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
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
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.
F. Ratliff, Mach bands: quantitative studies on neural networks in the retina, Holden-Day, San Francisco, 1967
T. N. Cornsweet, Visual perception, Academic Press, New York, 1970
H. R. Wilson and S. C. Giese, Threshold visibility of frequency gradient patterns, Vision Res. 17, 1177–1190 (1977)
J. O. Limb and C. B. Rubenstein, A model of threshold vision incorporating inhomogeneity of the visual field, Vision Res. 17, 571–584 (1977)
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)
J. E. Robson and N. Graham, Probability summation and regional variation in sensitivity across the visual field, Vision Res. (in press)
- Lawrence Sirovich, Boundary effects in neural networks, SIAM J. Appl. Math. 39 (1980), no. 1, 142–160. MR 585834, DOI https://doi.org/10.1137/0139012
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)
E. P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40, 749–759 (1932)
H. J. Groenewald, On the principles of quantum mechanics, Physica 12, 405–460 (1946)
- J. E. Moyal, Quantum mechanics as a statistical theory, Proc. Cambridge Philos. Soc. 45 (1949), 99–124. MR 29330
J. Bruer, The classical limit of quantum theory, Ph.D. thesis, The Rockefeller University, 1978
- J. Heading, An introduction to phase-integral methods, Methuen & Co., Ltd., London; John Wiley & Sons, Inc., New York, 1962. MR 0148995
- L. Sirovich, Techniques of asymptotic analysis, Applied Mathematical Sciences, vol. 2, Springer-Verlag, New York-Berlin, 1971. MR 0275034
J. Cole, Perturbation techniques in applied mathematics, Blaisdell, Waltham, Mass. 1968
M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, U.S. Government Printing Office,
- 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, DOI https://doi.org/10.1137/0142029
- 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, DOI https://doi.org/10.1137/0142028
- E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Part I, 2nd ed., Clarendon Press, Oxford, 1962. MR 0176151
A. Erdelyi, ed., Bateman manuscript project, higher transcendental functions, vol. 2, p. 194, McGraw-Hill, New York, 1953
F. Ratliff, Mach bands: quantitative studies on neural networks in the retina, Holden-Day, San Francisco, 1967
T. N. Cornsweet, Visual perception, Academic Press, New York, 1970
H. R. Wilson and S. C. Giese, Threshold visibility of frequency gradient patterns, Vision Res. 17, 1177–1190 (1977)
J. O. Limb and C. B. Rubenstein, A model of threshold vision incorporating inhomogeneity of the visual field, Vision Res. 17, 571–584 (1977)
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)
J. E. Robson and N. Graham, Probability summation and regional variation in sensitivity across the visual field, Vision Res. (in press)
L. Sirovich, Boundary effects in neutral networks, SIAM J. Appl. Math., 39, 142–160 (1980)
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)
E. P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40, 749–759 (1932)
H. J. Groenewald, On the principles of quantum mechanics, Physica 12, 405–460 (1946)
J. E. Moyal, Quantum mechanics as a statistical theory, Proc. Camb. Phil. Soc. 45, 99–124 (1949)
J. Bruer, The classical limit of quantum theory, Ph.D. thesis, The Rockefeller University, 1978
J. Heading, An introduction to phase-integral methods, Methuen & Co. Ltd., London, 1962
L. Sirovich, Techniques of asymptotic analysis, Springer-Verlag, New York, 1971
J. Cole, Perturbation techniques in applied mathematics, Blaisdell, Waltham, Mass. 1968
M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, U.S. Government Printing Office,
B. W. Knight and L. Sirovich, The Wigner transform and some exact properties of linear operators, submitted for publication
L. Sirovich and B. W. Knight, Contributions to the eigenvalue problem for slowly varying operators, submitted for publication
E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations, pt. 1, 2nd ed., Oxford University Press, London, 1962
A. Erdelyi, ed., Bateman manuscript project, higher transcendental functions, vol. 2, p. 194, McGraw-Hill, New York, 1953
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
47A70,
35B40,
45C05,
92A09
Retrieve articles in all journals
with MSC:
47A70,
35B40,
45C05,
92A09
Additional Information
Article copyright:
© Copyright 1981
American Mathematical Society