St. Petersburg Mathematical Society

On Instability of the absolutely continuous spectrum of dissipative Schrödinger operators and Jacobi matrices

Author(s): R. Romanov
Translated by: the author
Original publication: Algebra i Analiz, tom 17 (2005), vypusk 2.
Journal: St. Petersburg Math. J. 17 (2006), 325-341.
MSC (2000): Primary 34L10, 47B44
Posted: February 20, 2006
Retrieve article in: PDF DVI PostScript

Abstract: The absence of the absolutely continuous spectrum is proved for dissipative Schrödinger operators and Jacobi matrices with slowly decaying imaginary part of the potential.

1.: P. Deift and R. Killip, On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials, Comm. Math. Phys. 203 (1999), 341-347. MR 1697600 (2000i:34223)
2.: M. Christ and A. Kiselev, Scattering and wave operators for one-dimensional Schrödinger operators with slowly decaying nonsmooth potentials, Geom. Funct. Anal. 12 (2002), 1174-1234. MR 1952927 (2003m:47019)
3.: S. N. Naboko, Functional model of perturbation theory and its applications to scattering theory, Trudy Mat. Inst. Steklov. 147 (1980), 86-114; English transl. in Proc. Steklov Inst. Math. 1981, no. 2. MR 0573902 (82d:47012)
4.: -, On the conditions for existence of wave operators in the nonselfadjoint case, Wave Propagation. Scattering Theory, Probl. Mat. Fiz., vyp. 12, Leningrad. Univ., Leningrad, 1987, pp. 132-155; English transl., Amer. Math. Soc. Transl. Ser. 2, vol. 157, Amer. Math. Soc., Providence, RI, 1993, pp. 127-149. MR 0923975 (90b:47019)
5.: B. Sz.-Nagy and C. Foias, Analyse harmonique des opérateurs de l $^{\prime }$ espace de Hilbert, Masson et Cie, Paris; Akad. Kiadó, Budapest, 1967. MR 0225183 (37:778)
6.: B. S. Pavlov, Expansion in eigenfunctions of the absolutely continuous spectrum of a dissipative operator, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1975, vyp. 1, 130-137; English transl. in Vestnik Leningrad Univ. Math. 8 (1980). MR 0425657 (54:13611)
7.: L. A. Sakhnovich, Dissipative operators with absolutely continuous spectrum, Trudy Moskov. Mat. Obshch. 19 (1968), 211-270; English transl., Trans. Moscow Math. Soc. 19 (1968), 233-297. MR 0250109 (40:3350)
8.: B. S. Pavlov, Selfadjoint dilation of a dissipative Schrödinger operator, and expansion in its eigenfunction, Mat. Sb. (N.S.) 102 (144) (1977), no. 4, 511-536; English transl., Math. USSR-Sb. 31 (1977), no. 4, 457-478. MR 0510053 (58:23154)
9.: A. S. Tikhonov, Absolutely continuous spectrum of a linear operator and problems of scattering theory and of factorization of operator-functions, Ph. D. thesis, Simferopol. Gos. Univ., Simferopol, 1989. (Russian)
10.: R. Romanov, A remark on equivalence of weak and strong definitions of the absolutely continuous subspace for nonself-adjoint operators, Oper. Theory Adv. Appl., vol. 154, Birkhäuser, Basel, 2004, pp. 179-184. MR 2103375 (2005g:47064)
11.: A. R. Sims, Secondary conditions for linear differential operators of the second order, J. Math. Mech. 6 (1957), no. 2, 247-285. MR 0086224 (19:144g)
12.: M. S. P. Eastham, The asymptotic solution of linear differential systems, London Math. Soc. Monogr. New Ser., vol. 4, The Clarendon Press, Oxford Univ. Press, New York, 1989. MR 1006434 (91d:34001)
13.: D. B. Pearson, Quantum scattering and spectral theory, Techn. Phys., vol. 9, Acad. Press, London, 1988. MR 1099604 (91k:81198)
14.: Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999), no. 2, 329-367. MR 1666767 (2000f:47060)
15.: B. Simon, Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators, Proc. Amer. Math. Soc. 124 (1996), 3361-3369. MR 1350963 (97a:34223)
16.: N. K. Nikol'skii, Treatise of the shift operator. Spectral functionl theory, ``Nauka'', Moscow, 1980; English transl., Grundlehren Math. Wiss., vol. 273, Springer-Verlag, Berlin, 1986. MR 0827223 (87i:47042).

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 34L10, 47B44

R. Romanov
Affiliation: School of Computer Science, Cardiff University, Cardiff, Queen's Buildings, PO Box 916, Newport Road, Cardiff CF24 3XF, United Kingdom, and Laboratory of Quantum Networks, Institute for Physics, St. Petersburg State University, St. Petersburg 198504, Russia
Email: roma@rvr.stud.pu.ru

DOI: 10.1090/S1061-0022-06-00907-1
PII: S 1061-0022(06)00907-1
Keywords: Absolutely continuous subspace, dissipative operators, Schr\"odinger operator
Received by editor(s): 14/JAN/2004
Posted: February 20, 2006
Additional Notes: The author was supported in part by EPSRC Grant GR/R20885 and by RFBR (grant no. 00-01-00479).
Copyright of article: Copyright 2006, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN: 1547-7371(e) ISSN: 1061-0022(p)

Previous issue \| Table of contents \| Next issue Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS