University, 1978
  • [13] J. Heading, An introduction to phase-integral methods, Methuen & Co., Ltd., London; John Wiley & Sons, Inc., New York, 1962. MR 0148995
  • [14] L. Sirovich, Techniques of asymptotic analysis, Applied Mathematical Sciences, Vol. 2, Springer-Verlag, New York-Berlin, 1971. MR 0275034
  • [15] J. Cole, Perturbation techniques in applied mathematics, Blaisdell, Waltham, Mass. 1968
  • [16] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, U.S. Government Printing Office,
  • [17] B. W. Knight and L. Sirovich, The Wigner transform and some exact properties of linear operators, SIAM J. Appl. Math. 42 (1982), no. 2, 378–389. MR 650231, https://doi.org/10.1137/0142029
  • [18] L. Sirovich and B. W. Knight, Contributions to the eigenvalue problem for slowly varying operators, SIAM J. Appl. Math. 42 (1982), no. 2, 356–377. MR 650230, https://doi.org/10.1137/0142028
  • [19] E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Part I, Second Edition, Clarendon Press, Oxford, 1962. MR 0176151
  • [20] A. Erdelyi, ed., Bateman manuscript project, higher transcendental functions, vol. 2, p. 194, McGraw-Hill, New York, 1953

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DOI: https://doi.org/10.1090/qam/614554
Article copyright: © Copyright 1981 American Mathematical Society

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