# zbMATH — the first resource for mathematics

Quasiconformal mappings and periodic spectral problems in dimension two. (English) Zbl 1048.30010
Summary: We study spectral properties of second-order elliptic operators with periodic coefficients in dimension two. These operators act in periodic simply-connected waveguides, with either Dirichlet, or Neumann, or the third boundary condition. The main result is the absolute continuity of the spectra of such operators. The cornerstone of the proof is an isothermal change of variables, reducing the metric to a flat one and the waveguide to a straight strip. The main technical tool is the quasiconformal variant of the Riemann mapping theorem.

##### MSC:
 30C62 Quasiconformal mappings in the complex plane 35P05 General topics in linear spectral theory for PDEs
Full Text:
##### References:
 [1] L. V. Ahlfors,Lectures on Quasiconformal Mappings, D. Van Nostrand, New York, 1966. · Zbl 0138.06002 [2] L. V. Ahlfors and L. Bers,Riemann’s mapping theorem for variable metrics, Ann. of Math.72 (1960), 385–404. · Zbl 0104.29902 · doi:10.2307/1970141 [3] L. Bers, F. John and M. Schechter,Partial Differential Equations, Wiley, New York, 1964. · Zbl 0126.00207 [4] M. Birman, R. G. Shterenberg and T. Suslina,Absolute continuity of the spectrum of a two-dimensional Schrödinger operator with potential supported on a periodic system of curves, Algebra i Analiz12, no. 6 (2000), 140–177 (in Russian); Engl. transl.: St. Petersburg Math. J.12 (2001), no. 6, 983–1012. · Zbl 0998.35006 [5] M. Sh. Birman and M. Solomyak,On the negative discrete spectrum of a periodic elliptic operator in a waveguide-type domain, perturbed by a decaying potential, J. Analyse Math.83 (2001), 337–391. · Zbl 1200.35196 · doi:10.1007/BF02790267 [6] M. Birman and T. Suslina,Two-dimensional periodic magnetic Hamiltonian is absolutely continuous, Algebra i Analiz9, no. 1 (1997), 32–48 (in Russian); Engl. transl.: St Petersburg Math. J.9 (1998), 21–32. · Zbl 0890.35096 [7] M. Birman and T. Suslina,Absolute continuity of the two-dimensional periodic magnetic Hamiltonian with discontinuous vector-valued potential, Algebra i Analiz10, no. 4 (1998), 1–36 (in Russian); Engl. transl.: St. Petersburg Math. J.10, no. 4 (1999), 579–601. [8] M. Birman and T. Suslina,Periodic magnetic Hamiltonian with a variable metric. Problem of absolute continuity, Algebra i Analiz11, no. 2 (1999), 1–40 (in Russian); Engl. transl.: St. Petersburg Math. J.11, no. 2 (2000), 203–232. · Zbl 0941.35015 [9] V. I. Derguzov,The spectrum of a plane periodic dielectric waveguide, in ”Partial Diff. Eq. Spectral Theory”, Probl. Mat. Anal.8 (1981), 26–35 (in Russian); Sibirsk. Mat. Zh.21, no.5 (1980), 27–38 (in Russian); Engl. transl.: Siberian Math. J.21 (1981), 664–672. · Zbl 0484.35061 [10] N. D. Filonov,An elliptic equation of second order in the divergence form, having a solution with a compact support, Probl. Mat. Anal.22 (2001), 246–257 (in Russian); Engl. transl.: J. Math. Sci., New York106, no. 3 (2001), 3078–3086. · Zbl 0991.35020 [11] L. Friedlander,On the spectrum of a class of second order periodic elliptic differential operators, Comm. Math. Phys.229 (2002), 49–55. · Zbl 1014.35066 · doi:10.1007/s00220-002-0675-6 [12] D. Gilbarg and N. S. Trudinger,Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001. · Zbl 1042.35002 [13] V. M. Gol’dstein and Yu. Reshetnyak,Quasiconformal Mappings and Sobolev Spaces, Kluwer, Dordrecht, 1990. · Zbl 0687.30001 [14] G. M. Goluzin,Geometric Theory of Functions of a Complex Variable, Transl. of Math. Monographs, Vol. 26, Amer. Math. Soc, Providence, RI, 1969. · Zbl 0183.07502 [15] R. Hempel and I. Herbst,Bands