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On approximations of rank one ${\mathcal H}_{-2}$-perturbations

Authors: S. Albeverio, V. Koshmanenko, P. Kurasov and L. Nizhnik
Journal: Proc. Amer. Math. Soc. 131 (2003), 1443-1452
MSC (2000): Primary 47A55, 47B25; Secondary 81Q15
Published electronically: September 5, 2002
MathSciNet review: 1949874
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Abstract: Approximations of rank one ${\mathcal H}_{-2}$-perturbations of self-adjoint operators by operators with regular rank one perturbations are discussed. It is proven that in the case of arbitrary not semibounded operators such approximations in the norm resolvent sense can be constructed without any renormalization of the coupling constant. Approximations of semibounded operators are constructed using rank one non-symmetric regular perturbations.

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  • 1. V.M.Adamyan and B.S.Pavlov, Zero-radius potentials and M. G. Krein's formula for generalized resolvents, (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 149 (1986), Issled. Linein. Teor. Funktsii. XV,7-23 (translation in J. Soviet Math., 42 (1988), 1537-1550). MR 87h:47031
  • 2. N.I.Akhiezer and I.M.Glazman, Theory of linear operators in Hilbert space, vol. I,II, Piman, Boston, 1981 (Translation from Russian). MR 83i:47001a; MR 83i:47001b
  • 3. S.Albeverio, F.Gesztesy, R.Høegh-Krohn, and H.Holden, Solvable models in quantum mechanics, Springer, 1988. MR 90a:81021
  • 4. S.Albeverio and V.Koshmanenko, Singular rank one perturbations of self-adjoint operators and Krein theory of self-adjoint extensions, Potential Anal. 11 (1999), 279-287. MR 2000g:47011
  • 5. S.Albeverio and V.Koshmanenko, On form-sum approximations of singularly perturbed positive self-adjoint operators, J. Func. Anal., 169 (1999), 32-45. MR 2000j:47043
  • 6. S.Albeverio and P.Kurasov, Rank one perturbations, approximations and selfadjoint extensions, J. Func. Anal., 148 (1997), 152-169. MR 98g:47011
  • 7. S.Albeverio and P.Kurasov, Rank one perturbations of not semibounded operators, Integr. Equ. Oper. Theory, 27 (1997), 379-400. MR 98k:47025
  • 8. S.Albeverio and P.Kurasov, Finite rank perturbations and distribution theory, Proc. Amer. Math. Soc., 127 (1999), 1151-1161. MR 99m:47012
  • 9. S.Albeverio and P.Kurasov, Singular perturbations of differential operators. Solvable Schrödinger type operators, London Mathematical Society Lecture Note Series, 271, Cambridge University Press, Cambridge, 2000, xiv+429 pp. MR 2001g:47084
  • 10. S.Albeverio and L.Nizhnik, Approximation of general zero-range potentials, Ukrainian Math. J., 5 (2000). MR 2002c:81035
  • 11. A.Alonso and B.Simon, The Birman-Kre{\u{\i}}\kern.15emn-Vishik theory of selfadjoint extensions of semibounded operators, J. Operator Theory, 4 (1980), 251-270. MR 81m:47038
  • 12. F.A.Berezin and L.D.Faddeev, A remark on Schrödinger equation with a singular potential, Soviet Math. Dokl., 137 (1961), 1011-1014. MR 23:B2345
  • 13. J.Boman and P.Kurasov, Finite rank singular perturbations and distributions with discontinuous test functions, Proc. Amer. Math. Soc., 126 (1998), 1673-1683. MR 98g:47012
  • 14. Yu.N.Demkov and V.N.Ostrovsky, Zero-range potentials and their applications in atomic physics, Plenum, New York, 1988.
  • 15. J.F.van Diejen and A.Tip, Scattering from generalized point interactions using selfadjoint extensions in Pontryagin spaces, J. Math. Phys., 32 (1991), 630-641. MR 92e:81028
  • 16. A.Dijksma, H.Langer, Yu.Shondin, and C. Zeinstra, Self-adjoint operators with inner singularities and Pontryagin spaces, Operator theory and related topics, Vol. II (Odessa, 1997), 105-175, Oper. Theory Adv. Appl., 118, Birkhäuser, Basel, 2000. MR 2001f:47039
  • 17. F.Gesztesy and B.Simon, Rank-one perturbations at infinite coupling, J. Func. Anal., 128 (1995), 245-252. MR 95m:47014
  • 18. S.Hassi and Snoo, On rank one perturbations of selfadjoint operators, Integral Equations Operator Theory, 29 (1997), 288-300.MR 98k:47027
