Resolvent formulas of general type and its application to point interactions

Journal of Evolution Equations

Volumn 1, Issue 4, 2001, Pages 421-440

  Kuroda, S T ^a   Nagatani, Hiroshi ^b  

^a GAKUSHUIN UNIVERSITY   (Japan)

^b Information and Mathematical Science Laboratory Inc   (Japan)

Author keywords

Point interaction; Resolvent formula; Singular interaction

Indexed keywords

EID: 0347749338     PISSN: 14243199     EISSN: None     Source Type: Journal    
DOI: 10.1007/PL00001381     Document Type: Article

References (30)

1. ALBEVERIO, S., GESZTESY, F., HøEGH-KROHN, R. and HOLDEN, H. Solvable Models in Quantum Mechanics, Springer-Verlag, Berlin-Heidelberg-New York, 1988.
2. ALBEVERIO, S., KARWOWSKI, W. and KOSHMANENKO, V., Square powers of singularly perturbed operators, Math. Nachr. 173 (1995), 5-24.
3. ALBEVERIO, S. and KURASOV, P., Rank one perturbation of not semibounded operators, Integr. Equ. Oper. Theory 27 (1997), 379-400.
4. BERESIN, F. A. and FADDEEV, L. D., A remark on Schrödinger's equation with a singular potential, Soviet Math. Dokl. 2 (1961), 372-375.
5. CHERNOFF, P. R. and HUGHES, R. J., A new class of point interactions in one dimension, J. Functional Anal. 111 (1993), 97-117.
6. DELL'ANTONIO, G. F., FIGARI, R. and TETA, A., Hamiltonians for systems of N particles interacting through point interactions, Ann. Inst. H. Poincaré, Phys. Théor. 60 (1994), 253-290.
7. GESZTESY, F. and KIRSCH, W., One-dimensional Schrödinger operators with interactions singular on a discrete set, J. Reine Angew. Math. 362 (1985), 28-50.
8. GESZTESY, F. and SIMON, B., Rank one perturbation at infinite coupling, J. Funct. Anal. 128 (1995), 245-252.
9. HASSI, S. and DE SNOO, H., One-dimensional graph perturbations of selfadjoint relations, Ann. Acad. Sci. Fenn. 22 (1997), 123-164.
10. IKEBE, T., and SHIMADA, S., Spectral and scattering theory of Schrödinger operators with penetrable wall potentials, J. Math. Kyoto Univ. 31 (1991), 219-258.
11. KARWOWSKI, W. and KOSHMANENKO, V., Singular quadratic forms: regularization by restriction, J. Functional Anal. 143 (1997), 205-220.
12. KATO, T., Perturbation Theory for Linear Operators, 2nd ed., 1976, Springer.
13. KISELEV, A. and SIMON, B., Rank one perturbations with infinitesimal coupling, J. Functional Anal. 130 (1995), 345-356.
14. KURASOV, P., Distribution theory for discontinuous test functions and differential operators with generalized coefficients, J. Math. Anal. Appl. 201 (1996), 297-323.
15. KURASOV, P. and BOMAN, J., Finite rank singular perturbations and distributions with discontinuous test functions, Proc. Amer. Math. Soc. 126 (1998), 1673-1683.
16. KURASOV, P. and KURODA, S. T., Krein's formula and perturbation theory, Research Rep. in Math., No. 6, 2000, Department of Math., Stockholm Univ.
17. KURASOV, P. and KURODA, S. T., Krein's formula and perturbation theory, Revised version of [16], 2001.
18. -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., 2000.
19. -4-perturbations of self-adjoint operators, Research Rep. in Math., No. 16, 2000, Department of Math., Stockholm Univ.
20. KURODA, S. T., Resolvent formulas, special and general, to appear in Kōkyuroku RIMS, Kyoto Univ., Proc. Conf. "Tosio Kato's Method and Principles for Evolution Equations in Mathematical Physics".
21. -2-construction and some applications, Mathematical Results in Quantum Mechanics, J. Dittrich, P. Exner, M. Tartar, ed., Operator Theory: Advances and Applications, Birkhäuser Verlag, 1999, pp. 99-105.
22. -2 resolvent construction and non-local surface interactions of rank one, in preparation.
23. -2-construction and Hamiltonians with interactions of various type, J. Uchiyama, ed., Spectral and Scattering Theory and Related Topics, Kōkyuroku RIMS, Kyoto Univ., 1997 no. 994, 155-167, (in Japanese).
24. POSILICANO, A., A Krein-like formula for singular perturbations of self-adjoint operators and applications, J. Functional Anal. 183 (2001), 109-147.
25. SĚBA, P., The generalized point interaction in one dimension, Czechoslovak. J. Phys. B 36(1986), 111-120.
26. SHIMADA, S., A solvable model for line interaction, J. Uchiyama, ed., Spectral and Scattering Theory and Related Topics, Kōkyuroku RIMS, Kyoto Univ., 1997 no. 994, 168-183 (in Japanese).
27. SIMON, B., Spectral analysis of rank one perturbations and applications, CRM Proceedings and and Lecture Notes, vol. 8 (1995), 109-149.
28. TETA, A., Quadratic forms for singular perturbations of the Laplacian, Publ. RIMS, Kyoto Univ. 26 (1990), 803-817.
29. YAFAEV, D. R., Mathematical Scattering Theory, General Theory, Transi. Math. Monographs vol. 105, Amer. Math. Soc., Providence, 1992.
30. -2 construction, Preprint, 2001.