Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Conformal spectral theory for the monodromy matrix

Author: Evgeny Korotyaev
Journal: Trans. Amer. Math. Soc. 362 (2010), 3435-3462
MSC (2010): Primary 34L40
Published electronically: February 2, 2010
MathSciNet review: 2601596
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For any $ N\times N$ monodromy matrix we define the Lyapunov function which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the Hill operator. The Lyapunov function has (real or complex) branch points, which we call resonances. We determine the asymptotics of the periodic, anti-periodic spectrum and of the resonances at high energy. We show that the endpoints of each gap are periodic (anti-periodic) eigenvalues or resonances (real branch points). Moreover, the following results are obtained: 1) We define the quasimomentum as an analytic function on the Riemann surface of the Lyapunov function; various properties and estimates of the quasimomentum are obtained. 2) We construct the conformal mapping with imaginary part given by the Lyapunov exponent, and we obtain various properties of this conformal mapping, which are similar to the case of the Hill operator. 3) We determine various new trace formulae for potentials and the Lyapunov exponent. 4) We obtain a priori estimates of gap lengths in terms of the Dirichlet integral. We apply these results to the Schrödinger operators and to first order periodic systems on the real line with a matrix-valued complex self-adjoint periodic potential.

References [Enhancements On Off] (What's this?)

