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On Field Theory Methods in Singular Perturbation Theory

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Abstract

Singular and supersingular finite rank perturbations of self-adjoint operators are studied using methods from renormalization theory for quantum fields. It is shown that the ideas from dimensional and Pauli–Villars regulatizations can be applied to determine uniquely certain finite rank supersingular perturbations. Approach is based on the regularization of homogeneous singular quadratic forms.

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References

  1. Albeverio, S., Dabrowski, L. and Kurasov, P.: Symmetries of Schrödinger operators with point interactions, Lett. Math. Phys. 45 (1998), 33–47.

    Google Scholar 

  2. Albeverio, S., Gesztesy, F., Høegh-Krohn, R. and Holden, H.: Solvable Models in Quantum Mechanics, Springer, New York, 1988.

    Google Scholar 

  3. Albeverio, S., Koshmanenko, V., Kurasov, P. and Nizhnik, L.: On approximations of rank one H -2-perturbations, Proc. Amer. Math. Soc. 131 (2003), 1443–1452.

    Google Scholar 

  4. Albeverio, S. and Kurasov, P.: Rank one perturbations, approximations, and selfadjoint extensions, J. Funct. Anal. 148 (1997), 152–169.

    Google Scholar 

  5. Albeverio, S. and Kurasov, P.: Rank one perturbations of not semibounded operators, Integral Equations Operator Theory 27 (1997), 379–400.

    Google Scholar 

  6. Albeverio, S. and Kurasov, P.: Finite rank perturbations and distribution theory, Proc. Amer. Math. Soc. 127 (1999), 1151–1161.

    Google Scholar 

  7. Albeverio, S. and Kurasov, P.: Singular Perturbations of Differential Operators: Solvable Schrödinger Type Operators, Cambridge Univ. Press, Cambridge, 2000.

    Google Scholar 

  8. Berezin, F. A. and Faddeev, L. D.: A remark on Schrö dinger equation with a singular potential, Dokl. Akad. Nauk. SSSR. 137 (1961), 1011–1014 [English transl.: Soviet Math. Dokl. 2 (1961), 372–375].

    Google Scholar 

  9. Bogoliubov, N. N. and Shirkov, D. V.: Introduction to the Theory of Quantized Fields, Wiley, New York, 1980.

    Google Scholar 

  10. Boman, J. and Kurasov, P.: Finite rank singular perturbations and distributions with discontinuous test functions, Proc. Amer. Math. Soc. 126 (1998), 1673–1683.

    Google Scholar 

  11. Collins, J. C.: Renormalization, Cambridge Univ. Press, Cambridge, 1984.

    Google Scholar 

  12. Demkov, Yu. N. and Ostrovsky, V. N.: Zero-range Potentials and their Applications in Atomic Physics, Plenum Press, New York 1988.

    Google Scholar 

  13. Van Diejen, J. F. and Tip, A.: Scattering from generalized point interaction using selfadjoint extensions in Pontryagin spaces, J. Math. Phys. 32(3) (1991), 630–641.

    Google Scholar 

  14. Dijskma, A., Kurasov, P. and Shondin, Yu.: High order singular perturbations, in preparation.

  15. Dijksma, A., Langer, H., Shondin, Yu. and Zeinstra, C.: Self-adjoint operators with inner singularities and Pontryagin spaces, Operator Theory and Related Topics, Vol. II (Odessa, 1997), Operator Theory Adv. Appl. 118, Birkhäuser, Basel, 2000, pp. 105–175.

    Google Scholar 

  16. Dijksma, A. and Shondin, Yu.: Singular point-like perturbations of the Bessel operator in a Pontryagin space, J. Differential Equations 164 (2000), 49–91.

    Google Scholar 

  17. Dijksma, A. and de Snoo, H. S. V.: Symmetric and selfadjoint relations in Krein spaces I, Operator Theory 24, Birkhäuser, Basel, 1987, pp. 145–166.

    Google Scholar 

  18. Faddeev, L. D.: private communication.

  19. 't Hooft, G. and Veltman, M.: Regularization and renormalization of gauge fields, Nuclear. Phys. B 44, 189 (1972).

    Google Scholar 

  20. Kiselev, A. and Simon, B.: Rank one perturbations with infinitesimal coupling, J. Funct. Anal. 130 (1995), 345–356.

    Google Scholar 

  21. Krein, M. G. and Langer, H.: Ñber die Q-Funktion eines π-hermiteschen Operators in Raume ΠK, Acta Sci. Math. (Szeged) 34 (1973), 191–230.

    Google Scholar 

  22. Krein, M. G. and Langer, H.: Ñber einige Fortzetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume ΠK zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr. 77 (1977), 187–236.

    Google Scholar 

  23. Krein, M. G. and Langer, H.: Some propositions on analytic matrix functions related to the theory of operators on the space ΠK, Acta Sci. Math. (Szeged) 43 (1981), 181–205.

    Google Scholar 

  24. Kurasov, P.: Distribution theory for discontinuous test functions and differential operators with generalized coefficients, J. Math. Anal. Appl. 201 (1996), 297–323.

    Google Scholar 

  25. Kurasov, P.: H -n-perturbations of self-adjoint operators and Krein's resolvent formula, Integral Equation Operators Theory 45 (2003), 437–460.

    Google Scholar 

  26. Kurasov, P. and Watanabe, K.: On rank one H -3-perturbations of positive self-adjoint operators, Stochastic Processes, Physics and Geometry: NewInterplays, II (Leipzig, 1999), CMS Conf. Proc., 29, Amer. Math. Soc., Providence, RI, 2000, pp. 413–422.

    Google Scholar 

  27. Kurasov, P. and Watanabe, K.: On H -4-perturbations of self-adjoint operators, Partial Differential Equations and Spectral Theory (Clausthal, 2000), Operator Theory Adv. Appl. 126, Birkhäuser, Basel, 2001, pp. 179–196.

    Google Scholar 

  28. Langer, H.: A characterization of generalized zeros of negative type of functions of the class N K, In: Operator Theory: Adv. Appl. 17, Birkhäuser, Basel, 1986, pp. 201–212.

    Google Scholar 

  29. Pauli, W. and Villars, F.: On the invariant regularization in relativistic quantum theory, Rev. Mod. Phys. 21 (1949), 434–444.

    Google Scholar 

  30. Pavlov, B.: The theory of extensions and explicitly solvable models, Uspekhi Mat. Nauk 42 (1987), 99–131.

    Google Scholar 

  31. Pavlov, B. S. and Popov, I.: Scattering by resonators with small and point holes, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 3 (1984), 116–118.

    Google Scholar 

  32. Pavlov, B. S. and Popov, I.: An acoustic model of zero-width slits and the hydrodynamic stability of a boundary layer, Teoret. Mat. Fiz. 86 (1991), 391–401.

    Google Scholar 

  33. Pavlov, Yu. V.: Dimensional regularization and n-wave procedure for scalar fields in many-dimensional quasi-Euclidean spaces, Teor. Matem. Fiz. 128 (2001), 236–248 [English transl.: Theor. Math. Phys. 128 (2001), 1034–1045].

    Google Scholar 

  34. Popov, I.: The Helmholtz resonator and operator extension theory in a space with inde-finite metric, Mat. Sb. 183 (1992), 3–27.

    Google Scholar 

  35. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975.

    Google Scholar 

  36. Shondin, Yu. G.: Quantum-mechanical models in R n associated with extension of the energy operator in a Pontryagin space, Teor. Mat. Fiz. 74 (1988), 331–344 (Russian) [English translation: Theor. Math. Phys. 74 (1988), 220–230].

    Google Scholar 

  37. Shondin, Yu. G.: Perturbation of elliptic operators supported on subsets of high codimension, and extension theory in indefinite metric spaces, Seminars of St. Petersburg Math. Inst. 222, Researches in linear operators and function theory, 23 (1995), 246–292 (Russian).

    Google Scholar 

  38. Simon, B.: Spectral analysis of rank one perturbations and applications, in CRM Proc. and Lecture Notes, Vol. 8 (1995), 109–149.

    Google Scholar 

  39. Wilson, K. G.: Quantum field-theory models in less than 4 dimensions, Phys. Rev. D 7 (1973), 2911–2926.

    Google Scholar 

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Kurasov, P., Pavlov, Y.V. On Field Theory Methods in Singular Perturbation Theory. Letters in Mathematical Physics 64, 171–184 (2003). https://doi.org/10.1023/A:1025750025624

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