Description of the structure of singular spectrum for Friedrichs model operator near singular point (original) (raw)

2001, International Journal of Mathematics and Mathematical Sciences

Abstract

The study of the point spectrum and the singular continuous one is reduced to investigating the structure of the real roots set of an analytic function with positive imaginary partM(λ). We prove a uniqueness theorem for such a class of analytic functions. Combining this theorem with a lemma on smoothness ofM(λ)near its real roots permits us to describe the density of the singular spectrum.

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References (15)

  1. L. Carleson, Sets of uniqueness for functions regular in the unit circle, Acta Math. 87 (1952), 325-345. MR 14,261a. Zbl 046.30005.
  2. E. M. Dynkin, S. N. Naboko, and S. I. Yakovlev, A finiteness bound for the singular spec- trum in a selfadjoint Friedrichs model, Algebra i Analiz 3 (1991), no. 2, 77-90. MR 93d:47028. Zbl 764.47001.
  3. L. D. Faddeev, On a model of Friedrichs in the theory of perturbations of the con- tinuous spectrum, Trudy Mat. Inst. Steklov 73 (1964), 292-313. MR 31#2620. Zbl 148.12801.
  4. R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for the conju- gate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227-251. MR 47#701. Zbl 262.44004.
  5. P. Koosis, Introduction to H p Spaces, London Mathematical Society Lecture Note Series, vol. 40, Cambridge University Press, Cambridge, 1980. MR 81c:30062. Zbl 435.30001.
  6. Y. V. Mikityuk, The singular spectrum of selfadjoint operators, Dokl. Akad. Nauk SSSR 303 (1988), no. 1, 33-36, translation in Soviet Math. Dokl. 38 (1989), no. 3, 472-475. MR 90f:47032. Zbl 733.47025.
  7. S. N. Naboko, Functional model of perturbation theory and its applications to scattering theory, Trudy Mat. Inst. Steklov. 147 (1980), 86-114, 203, English translation in Proc. Steklov Inst. Math. (1981), no. 2, 85-116. MR 82d:47012. Zbl 445.47010.
  8. Uniqueness theorems for operator-valued functions with positive imaginary part and the singular spectrum in the selfadjoint Friedrichs model, Dokl. Akad. Nauk SSSR 275 (1984), no. 6, 1310-1313. MR 86a:47027. Zbl 587.47004.
  9. Uniqueness theorems for operator-valued functions with positive imaginary part, and the singular spectrum in the selfadjoint Friedrichs model, Ark. Mat. 25 (1987), no. 1, 115-140. MR 89e:47028. Zbl 632.47012.
  10. S. N. Naboko and S. I. Yakovlev, Conditions for the finiteness of the singular spectrum in a selfadjoint Friedrichs model, Funktsional. Anal. i Prilozhen. 24 (1990), no. 4, 88-89. CMP 1 092 812. Zbl 724.47010.
  11. B. S. Pavlov, On a nonselfadjoint Schrödinger operator. III, Problems of Mathematical Physics, No. 3: Spectral Theory (Russian), Izdat. Leningrad. Univ., Leningrad, 1968, pp. 59-80. MR 50#1052.
  12. B. S. Pavlov and S. V. Petras, The singular spectrum of a weakly perturbed multiplica- tion operator, Funkcional. Anal. i Priložen. 4 (1970), no. 2, 54-61. MR 42#892. Zbl 206.43801.
  13. S. I. Yakovlev, Perturbations of a singular spectrum in a selfadjoint Friedrichs model, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. (1990), no. 1, 116-117, transla- tion in Vestnik Leningrad Univ. Math. 23 (1990), no. 1, 73-75. MR 92e:47036. Zbl 715.47009.
  14. On the structure of the singular spectrum in selfadjoint Friedrichs model, Depon. VINITI 2050-B (1991), 1-25.
  15. A finiteness bound for the singular spectrum in a neighborhood of a singular point of operators of the Friedrichs model, Algebra i Analiz 10 (1998), no. 4, 210-237, translation in St. Petersburg Math. J. 10 (1999), no. 4, 715-731. MR 99k:47058.