and gaps for periodic magnetic Hamiltonians, in Operator Theory: Advances and Applications, Vol. 78, Birkhäuser, Basel, 1995, pp. 175–184. · Zbl 0849.35091 [16] G. Khuskivadze, V. Kokilashvili and V. Paatashvili,Boundary value problems for analytic and harmonic functions in domains with nonsmooth boundaries. Applications to conformal mappings, Mem. Differential Equations Math. Phys.14 (1998), 1–195. · Zbl 0947.30030 [17] S. L. Krushkal,Quasiconformal Mappings and Riemann Surfaces, Wiley, New York, 1979. [18] P. Kuchment,Floquet Theory for Partial Differential Eequations, Birkhäuser, Basel, 1993. · Zbl 0789.35002 [19] P. Kuchment and S. Levendorski,On the structure of spectra of periodic elliptic operators, Trans. Amer. Math. Soc.354, (2002), 537–569. · Zbl 1058.35174 · doi:10.1090/S0002-9947-01-02878-1 [20] M. A. Lavrent’ev,Variantional Methods for Boundary Value Problems for Systems of Elliptic Equations, Noordhoff, Groningen, 1963. [21] M. A. Lavrent’ev and B. V. Shabat,Metody teorii funktsii kompleksnogo peremennogo, GITTL, 1951 (in Russian); Transl.:Methoden der komplexen Funktionentheorie, Mathematik fur Naturwissenschaft und Technik, Bd. 13, Deutscher Verlag der Wissenschaften, Berlin, 1967. [22] P. MacManus,Bi-Lipschitz extensions in the plane, J. Analyse Math.66 (1995), 85–115. · Zbl 0907.54028 · doi:10.1007/BF02788819 [23] V. G. Maz’ya,Sobolev Spaces, Springer, Berlin, 1985. [24] V. G. Maz’ya and S. V. Poborchi,Differentiable Functions on Bad Domains, World Sci., Singapore, 1997. · Zbl 0918.46033 [25] A. Morame,Absence of singular spectrum for a perturbation of a two-dimensional Laplace-Beltrami operator with periodic electromagnetic potential, J. Phys. A: Math. Gen.31 (1998), 7593–7601. · Zbl 0931.35143 · doi:10.1088/0305-4470/31/37/017 [26] Ch. Pommerenke,Boundary Behaviour of Conformal Maps, Springer, Berlin, 1992. · Zbl 0762.30001 [27] R. Shterenberg,Absolute continuity of the two-dimensional magnetic Schrödinger operator with a potential of the measure density type, Zap. Nauchn. Sem. POMI271 (2000), 276–312 (in Russian); Preprint KTH, Stockholm (2000). [28] R. Shterenberg,Absolute continuity of the spectrum of the two-dimensional Schrödinger operator with a positive electric potential, Algebra i Analiz13, no. 4 (2001), 196–228 (in Russian); Engl. transl.: St Petersburg Math. J.13, no. 4 (2002), 659–685. [29] R. Shterenberg and T. Suslina,Absolute continuity of the spectrum of a magnetic Schrödinger operator with metric in a two-dimensional periodic waveguide, Algebra i Analiz14, no. 2 (2002), 159–206 (in Russian). · Zbl 1057.35024 [30] A. V. Sobolev,Absolute continuity of the periodic magnetic Schrödinger operator, Invent. Math.137 (1999), 85–112. · Zbl 0932.35049 · doi:10.1007/s002220050324 [31] A. V. Sobolev and J. Walthoe,Absolute continuity in periodic waveguides, Proc. London Math. Soc.85 (2002), 717–741. · Zbl 1247.35077 · doi:10.1112/S0024611502013631 [32] T. Suslina,Absolute continuity of the spectrum of periodic operators of mathematical physics, in Proc. Journées Equation aux Derivées Partielles, Nantes 2000, Exposé XVIII. · Zbl 1213.35194 [33] L. E. Thomas,Time dependent approach to scattering from impurities in a crystal, Comm. Math. Phys.33 (1973), 335–343. · doi:10.1007/BF01646745 [34] P. Tukia,The planar Schönflies theorem for Lipschitz maps, Ann. Acad. Sci. Fenn. Series A I Math.5 (1980), 49–72. · Zbl 0411.57015 [35] I. N. Vekua,Generalized Analytic Functions, Pergamon Press, Oxford, 1962. · Zbl 0127.03505 [36] C. H. Wilcox,Theory of Bloch waves, J. Analyse Math.33 (1978), 146–167. · Zbl 0408.35067 · doi:10.1007/BF02790171
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.