1D Dirac Operators with Special Periodic Potentials

• Autorzy: Djakov P. , Mityagin B.

Identyfikatory

DOI

10.4064/ba60-1-5

• Języki publikacji: EN

Abstrakty

EN

For one-dimensional Dirac operators of the form [formula] we single out and study a class X of π-periodic potentials v whose smoothness is determined only by the rate of decay of the related spectral gaps γn=|λ+n−λ−n|, where λ±n are the eigenvalues of L=L(v) considered on [0,π] with periodic (for even n) or antiperiodic (for odd n) boundary conditions.

Słowa kluczowe

EN

one-dimensional Dirac periodic operator; spectral gaps; Sobolev spaces; potential smoothness

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2012
• Tom: Vol. 60, no 1
• Strony: 59--75
• Opis fizyczny: Bibliogr. 38 poz.

Twórcy

autor

Djakov P.

• Sabanci University Orhanli, 34956 Tuzla, Istanbul, Turkey

autor

Mityagin B.

• Department of Mathematics The Ohio State University 231 West 18th Ave. Columbus, OH 43210, U.S.A.

Bibliografia

• [1] P. Djakov and B. Mityagin, Smoothness of Schrödinger operator potential in the case of Gevrey type asymptotics of the gaps, J. Funct. Anal. 195 (2002), 89-128.
• [2] -, -, Spectral triangles of Schrödinger operators with complex potentials, Selecta Math. (N.S.) 9 (2003), 495-528.
• [3]-, -, Spectra of 1-D periodic Dirac operators and smoothness of potentials, C. R. Math. Acad. Sci. Soc. R. Can. 25 (2003), 121-125.
• [4] -, -, Instability zones of a periodic 1D Dirac operator and smoothness of its potential, Comm. Math. Phys. 259 (2005), 139-183.
• [5] -, -, Instability zones of periodic 1D Schrdinger and Dirac operators, Uspekhi Mat. Nauk 61 (2006), no. 4, 77-182 (in Russian); English transl.: Russian Math. Surveys 61 (2006), 663-766.
• [6] -, -, Asymptotics of instability zones of the Hill operator with a two term potential, J. Funct. Anal. 242 (2007), 157-194.
• [7] B. A. Dubrovin, The inverse problem of scattering theory for periodic finite-zone potentials, Funktsional. Anal. i Prilozhen. 9 (1975), no. 1, 65-66 (in Russian).
• [8] B. A. Dubrovin, V. B. Matveev and S. P. Novikov, Nonlinear equations of Kortewegde Vries type, finite-band linear operators and Abelian varieties, Uspekhi Mat. Nauk 31 (1976), no. 1, 55-136 (in Russian).
• [9] M. S. P. Eastham, The Spectral Theory of Periodic Differential Operators, Hafner, New York, 1974.
• [10] M. G. Gasymov, Spectral analysis of a class of second order nonselfadjoint differential operators, Funktsional. Anal. i Prilozhen. 14 (1980), no. 1, 14-19 (in Russian).
• [11] I. M. Gelfand and B. M. Levitan, On the determination of a differential equation by its spectral function, Dokl. Akad. Nauk SSSR 77 (1951), 557-560 (in Russian).
• [12] B. Grébert and T. Kappeler, Estimates on periodic and Dirichlet eigenvalues for the Zakharov-Shabat system, Asymptot. Anal. 25 (2001), 201-237; Erratum, ibid. 29 (2002), 183.
• [13] -, -, Density of finite gap potentials for the Zakharov-Shabat system, ibid. 33 (2003), 1-8.
• [14] B. Grébert, T. Kappeler and B. Mityagin, Gap estimates of the spectrum of the Zakharov-Shabat system, Appl. Math. Lett. 11 (1998), 95-97.
• [15] H. Hochstadt, Estimates on the stability intervals for the Hill's equation, Proc. Amer. Math. Soc. 14 (1963), 930-932.
• [16] -, On the determination of a Hill's equation from its spectrum, Arch. Ration. Mech. Anal. 19 (1965), 353-362.
• [17] T. Kappeler and B. Mityagin, Gap estimates of the spectrum of Hill's equation and action variables for KdV, Trans. Amer. Math. Soc. 351 (1999), 619-646.
• [18] -, -, Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator, SIAM J. Math. Anal. 33 (2001), 113-152.
• [19] T. Kappeler, F. Serier and P. Topalov, On the characterization of the smoothness of skew-symmetric potentials in periodic Dirac operators, J. Funct. Anal. 256 (2009), 2069-2112.
• [20] V. F. Lazutkin and T. F. Pankratova, Asymptotics of the width of gaps in the spectrum of the Sturm-Liouville operators with periodic potential, Soviet Math. Dokl. 15 (1974), 649-653.
• [21] V. A. Marchenko, Sturm-Liouville Operators and Applications, Oper. Theory Adv. Appl. 22, Birkhäuser, 1986.
• [22] -, Spektraltheorie und nichtlineare Gleichungen, in: Differential Equations (Uppsala, 1977), Sympos. Univ. Upsaliensis Ann. Quingentesimum Celebrantis, No. 7, Almqvist & Wiksell, Stockholm, 1977, 134-143.
• [23] V. A. Marchenko and I. V. Ostrovskii, Characterization of the spectrum of Hill's operator, Mat. Sb. 97 (1975), 540-606 (in Russian); English transl.: Math. USSR-Sb. 26 (1975), 493-554.
• [24] -, -, Approximation of periodic potentials by finite zone potentials, Vestnik Khar'kov. Gos. Univ. 205 (1980), 4-40, 139 (in Russian).
• [25] T. V. Misyura, Characterization of the spectra of the periodic and antiperiodic boundary value problems that are generated by the Dirac operator, Teor. Funktsii Funktsional. Anal. i Prilozhen. 30 (1978), 90-101 (in Russian).
• [26] -, Characteristics of the spectra of the periodic and antiperiodic boundary value problems that are generated by the Dirac operator. II, ibid. 31 (1979), 102-109, 168 (in Russian).
• [27] -, Finite-zone Dirac operators, ibid. 33 (1980), 107-111 (in Russian).
• [28] -, Approximation of the periodic potential of the Dirac operator by finite-zone potentials, ibid. 36 (1981), 55-65, 127 (in Russian).
• [29] H. McKean and E. Trubowitz, Hill's operator and hyperelliptic function theory in the presence of infginitely many branch points, Comm. Pure Appl. Math. 29 (1976), 143-226.
• [30] B. Mityagin, manuscript, 2000.
• [31] -, Convergence of expansions in eigenfunctions of the Dirac operator, Dokl. Akad. Nauk 393 (2003), 456-459 (in Russian).
• [32] -, Spectral expansions of one-dimensional periodic Dirac operators, Dynam. Partial Differential Equations 1 (2004), 125-191.
• [33] S. P. Novikov, The periodic problem for the Korteweg-de Vries equation, Funktsional. Anal. i Prilozhen. 8 (1974), no. 3, 54-66 (in Russian); English transl.: Funct. Anal. Appl. 8 (1975), 236-246.
• [34] J. J. Sansuc and V. Tkachenko, Spectral parametrization of non-selfadjoint Hill's operators, J. Differential Equations 125 (1996), 366-384.
• [35] V. Tkachenko, Spectral analysis of the nonselfadjoint Hill operator, Dokl. Akad. Nauk SSSR 322 (1992), 248-252 (in Russian); English transl.: Soviet Math. Dokl. 45 (1992), 78-82.
• [36] -, Discriminants and generic spectra of nonselfadjoint Hill's operators, in: Spectral Operator Theory and Related Topics, Adv. Soviet Math. 19, Amer. Math. Soc., Providence, RI, 1994, 41-71.
• [37] -, Non-selfadjoint periodic Dirac operators with finite-band spectra, Integral Equations Operator Theory 36 (2000), 325-348.
• [38] E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (1977), 321-342.