Book Review
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MathSciNet review:
1568162
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Book Information:
Author:
Peter Kuchment
Title:
Floquet theory for partial differential equations
Additional book information:
Operator Theory Advances and Applications, vol. 60, Birkh\"auser Verlag, Basel and Boston, 1993, xiv+350 pp., US$108.50. ISBN 0-8176-2901-7.
J. Avron, A. Grossmann, and R. Rodriguez, Hamiltonians in one-electron theory of solids. I, Rep. Mathematical Phys. 5 (1974), no. 1, 113–120. MR 413859, DOI 10.1016/0034-4877(74)90020-2
J. E. Avron, A. Grossmann, and R. Rodriguez, Spectral properties of reduced Bloch Hamiltonians, Ann. Physics 103 (1977), no. 1, 47–63. MR 428990, DOI 10.1016/0003-4916(77)90259-7
[B1] F. Bloch, Über die Quantenmechanik der Elektronen in Kristallgittern, Z. Phys. 52 (1929), 555-600.
Ju. L. Dalec′kiĭ and M. G. Kreĭn, Stability of solutions of differential equations in Banach space, Translations of Mathematical Monographs, Vol. 43, American Mathematical Society, Providence, R.I., 1974. Translated from the Russian by S. Smith. MR 0352639
G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Ann. Sci. École Norm. Sup. (2) 12 (1883), 47–88 (French). MR 1508722
I. M. Gel′fand, Expansion in characteristic functions of an equation with periodic coefficients, Doklady Akad. Nauk SSSR (N.S.) 73 (1950), 1117–1120 (Russian). MR 0039154
Philip Hartman, Ordinary differential equations, S. M. Hartman, Baltimore, Md., 1973. Corrected reprint. MR 0344555
G. W. Hill, On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon, Acta Math. 8 (1886), no. 1, 1–36. MR 1554690, DOI 10.1007/BF02417081
James S. Howland, Scattering theory for Hamiltonians periodic in time, Indiana Univ. Math. J. 28 (1979), no. 3, 471–494. MR 529679, DOI 10.1512/iumj.1979.28.28033
[Ho2] -, Floquet operators with singular spectrum. I, Ann. Inst. H. Poincaré 49 (1989), 309-323.
[Ho3] -, Floquet operators with singular spectrum. II, Ann. Inst. H. Poincaré 49 (1989), 325-334.
P. A. Kuchment, Floquet theory for partial differential equations, Uspekhi Mat. Nauk 37 (1982), no. 4(226), 3–52, 240 (Russian). MR 667973
A. I. Miloslavskiĭ, On the Floquet theory for parabolic equations, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 80–81 (Russian). MR 0473390
[MS] J. Massera and J. Schäffer, Differential equations and function spaces, Academic Press, New York, 1966.
Farouk Odeh and Joseph B. Keller, Partial differential equations with periodic coefficients and Bloch waves in crystals, J. Mathematical Phys. 5 (1964), 1499–1504. MR 168924, DOI 10.1063/1.1931182
Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
M. M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Trudy Mat. Inst. Steklov. 171 (1985), 122 (Russian). MR 798454
Lawrence E. Thomas, Time dependent approach to scattering from impurities in a crystal, Comm. Math. Phys. 33 (1973), 335–343. MR 334766
Kenji Yajima, Scattering theory for Schrödinger equations with potentials periodic in time, J. Math. Soc. Japan 29 (1977), no. 4, 729–743. MR 470525, DOI 10.2969/jmsj/02940729
- [AGR1]
- J. E. Avron, A. Grossmann, and R. Rodriguez, Hamiltonians in the one-electron theory of solids. I, Rep. Math. Phys. 5 (1974), 113-120. MR 0413859 (54:1971)
- [AGR2]
- -, Spectral properties of Bloch Hamiltonians, Ann. Physics 103 (1977), 47-63. MR 0428990 (55:2010)
- [B1]
- F. Bloch, Über die Quantenmechanik der Elektronen in Kristallgittern, Z. Phys. 52 (1929), 555-600.
- [DK]
- Yu. L. Daletskii and M. G. Krein, Stability of solutions of differential equations in Banach space, Amer. Math. Soc., Providence, RI, 1974. MR 0352639 (50:5126)
- [F1]
- G. Floquet, Sur les équations différentielles linéarles à coefficients périodiques, Ann. École Norm. Sup. 12 (1883), 47-88. MR 1508722
- [Ge]
- I. M. Gel'fand, Expansion in eigenfunctions of an equation with periodic coefficients, Dokl. Akad. Nauk SSSR 73 (1950), 1117-1120. MR 0039154 (12:503a)
- [Ha]
- P. Hartman, Ordinary differential equations, Hartman, Baltimore, MD, 1973. MR 0344555 (49:9294)
- [Hi]
- G. W. Hill, On the path of motion of the lunar perigee, Acta Math 8 (1886), 1-36. (This is a reprinting of an article which first appeared in 1877.) MR 1554690
- [Ho1]
- J. S. Howland, Scattering theory for Hamiltonians periodic in time, Indiana Univ. Math. J. 28 (1979), 471-494. MR 529679 (80e:47008)
- [Ho2]
- -, Floquet operators with singular spectrum. I, Ann. Inst. H. Poincaré 49 (1989), 309-323.
- [Ho3]
- -, Floquet operators with singular spectrum. II, Ann. Inst. H. Poincaré 49 (1989), 325-334.
- [Ku]
- P. A. Kuchment, Floquet theory for partial differential equations, Russian Math. Surveys 37 (1982), 1-60. MR 667973 (84b:35018)
- [Mi]
- A. I. Miloslavskii, On the Floquet theory for parabolic differential equations, Functional Anal. Appl. 10 (1976), 151-153. MR 0473390 (57:13057)
- [MS]
- J. Massera and J. Schäffer, Differential equations and function spaces, Academic Press, New York, 1966.
- [OK]
- F. Odeh and J. B. Keller, Partial differential equations with periodic coefficients and Bloch waves in crystals, J. Math. Phys. 5 (1964), 1499-1504. MR 0168924 (29:6180)
- [RS]
- M. Reed and B. Simon, Analysis of operators, Methods of Modern Mathematical Physics, vol. IV, Academic Press, New York, 1978. MR 0493421 (58:12429c)
- [Sk]
- M. M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Proceedings of the Steklov Institute of Mathematics, no. 171, Amer. Math. Soc., Providence, RI, 1987. MR 798454 (87h:47110)
- [Th]
- L. E. Thomas, Time dependent approach to scattering from impurities in a crystal, Comm. Math. Phys. 33 (1973), 335-343. MR 0334766 (48:13084)
- [Ya]
- K. Yajima, Scattering theory for Schroedinger equation with potential periodic in time, J. Math. Soc. Japan 29 (1977), 729-743. MR 0470525 (57:10276)
Review Information:
Reviewer:
Evans M. Harrell II
Journal:
Bull. Amer. Math. Soc.
32 (1995), 158-162
DOI:
https://doi.org/10.1090/S0273-0979-1995-00566-5