  • [Ah] Akhiezer, N. The classical moment problem and some related questions in analysis. Hafner Publishing Co., New York, 1965. MR 0184042 (32:1518)
  • [A] Atkinson, F. Discrete and continuous boundary problems. Mathematics in Science and Engineering. Vol. 8, Academic Press, New York-London, 1964. MR 0176141 (31:416)
  • [BBK] Badanin, A.; Brüning, J.; Korotyaev, E. The Lyapunov function for Schrödinger operator with a periodic $ 2\times 2$ matrix potential. J. Funct. Anal. 234 (2006), 106-126. MR 2214141 (2006k:47090)
  • [BBKL] Badanin, A.; Brüning, J.; Korotyaev, E.; Lobanov, I. Schrödinger operators on armchair nanotubes, in preparation.
  • [BK] Badanin, A.; Korotyaev, E. Spectral asymptotics for periodic fourth-order operators. Int. Math. Res. Not. 45 (2005), 2775-2814. MR 2182471 (2006f:34064)
  • [CD1] F.Calogero; A.Degasperis, Nonlinear evolution equations solvable by the inverse spectral transform. II. Nuovo Cimento B (11) 39 (1977), no. 1, 1-54. MR 0456025 (56:14257)
  • [CD2] F.Calogero; A.Degasperis, Nonlinear evolution equations solvable by the inverse spectral transform. I. Nuovo Cimento B (11) 32 (1976), no. 2, 201-242. MR 0426642 (54:14581)
  • [C1] Carlson, R. A spectral transform for the matrix Hill's equation. Rocky Mountain J. Math. 34 (2004), no. 3, 869-895. MR 2087436 (2005e:34013)
  • [C2] Carlson, R. Eigenvalue estimates and trace formulas for the matrix Hill's equation. J. Differential Equations 167 (2000), no. 1, 211-244. MR 1785119 (2001e:34157)
  • [CK] Chelkak, D.; Korotyaev, E. Spectral estimates for Schrödinger operators with periodic matrix potentials on the real line. Int. Math. Res. Not. 2006, Art. ID 60314, 41 pp. MR 2219217 (2007g:47071)
  • [CG] Clark, S.; Gesztesy, F. Weyl-Titchmarsh $ M$-function asymptotics, local uniqueness results, trace formulas, and Borg-type theorems for Dirac operators. Trans. Amer. Math. Soc. 354 (2002), no. 9, 3475-3534. MR 1911509 (2003i:34191)
  • [CG1] Clark, S.; Gesztesy, F. Weyl-Titchmarsh $ M$-function asymptotics for matrix-valued Schrödinger operators, Proc. London Math. Soc. (3) 82 (2001), no. 3, 701-724. MR 1816694 (2002c:34144)
  • [CHGL] Clark S.; Holden H.; Gesztesy, F.; Levitan, B. Borg-type theorem for matrix-valued Schrödinger and Dirac operators. J. Diff. Eqs. 167 (2000), 181-210. MR 1785118 (2002d:34019)
  • [DS] Dunford, N.; Schwartz, J. Linear Operators. Part II. Spectral Theory, Interscience, New York, 1988. MR 1009163 (90g:47001b)
  • [Fo] Forster, O. Lectures on Riemann surfaces. Graduate Texts in Mathematics, 81. Springer-Verlag, New York, 1991. MR 1185074 (93h:30061)
  • [GT] Garnett, J.; Trubowitz, E. Gaps and bands of one dimensional periodic Schrödinger operators. II. Comment. Math. Helv. 62 (1987), 18-37. MR 882963 (88g:34028)
  • [GL] Gel'fand I.; Lidskii, V. On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients. (Russian) Uspehi Mat. Nauk (N.S.) 10 (1955), no. 1(63), 3-40. MR 0073767 (17:482g)
  • [Ge] Gel'fand, I. Expansion in characteristic functions of an equation with periodic coefficients. (Russian) Doklady Akad. Nauk SSSR (N.S.) 73 (1950), 1117-1120. MR 0039154 (12:503a)
  • [GKM] Gesztesy, F.; Kiselev A.; Makarov, K. Uniqueness results for matrix-valued Schrodinger, Jacobi, and Dirac-type operators. Math. Nachr. 239/240 (2002), 103-145. MR 1905666 (2003i:47047)
  • [GS] Gesztesy, F.; Sakhnovich, L. A. A class of matrix-valued Schrodinger operators with prescribed finite-band spectra. Reproducing kernel spaces and applications, 213-253, Oper. Theory Adv. Appl., 143, Birkhauser, Basel, 2003. MR 2019352 (2005g:47091)
  • [Ka] Kato, T. Perturbation theory for linear operators. Springer-Verlag, Berlin, 1995. MR 1335452 (96a:47025)
  • [KK] Kargaev, P.; Korotyaev, E. The inverse problem for the Hill operator, a direct approach. Invent. Math. 129, (1997), no. 3, 567-593. MR 1465335 (98i:34024)
  • [KK1] Kargaev, P.; Korotyaev, E. Identities for the Dirichlet integral of subharmonic functions from the Cartright class. Complex Var. Theory Appl. 50 (2005), no. 1, 35-50. MR 2114352 (2005i:30010)
  • [KK2] Kargaev, P.; Korotyaev, E. Effective masses and conformal mappings. Comm. Math. Phys. 169 (1995), no. 3, 597-625. MR 1328738 (96f:81021)
  • [Koo] Koosis, P. The logarithmic integral. I. Cambridge Univ. Press, Cambridge, London, New York, 1988. MR 961844 (90a:30097)
