High order singular rank one perturbations of a positive operator

Integral Equations and Operator Theory

Volumn 53, Issue 2, 2005, Pages 209-245

  Dijksma, A ^a   Kurasov, P ^b   Shondin, Yu ^c  

^a UNIVERSITY OF GRONINGEN   (Netherlands)

^b LUND INSTITUTE OF TECHNOLOGY   (Sweden)

^c LOBACHEVSKY STATE UNIVERSITY OF NIZHNY NOVGOROD   (Russian Federation)

Author keywords

Defect function; Gelfand triple; Hilbert space; Pontryagin space; Q function; Rank one perturbation; Scale of Hilbert spaces; Self adjoint extension; Symmetric operator

Indexed keywords

EID: 25644438431     PISSN: 0378620X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00020-005-1357-5     Document Type: Article

References (38)

1. N.I. Achieser and I.M. Glasmann, Theorie der linearen Operatoren im Hilbertraum, Akademie Verlag, Berlin, 1981.
2. S. Albeverio and P. Kurasov, Rank one perturbations, approximations, and self-adjoint extensions, J. Funct. Anal. 148, 1997, 152-169.
3. S. Albeverio and P. Kurasov, Singular perturbations of differential operators, Cambridge Univ. Press, 2000 (London Mathematical Society Lecture Notes 271).
4. T.Ya. Azizov and I.S. Iokhvidov, Foundations of the theory of linear operators in spaces with an indefinite metric, Nauka, Moscow, 1986 (Russian);
5. English translation: Linear operators in spaces with an indefinite metric, Wiley, New York, 1989.
6. Yu.M. Berezanskii, Expansions in eigenfunctions of self-adjoint operators, Transl. Amer. Math. Soc. 17, Providence, Rhode Island, 1968.
7. F.A. Berezin, On the Lee model, Matem. Sborn. 60, 1963, 425-453 (Russian).
8. F.A. Berezin and L.D. Faddeev, Remark on the Schrödinger equation with singular potential, Dokl. Akad. Nauk SSSR 137, 1961, 1011-1014.
9. J. Bognar, Indefinite inner product spaces, Springer-Verlag, Berlin, 1974.
10. N. Bogolubuv, D. Shirkov, Introduction to theory of quantized fields, Interscience Publishers, New York, 1959.
11. W. Caspers, On point interactions, Thesis, Technical University Delft, 1992.
12. A. Dijksma and H. Langer, Operator theory and ordinary differential operators, Lecture Series 2 in: Albrecht Böttcher et al., Lectures on Operator theory and its applications, Fields Institute Monographs, Amer. Math. Soc., Providence RI, 1995, 73-139.
13. A. Dijksma, H. Langer, A. Luger, and Yu. Shondin, Minimal realizations of scalar generalized Nevanlinna functions related to their basic factorization, Operator Theory: Adv., Appl., vol. 154, Birkhäuser Verlag, Basel, 2004, 69-90.
14. A. Dijksma, H. Langer, Yu. Shondin, and C. Zeinstra, Self-adjoint operators with inner singularities and Pontryagin spaces, Operator Theory: Adv. Appl., vol. 118, Birkhäuser Verlag, Basel, 2000, 105-175.
15. A. Dijksma and Yu. Shondin, Singular point-like perturbations of the Bessel operator in a Pontryagin space, J. Diff. Equations 164, 2000, 49-91.
16. A. Dijksma and Yu. Shondin, Singular point-like perturbation of the Laguerre operator, Operator Theory: Adv., Appl., vol 132, Birkhäuser Verlag, Basel, 2002, 141-181.
17. A. Dijksma and H.S.V. de Snoo, Symmetric and self-adjoint relations in Krein spaces I, Operator Theory: Adv. Appl., vol. 24, Birkhäuser Verlag, Basel, 1987, 145-166.
18. J.F. van Diejen and A. Tip, Scattering from generalized point interaction using self-adjoint extensions in Pontryagin spaces, J. Math. Phys. 32(3), 1991, 630-641.
19. I.S. Iokhvidov, M.G. Krein, and H. Langer, Introduction to the spectral theory of operators in spaces with an indefinite metric, Mathematical Research, vol. 9, Academie-Verlag, Berlin, 1982.
20. A.A. Kiselev and I.Yu. Popov, Higher moments in the model of zero-width slits, Teor. Mat. Fiz. 89(1), 1991, 11-17 (Russian);
21. English translation: Theor. Math. Phys. 89, 1991, 1019-1024.
22. A. Kiselev and I. Popov, An indefinite metric and scattering by regions with a small aperture, Mat. Zametki 58, 1995, 837-850, 959.
23. k, Acta Sci. Math. (Szeged) 34, 1973, 191-230.
24. K zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr. 77, 1977, 187-236.
25. K, Operator Theory: Adv. Appl., vol. 17, Birkhäuser Verlag, Basel, 1986, 201-212.
26. -n-perturbations of self-adjoint operators and Krein's resolvent formula, Integr. Eq. Oper. Theory 45, 2003, 437-460.
27. P. Kurasov, Singular and supersingular perturbations: Hilbert space methods, Spectral theory of Schrdinger operators, 185-216, Contemp. Math., 340, Amer. Math. Soc., Providence, RI, 2004.
28. P. Kurasov and Yu. Pavlov, On field theory methods in singular perturbation theory, Lett. Math. Phys. 64 (2003), no. 2, 171-184.
29. -3-perturbations of positive self-adjoint operators, Stochastic processes, physics and geometry: new interplays, II (Leipzig, 1999), 413-422, CMS Conf. Proc., 29, Amer. Math. Soc., Providence, RI, 2000.
30. -4-perturbations of self-adjoint operators, Operator Theory: Adv. Appl., vol. 126, Birkhäuser Verlag, Basel, 2001, 179-196.
31. B. Pavlov, The theory of extensions and explicitly solvable models, Uspekhi Mat. Nauk 42, 1987, 99-131.
32. I.Yu. Popov, Helmholtz resonator and operator extension theory in spaces with indefinite metrics, Matem. Sb. 183 3, 1992, 3-37 (Russian);
33. English translation: Russian Acad. Sci. Sb Math. 75 2, 1993, 285-315.
34. n associated with extension of the energy operator in a Pontryagin space, Teor. Mat. Fiz. 74, 1988, 331-344 (Russian);
35. English translation: Theor. Math. Phys. 74, 1988, 220-230.
36. Yu.G. Shondin, Perturbation of differential operators on high-codimensional manifold and the extension theory for symmetric linear relations in an indefinite metric space, Teor. Mat. Fiz. 92(3), 1992, 466-472 (Russian);
37. English translation: Theor. Math. Phys. 92, 1992, 1032-1037.
38. B. Simon, Spectral analysis of rank one perturbations and applications, in: Mathematical quantum theory. II. Schrödinger operators (Vancouver, BC, 1993), 109-149, CRM Proc. Lecture Notes 8, AMS, Providence, RI, 1995.