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Singular Weyl–Titchmarsh–Kodaira theory for one-dimensional Dirac operators

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Abstract

We develop singular Weyl–Titchmarsh–Kodaira theory for one-dimensional Dirac operators. In particular, we establish existence of a spectral transformation as well as local Borg–Marchenko and Hochstadt–Lieberman type uniqueness results. Finally, we give some applications to the case of radial Dirac operators.

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References

  1. Albeverio, S., Hryniv, R., Mykytyuk, Ya.: Inverse spectral problems for Dirac operators with summable potentials. Russ. J. Math. Phys. 12, 406–423 (2005)

    MATH  MathSciNet  Google Scholar 

  2. Albeverio, S., Hryniv, R., Mykytyuk, Ya.: Reconstruction of radial Dirac operators. J. Math. Phys. 48, 043501 (2007)

    Article  MathSciNet  Google Scholar 

  3. Albeverio, S., Hryniv, R., Mykytyuk, Ya.: Reconstruction of radial Dirac and Schrödinger operators from two spectra. J. Math. Anal. Appl. 339, 45–57 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Balinsky, A.A., Evans, W.D.: Spectral Analysis of Relativistic Operators. Imperial College Press, London (2011)

    MATH  Google Scholar 

  5. Bennewitz, C.: A proof of the local Borg-Marchenko theorem. Commun. Math. Phys. 218, 131–132 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Borg, G.: Uniqueness theorems in the spectral theory of \(y^{\prime \prime }+(\lambda -q(x))y=0\). Den 11te Skandinaviske Matematikerkongress, Trondheim, 1949, pp. 276–287. Johan Grundt Tanums Forlag, Oslo (1952)

  7. Clark, S., Gesztesy, F.: Weyl–Titchmarsh \(M\)-function asymptotics, local uniqueness results, trace formulas, and Borg-type theorems for Dirac operators. Trans. Am. Math. Soc. 358, 3475–3534 (2002)

    Article  MathSciNet  Google Scholar 

  8. Brunnhuber, R.: Weyl-Titchmarsh-Kodaira theory for Dirac operators with strongly singular potentials, master thesis, University of Vienna (2012). http://othes.univie.ac.at/20726/

  9. Eckhardt, J.: Inverse uniqueness results for Schrödinger operators using de Branges theory. Complex Anal. Oper. Theory (2013, to appear). doi:10.1007/s11785-012-0265-3

  10. Eckhardt, J.: Direct and inverse spectral theory of singular left-definite Sturm–Liouville operators. J. Differ. Equ. 253(2), 604–634 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Eckhardt, J., Gesztesy, F., Nichols, R., Teschl, G.: Supersymmetry and Schrödinger-type operators with distributional matrix-valued, potentials, arxiv:1206.4966

  12. Eckhardt, J., Gesztesy, F., Nichols, R., Teschl, G.: Weyl–Titchmarsh theory for Sturm-Liouville operators with distributional potential potentials. Opuscula Math. 33(3), 467–563 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. Eckhardt, J., Gesztesy, F., Nichols, R., Teschl, G.: Inverse spectral theory for Sturm–Liouville operators with distributional potentials, J. Lond. Math. Soc. (2) (2013, to appear). doi:10.1112/jlms/jdt041. arxiv:1210.7628

  14. Eckhardt, J., Kostenko, A., Teschl, G.: Inverse uniqueness results for one-dimensional weighted Dirac operators. In: Khruslov, E., Pastur, L. (eds.) Spectral Theory and Differential Equations. Advances in Mathematical Sciences. Amer. Math. Soc., Providence (2013, to appear). arxiv:1305.3100

  15. Eckhardt, J., Kostenko, A., Teschl, G.: Spectral asymptotics for one-dimensional weighted Dirac operators (2013, in preparation)

  16. Eckhardt, J., Teschl, G.: Uniqueness results for one-dimensional Schrödinger operators with purely discrete spectra. Trans. Am. Math. Soc. 365, 3923–3942 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  17. Eckhardt, J., Teschl, G.: Singular Weyl-Titchmarsh-Kodaira theory for Jacobi operators. Oper. Matrices (2013, to appear)

  18. Fulton, C.: Titchmarsh–Weyl \(m\)-functions for second order Sturm–Liouville problems. Math. Nachr. 281, 1417–1475 (2008)

