Uniqueness theorem and singular spectrum in the Friedrichs model near a singular point

Yakovlev, S.

doi:10.1090/S1061-0022-03-00807-0

Uniqueness theorem and singular spectrum in the Friedrichs model near a singular point
HTML articles powered by AMS MathViewer

by S. I. Yakovlev
Translated by: the author

St. Petersburg Math. J. 15 (2004), 149-164

DOI: https://doi.org/10.1090/S1061-0022-03-00807-0

Published electronically: December 31, 2003

PDF | Request permission

Abstract:

A uniqueness theorem is proved for a class of analytic functions with positive imaginary part that admit representation in a special form. This theorem imposes some restrictions on the character of decay of these functions in the vicinity of their zeros. As an application, the density of the point spectrum and the singular continuous spectrum are described for selfadjoint operators in the Friedrichs model near a singular point.

References

G. A. Miller, Groups which contain ten or eleven proper subgroups, Proc. Nat. Acad. Sci. U.S.A. 25 (1939), 540–543. MR 31, DOI 10.1073/pnas.25.10.540
Julian Bonder, Über die Darstellung gewisser, in der Theorie der Flügelschwingungen auftretender Integrale durch Zylinderfunktionen, Z. Angew. Math. Mech. 19 (1939), 251–252 (German). MR 42, DOI 10.1002/zamm.19390190409
R. H. J. Germay, Généralisation de l’équation de Hesse, Ann. Soc. Sci. Bruxelles Sér. I 59 (1939), 139–144 (French). MR 86
S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089
John W. Green, Harmonic functions in domains with multiple boundary points, Amer. J. Math. 61 (1939), 609–632. MR 90, DOI 10.2307/2371316
W. J. Trjitzinsky, General theory of singular integral equations with real kernels, Trans. Amer. Math. Soc. 46 (1939), 202–279. MR 92, DOI 10.1090/S0002-9947-1939-0000092-6
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; English transl., Funct. Anal. Appl. 24 (1990), no. 4, 338–340 (1991).
S. I. Yakovlev, On the structure of the singular spectrum in a selfadjoint Friedrichs model, Leningrad, 1991. (Manuscript deposited in VINITI, no. 2050-V, 17.05.91). (Russian)
Erich Rothe, Topological proofs of uniqueness theorems in the theory of differential and integral equations, Bull. Amer. Math. Soc. 45 (1939), 606–613. MR 93, DOI 10.1090/S0002-9904-1939-07048-1
Lawrence M. Graves, The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5 (1939), 656–660. MR 99
Lawrence M. Graves, The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5 (1939), 656–660. MR 99
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
T. Radó and P. Reichelderfer, On a stretching process for surfaces, Amer. J. Math. 61 (1939), 645–650. MR 47, DOI 10.2307/2371318
I. M. Sheffer, Some properties of polynomial sets of type zero, Duke Math. J. 5 (1939), 590–622. MR 81
Julian Bonder, Über die Darstellung gewisser, in der Theorie der Flügelschwingungen auftretender Integrale durch Zylinderfunktionen, Z. Angew. Math. Mech. 19 (1939), 251–252 (German). MR 42, DOI 10.1002/zamm.19390190409