Theory of Bloch waves. (English) Zbl 0408.35067


35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
35L05 Wave equation
35Q99 Partial differential equations of mathematical physics and other areas of application
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
82D20 Statistical mechanics of solids
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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[1] S. Agmon,Lectures on Elliptic Boundary Value Problems, Van Nostrand, New York, 1965. · Zbl 0142.37401
[2] F. Bloch,Über die Quantenmechanik der Electronen in Kristallgittern, Z. Phys.52 (1928), 555–600. · JFM 54.0990.01
[3] S. Bochner and W. T. Martin,Several Complex Variables, Princeton Univ. Press, Princeton, 1948. · Zbl 0041.05205
[4] T. Carleman,Zur Theorie der linearen Integralgleichungen, Math. Z.9 (1921), 196–217. · JFM 48.1249.01
[5] H. Cartan,Variétés analytiques-réelles et variétés analytiques-complexes, Bull. Soc. Math. France85 (1957), 77–100.
[6] A. P. Cracknell and K. C. Wong,The Fermi Surface, Clarendon Press, Oxford, 1973.
[7] M. S. P. Eastham,The Schrödinger equation with a periodic potential, Proc. Roy. Soc. Edinburgh69A (1971), 125–131. · Zbl 0225.35083
[8] M. S. P. Eastham,The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh, 1973. · Zbl 0287.34016
[9] I. M. Gelfand,Expansion in series of eigenfunctions of an equation with periodic coefficients, Dokl. Akad. Nauk SSSR73 (1950), 1117–1120.
[10] E. Goursat,A Course in Mathematical Analysis, V. III Part 2, Dover Publications, Inc., New York, 1964. · Zbl 0144.04502
[11] E. Hille and R. S. Phillips,Functional Analysis and Semi-groups, AMS Colloquium Publications V. 31, Providence, 1957. · Zbl 0078.10004
[12] T. Kato,Perturbation Theory for Linear Operators, Springer, New York, 1966. · Zbl 0148.12601
[13] J. L. Lions and E. Magenes,Non-Homogeneous Boundary Value Problems and Applications I, Springer, New York, 1972.
[14] S. G. Mikhlin,Integral Equations, 2nd revised ed., Pergamon Press, Oxford, 1964. · Zbl 0117.31902
[15] F. Odeh and J. B. Keller,Partial differential equations with periodic coefficients and Bloch waves in crystals, J. Math. Phys.5 (1964), 1499–1504. · Zbl 0129.46004
[16] F. Rellich,Ein Satz über mittlere Konvergenz, Gött. Nachr. (math. phys.) (1930), 30–35. · JFM 56.0224.02
[17] R. Sikorski,The determinant theory in Banach spaces, Colloq. Math.8 (1961), 141–198. · Zbl 0103.33202
[18] R. Sikorski,On the Carleman determinants, Studia Math.20 (1961), 327–346. · Zbl 0103.33401
[19] F. Smithies,The Fredholm theory of integral equations, Duke Math. J.8 (1941), 107–130. · JFM 67.0376.02
[20] L. E. Thomas,Time dependent approach to scattering from impurities in a crystal, Comm. Math. Phys.33 (1973), 335–343.
[21] E. C. Titchmarsh,Eigenfunction Expansions Part II, Clarendon Press, Oxford, 1958. · Zbl 0097.27601
[22] H. Whitney,Elementary structure of real algebraic varieties. Ann. of Math.66 (1957), 545–556. · Zbl 0078.13403
[23] H. Whitney and F. Bruhat,Quelques propriétés fondamentales des ensembles analytiques-réels, Comm. Math. Helvetici33 (1959), 132–160. · Zbl 0100.08101
[24] C. H. Wilcox,Uniform asymptotic estimates for wave packets in the quantum theory of scattering, J. Math. Phys.6 (1965), 611–620. · Zbl 0125.46005
[25] C. H. Wilcox,Measurable eigenvectors for Hermitian matrix-valued polynomials, J. Math. Anal. Appl.40 (1972), 12–19. · Zbl 0237.35067
[26] J. M. Ziman,Principles of the Theory of Solids, 2nd ed., Cambridge Univ. Press, 1972. · Zbl 0121.44801
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