    Article  MathSciNet  Google Scholar 

  19. Fulton, C., Langer, H.: Sturm–Liouville operators with singularities and generalized Nevanlinna functions. Complex Anal. Oper. Theory 4, 179–243 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  20. Fulton, C., Langer, H., Luger, A.: Mark Krein’s method of directing functionals and singular potentials. Math. Nachr. 285, 1791–1798 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gesztesy, F., Kiselev, A., Makarov, K.A.: Uniqueness results for matrix-valued Schrödinger, Jacobi, and Dirac-type operators. Math. Nachr. 239–240, 103–145 (2002)

    Article  MathSciNet  Google Scholar 

  22. Gesztesy, F., Zinchenko, M.: On spectral theory for Schrödinger operators with strongly singular potentials. Math. Nachr. 279, 1041–1082 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Gitman, D.M., Tyutin, I.V., Voronov, B.L.: Self-adjoint Extensions in Quantum Mechanics. Birkhäuser, New York (2012)

    Book  MATH  Google Scholar 

  24. Hinton, D.B., Shaw, J.K.: On Titchmarsh-Weyl \(M(\lambda )\)-functions for linear Hamiltonian systems. J. Differ. Equ. 40(3), 316–342 (1981)

    Google Scholar 

  25. Hinton, D.B., Shaw, J.K.: On the spectrum of a singular Hamiltonian system. Quaestiones Math. 5(1), 29–81 (1982/1983)

    Google Scholar 

  26. Hinton, D.B., Shaw, J.K.: Hamiltonian systems of limit point or limit circle type with both endpoints singular. J. Differ. Equ. 50(3), 444–464 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  27. Hinton, D.B., Shaw, J.K.: On boundary value problems for Hamiltonian systems with two singular points. SIAM J. Math. Anal. 15(2), 272–286 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  28. Hinton, D.B., Shaw, J.K.: On the spectrum of a singular Hamiltonian system, II. Quaestiones Math. 10(1), 1–48 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  29. Hinton, D.B., Schneider, A.: On the Titchmarsh-Weyl coefficients for singular \(S\)-Hermitian Systems I. Math. Nachr. 163, 323–342 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  30. Hinton, D.B., Schneider, A.: On the Titchmarsh-Weyl coefficients for singular \(S\)-Hermitian Systems II. Math. Nachr. 185, 67–84 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  31. Hochstadt, H., Lieberman, B.: An inverse Sturm-Liouville problem with mixed given data. SIAM J. Appl. Math. 34, 676–680 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  32. Kac, I.S.: The existence of spectral functions of generalized second order differential systems with boundary conditions at the singular end. AMS Transl. 62(2), 204–262 (1967)

    Google Scholar 

  33. Kodaira, K.: The eigenvalue problem for ordinary differential equations of the second order and Heisenberg’s theory of \(S\)-matrices. Am. J. Math. 71, 921–945 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  34. Kostenko, A., Sakhnovich, A., Teschl, G.: Inverse eigenvalue problems for perturbed spherical Schrödinger operators. Inverse Problems 26, 105013 (2010)

    Article  MathSciNet  Google Scholar 

  35. Kostenko, A., Sakhnovich, A., Teschl, G.: Weyl–Titchmarsh theory for Schrödinger operators with strongly singular potentials. Int. Math. Res. Not. 2012, 1699–1747 (2012)

    MATH  MathSciNet  Google Scholar 

  36. Kostenko, A., Sakhnovich, A., Teschl, G.: Commutation methods for Schrödinger operators with strongly singular potentials. Math. Nachr. 285, 392–410 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  37. Kostenko, A., Teschl, G.: On the singular Weyl-Titchmarsh function of perturbed spherical Schrödinger operators. J. Differ. Equ. 250, 3701–3739 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  38. Kostenko, A., Teschl, G.: Spectral asymptotics for perturbed spherical Schrödinger operators and applications to quantum scattering. Comm. Math. Phys. 322, 255–275 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  39. Kurasov, P., Luger, A.: An operator theoretic interpretation of the generalized Titchmarsh–Weyl coefficient for a singular Sturm–Liouville problem. Math. Phys. Anal. Geom. 14, 115–151 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  40. Levin, B.Ya.: Lectures on Entire Functions. Transl. Math. Monographs, vol. 150. Amer. Math. Soc., Providence (1996)

  41. Levitan, B.M., Sargsjan, I.S.: Sturm–Liouville and Dirac Operators. Kluwer, Dordrecht (1991)

    Book  Google Scholar 

  42. Lesch, M., Malamud, M.M.: The inverse spectral problem for first order systems on the half line. Oper. Theory Adv. Appl. 117, 199–238 (2000)

    MathSciNet  Google Scholar 

  43. Malamud, M.M.: Uniqueness questions in inverse problems for systems of differential equations on a finite interval. Trans. Moscow Math. Soc. 60, 173–224 (1999)

    MathSciNet  Google Scholar 

  44. Malamud, M.M.: Borg type theorems for first-order systems on a finite interval. Funct. Anal. Appl. 33, 64–68 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  45. Marčenko, V.A.: Marčenko, Some questions of the theory of one-dimensional linear differential operators of the second order. I, Trudy Moskov. Mat. Obšč. 1, 327–340 (1952).