  • [K1] Korotyaev, E. The inverse problem for the Hill operator. I. Internat. Math. Res. Notices, 3 (1997), 113-125. MR 1434904 (98c:34137)
  • [K2] Korotyaev, E. Estimates of periodic potentials in terms of gap lengths. Comm. Math. Phys. 197 (1998), no. 3, 521-526. MR 1652779 (99h:34125)
  • [K3] Korotyaev, E. Spectral estimates for matrix-valued periodic Dirac operators. Asymptotic Analysis, 59 (2008), no. 3-4, 195-225. MR 2450359
  • [K4] Korotyaev, E. Marchenko-Ostrovki mapping for periodic Zakharov-Shabat systems. J. Differential Equations, 175 (2001), no. 2, 244-274. MR 1855971 (2002f:34206)
  • [K5] Korotyaev, E. Inverse Problem and Estimates for Periodic Zakharov-Shabat systems. J. Reine Angew. Math. 583 (2005), 87-115. MR 2146853 (2005m:35313)
  • [K6] Korotyaev, E. Characterization of the spectrum of Schrödinger operators with periodic distributions. Int. Math. Res. Not. (2003) no. 37, 2019-2031. MR 1995145 (2004e:34134)
  • [K7] Korotyaev, E. The estimates of periodic potentials in terms of effective masses. Comm. Math. Phys. 183 (1997), no. 2, 383-400. MR 1461964 (98f:34127)
  • [K8] Korotyaev, E. Metric properties of conformal mappings on the complex plane with parallel slits, Internat. Math. Res. Notices 10 (1996), 493-503. MR 1399414 (97e:30014)
  • [K9] Korotyaev, E. Estimates for the Hill operator. I. J. Differential Equations 162 (2000), 1-26. MR 1741871 (2000m:34194)
  • [K10] Korotyaev, E. Estimates for the Hill operator. II. J. Differential Equations 223 (2006), 229-260. MR 2214934 (2006k:34234)
  • [K11] Korotyaev, E. Inverse resonance scattering on the real line. Inverse Problems 21 (2005), 1-17. MR 2146179 (2006a:34033)
  • [KKu] Korotyaev, E.; Kutsenko, A. Lyapunov functions for periodic matrix-valued Jacobi operators, ``Spectral Theory of Differential Operators: M. Sh. Birman 80th Anniversary Collection''. AMS Translations, 225 (2008), no 2, 117-131.
  • [KL1] Korotyaev, E.; Lobanov, I. Operators on Zigzag Nanotubes. Annales Henri Poincare, 8 (2007), no. 6, 1151-1176. MR 2355344 (2008g:81076)
  • [KL2] Korotyaev, E.; Lobanov I. Zigzag periodic nanotube in magnetic field, preprint 2006.
  • [Kr] Krein, M. The basic propositions of the theory of $ \lambda$-zones of stability of a canonical system of linear differential equations with periodic coefficients. In memory of A. A. Andronov, pp. 413-498. Izdat. Akad. Nauk SSSR, Moscow, 1955. MR 0075382 (17:738c)
  • [MV] Maksudov, F.; Veliev, O. Spectral analysis of differential operators with periodic matrix coefficients. (Russian) Differentsial'nye Uravneniya 25 (1989), no. 3, 400-409, 547. MR 994320 (90d:47045)
  • [Ma] Manakov, S. On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Soviet JETP 38 (1974), 248-253.
  • [M] Marchenko V. Sturm-Liouville operator and applications. Birkhäuser, Basel, 1986. MR 897106 (88f:34034)
  • [MO] Marchenko V.; Ostrovski, I. A characterization of the spectrum of the Hill operator. Math. USSR Sb. 26 (1975), 493-554.
  • [RS] Reed, R. ; Simon, B. Methods of modern mathematical physics. IV. Analysis of operators. Academic Press, New York-London, 1978. MR 0493421 (58:12429c)
  • [Sh] Shen, Chao-Liang, Some eigenvalue problems for the vectorial Hill's equation. Inverse Problems 16 (2000), no. 3, 749-783. MR 1766223 (2001d:34025)
  • [Ti] Titchmarsh, E. Eigenfunction expansions associated with second-order differential equations. 2. Clarendon Press, Oxford, 1958. MR 0094551 (20:1065)
  • [YS] Yakubovich, V.; Starzhinskii, V. Linear differential equations with periodic coefficients. 1, 2. Halsted Press [John Wiley & Sons] New York-Toronto, 1975. Vol. 1, Vol. 2.
  • [Z] Zworski, M., Distribution of poles for scattering on the real line, J. Funct. Anal. 73 (1987), 277-296. MR 899652 (88h:81223)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 34L40

Retrieve articles in all journals with MSC (2010): 34L40

Additional Information

Evgeny Korotyaev
Affiliation: School of Mathematics, Cardiff University, Senghennydd Road, CF24 4AG Cardiff, Wales, United Kingdom
Address at time of publication: Saint Petersburg State University of Technology and Design, St. Petersburg, Bolshaya Morskaya 18, Russia

Received by editor(s): January 23, 2008
Published electronically: February 2, 2010
Additional Notes: This work was partially supported by EPSRC grant EP/D054621 and by DFG project BR691/23-1
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society