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Generalized eigenfunctions for waves in inhomogeneous media. (English) Zbl 1043.35097
This paper is devoted to the study of generalized eigenfunctions of classical wave operators with nonsmooth coefficients. Note that physically interesting inhomogeneous media give rise to nonsmooth coefficients in the classical wave equations, and hence in their classical wave operators. The authors construct a generalized eigenfunction expansion for classical wave operators which yields polynomially bounded generalized eigenfunctions, the set of generalized eigenvalues forming a subset of the operator’s spectrum with full spectral measure.

35L05 Wave equation
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
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[1] Berezanskii, Ju.M., On expansion according to eigenfunctions of general self-adjoint differential operators, Dokl. akad. nauk SSSR, 108, 379-382, (1956)
[2] Berezanskii, Ju.M., Expansions in eigenfuncions of selfadjoint operators, (1968), Amer. Math. Soc Providence
[3] Berezansky, Y.M.; Sheftel, Z.G.; Us, G.F., Functional analysis, (1996), Birkhäuser Basel · Zbl 0859.46001
[4] Browder, F., Eigenfunction expansions for formally self-adjoint partial differential operators, I, II, Proc. nat. acad. sci. U.S.A., 42, 769-872, (1956) · Zbl 0072.33301
[5] Diestel, J.; Uhl, J.J., Vector measures, Mathematical surveys, 15, (1977), Amer. Math. Soc Providence · Zbl 0369.46039
[6] Dirac, P.A.M., The principles of quantum mechanics, (1958), Oxford Univ. Press London · Zbl 0080.22005
[7] von Dreifus, H.; Klein, A., A new proof of localization in the Anderson tight binding model, Comm. math. phys., 124, 285-299, (1989) · Zbl 0698.60051
[8] Figotin, A.; Klein, A., Localization of classical waves. I. acoustic waves, Commun. math. phys., 180, 439-482, (1996) · Zbl 0878.35109
[9] Figotin, A.; Klein, A., Localization of classical waves. II. electromagnetic waves, Comm. math. phys., 184, 411-441, (1997) · Zbl 0878.35110
[10] Fröhlich, J.; Martinelli, F.; Scoppola, E.; Spencer, T., Constructive proof of localization in the Anderson tight binding model, Comm. math. phys., 101, 21-46, (1985) · Zbl 0573.60096
[11] Gel’fand, I.M.; Kostyucenko, A.G., Expansion in eigenfunctions of differential and other operators, Dokl. akad. nauk SSSR, 103, 349-352, (1955)
[12] Germinet, F.; Klein, A., Bootstrap multiscale analysis and localization in random media, Comm. math. phys., 222, 415-448, (2001) · Zbl 0982.82030
[13] Kac, G.I., Expansion in characteristic functions of self-adjoint operators, Dokl. akad. nauk SSSR, 119, 19-22, (1958) · Zbl 0080.10602
[14] Klein, A.; Koines, A., A general framework for localization of classical waves. I. inhomogeneous media and defect eigenmodes, Math. phys. anal. geom., 4, 97-130, (2001) · Zbl 0987.35154
[15] Martinelli, F.; Scoppola, E., Remark on the absence of absolutely continuous spectrum for d-dimensional Schrödinger operators with random potential for large disorder or low energy, Comm. math. phys., 97, 465-471, (1985) · Zbl 0603.60060
[16] Pastur, L., Spectral properties of disordered systems in one-body approximation, Comm. math. phys., 75, 179-196, (1980) · Zbl 0429.60099
[17] Poerschke, T.; Stolz, G., On eigenfunction expansions and scattering theory, Math. Z., 212, 337-357, (1993) · Zbl 0793.47007
[18] Poerschke, T.; Stolz, G.; Weidman, J., Expansions in generalized eigenfunctions of self-adjoint operators, Math. Z., 202, 397-408, (1989) · Zbl 0661.47021
[19] Reed, M.; Simon, B., Methods of modern mathematical physics, vol. I, analysis of operators, (1972), Academic Press San Diego
[20] Schiff, L.I., Quantum mechanics, (1955), McGraw-Hill New York · Zbl 0068.40202
[21] Schulenberger, J.; Wilcox, C., Coerciveness inequalities for nonelliptic systems of partial differential equations, Arch. rational mech. anal., 88, 229-305, (1971) · Zbl 0215.45302
[22] Simon, B., Schrödinger semigroups, Bull. amer. math. soc., 7, 447-526, (1982) · Zbl 0524.35002
[23] Wilcox, C., Wave operators and asymptotic solutions of wave propagation problems of classical physics, Arch. rational mech. anal., 22, 37-78, (1966) · Zbl 0159.14302
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