  46. Mykytyuk, Ya V., Puyda, D.V.: Inverse spectral problems for Dirac operators on a finite interval. J. Math. Anal. Appl 386, 177–194 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  47. Olver, F.W.J., et al.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  48. Puyda, P.V.: Inverse spectral problems for Dirac operators with summable matrix-valued potentials. Integr. Equ. Oper. Theory 74, 417–450 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  49. Sakhnovich, A.: Dirac type and canonical systems: spectral and Weyl–Titchmarsh matrix functions, direct and inverse problems. Inverse Problems 18(2), 331–348 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  50. Sakhnovich, A.: Skew-self adjoint discrete and continuous Dirac-type systems: inverse problems and Borg–Marchenko theorems. Inverse Problems 22(6), 2083–2101 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  51. Sakhnovich, A.L., Sakhnovich, L.A., Roitberg, I.Ya.: Inverse Problems and Nonlinear Evolution Equations. Solutions, Darboux Matrices and Weyl–Titchmarsh Functions. Walter de Gruyter, Berlin (2013)

  52. Serier, F.: Inverse spectral problem for singular Ablowitz–Kaup–Newell–Segur operators on \([0, 1]\). Inverse Problems 22, 1457–1484 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  53. Stadler, R., Teschl, G.: Relative oscillation theory for Dirac operators. J. Math. Anal. Appl. 371, 638–648 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  54. Teschl, G.: Renormalized oscillation theory for Dirac operators. Proc. Am. Math. Soc. 126, 1685–1695 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  55. Teschl, G.: Mathematical Methods in Quantum Mechanics. With Applications to Schrödinger Operators. Am. Math. Soc, Providence (2009)

    MATH  Google Scholar 

  56. Teschl, G.: Ordinary Differential Equations and Dynamical Systems. Amer. Math. Soc, Providence (2012)

    MATH  Google Scholar 

  57. Thaller, B.: The Dirac Equation. Springer, Berlin (1991)

    MATH  Google Scholar 

  58. Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1944)

    MATH  Google Scholar 

  59. Weidmann, J.: Spectral Theory of Ordinary Differential Operators. In: Lecture Notes in Mathematics, vol. 1258. Springer, Berlin (1987)

  60. Weidmann, J.: Lineare Operatoren in Hilberträumen, Teil 2: Anwendungen. B.G. Teubner, Stuttgart (2003)

    Book  Google Scholar 

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Acknowledgments

We thank Fritz Gesztesy and Alexander Sakhnovich for hints with respect to the literature. J.E. and G.T. gratefully acknowledge the stimulating atmosphere at the Institut Mittag-Leffler during spring 2013 where parts of this paper were written during the international research program on Inverse Problems and Applications.

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Authors and Affiliations

  1. Institute of Mathematics, University of Klagenfurt, Universitätsstraße 65-67, 9020 , Klagenfurt, Austria

    Rainer Brunnhuber

  2. Institut Mittag-Leffler, Auravägen 17, 182 60 , Djursholm, Sweden

    Jonathan Eckhardt

  3. Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 , Wien, Austria

    Aleksey Kostenko & Gerald Teschl

  4. International Erwin Schrödinger Institute for Mathematical Physics, Oskar-Morgenstern-Platz 1, 1090 , Wien, Austria

    Gerald Teschl

Authors
  1. Rainer Brunnhuber
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  2. Jonathan Eckhardt
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  3. Aleksey Kostenko
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  4. Gerald Teschl
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Correspondence to Gerald Teschl.

Additional information

Communicated by A. Constantin.

Research supported by the Austrian Science Fund (FWF) under Grant No. Y330 and M1309 as well as by the AXA Mittag-Leffler Fellowship Project, funded by the AXA Research Fund.

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Brunnhuber, R., Eckhardt, J., Kostenko, A. et al. Singular Weyl–Titchmarsh–Kodaira theory for one-dimensional Dirac operators. Monatsh Math 174, 515–547 (2014). https://doi.org/10.1007/s00605-013-0563-5

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