  • 19. S.Hassi, Snoo, and A.Willemsma, Smooth rank one perturbations of selfadjoint operators, Proc. Amer. Math. Soc., 126 (1998), 2663-2675. MR 98k:47028
  • 20. A.Kiselev and B.Simon, Rank one perturbations with infinitesimal coupling, J. Func. Anal., 130 (1995), 345-356. MR 96e:47012
  • 21. V. Koshmanenko, Towards the rank-one singular perturbations of self-adjoint operators, Ukrainian Math. J., 43 (1991), 1559-1566.
  • 22. V.Koshmanenko, Singular quadratic forms in perturbation theory, Kluwer, 1999. MR 2001a:47026
  • 23. M.G.Krein, On Hermitian operators whose deficiency indices are $ 1$, C. R. (Doklady) Acad. Sci URSS (N.S.), 43 (1944), 323-326. MR 6:131a
  • 24. M.G.Krein, On Hermitian operators whose deficiency indices equal to one. II, C. R. (Doklady) Acad. Sci URSS (N.S.), 44 (1944), 131-134. MR 6:179a
  • 25. P.Kurasov, Distribution theory for discontinuous test functions and differential operators with generalized coefficients, J. Math. Anal. Appl., 201 (1996), 297-323. MR 97g:46050
  • 26. P.Kurasov, $H_{-n}$-perturbations of self-adjoint operator and Krein's resolvent formula, Research Report N4, Stockholm Univ., 2001; accepted for publication in Integr. Eq. Oper. Theory.
  • 27. P.Kurasov and K.Watanabe, On rank one $H_{-3}$-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. MR 2001j:47012
  • 28. P.Kurasov and K.Watanabe, On $H_{-4}$-perturbations of self-adjoint operators, Operator Theory: Advances and Applications, 126 (2001), 179-196.
  • 29. S.T.Kuroda and H.Nagatani, ${\mathcal H}_{-2} $-construction and some applications, Mathematical results in quantum mechanics (Prague, 1998), 99-105, Oper. Theory Adv. Appl., 108, Birkhäuser, Basel, 1999. MR 2000f:47022
  • 30. L.Nizhnik, On point interactions in quantum mechanics, Ukrainian Math. J., 49 (1997), 1557-1560.
  • 31. B.S.Pavlov, The theory of extensions, and explicitly solvable models, (Russian), Uspekhi Mat. Nauk, 42 (1987), 99-131. MR 89b:47009
  • 32. B.S.Pavlov, Boundary conditions on thin manifolds and the semiboundedness of the three-body Schrödinger operator with point potential, (Russian), Mat. Sb. (N.S.), 136(178) (1988), 163-177 (translation in Math. USSR-Sb., 64 (1989), 161-175). MR 90g:35120
  • 33. Yu.Shondin, Quantum mechanical models in ${R}\sp n$ connected with extensions of the energy operator in a Pontryagin space, (Russian) Teoret. Mat. Fiz., 74 (1988), 331-344 (translation in Theoret. and Math. Phys., 74 (1988), 220-230). MR 89f:81042
  • 34. Yu.Shondin, Perturbation of elliptic operators on thin sets of high codimension, and extension theory in a space with an indefinite metric, (Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 222 (1995), Issled. po Linein. Oper. i Teor. Funktsii. 23, 246-292 (translation in J. Math. Sci. (New York) 87 (1997), 3941-3970). MR 97e:47084
  • 35. B.Simon, Spectral analysis of rank one perturbations and applications, in ``CRM Proceedings and Lecture Notes", 8 (1995), 109-149. MR 97c:47008

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Additional Information

S. Albeverio
Affiliation: Institute für Angewandte Mathematik, Univ. Bonn, Wegelerstr. 6, 53155 Bonn, Germany; SFB 256 Bonn, BiBoS, Bielefeld-Bonn, CERFIM, Locarno and USI (Switzerland)

V. Koshmanenko
Affiliation: Institute of Mathematics, vul. Tereschenkivs’ka, 3, Kyiv, 01601 Ukraine

P. Kurasov
Affiliation: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden
Address at time of publication: Department of Mathematics, Lund Institute of Technology, Box 118, 221 00 Lund, Sweden

L. Nizhnik
Affiliation: Institute of Mathematics, vul. Tereschenkivs’ka, 3, Kyiv, 01601 Ukraine

Keywords: Self-adjoint operators, singular interactions
Received by editor(s): July 19, 2001
Received by editor(s) in revised form: December 7, 2001
Published electronically: September 5, 2002
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2002 American Mathematical Society

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