Symmetries of Schrödinger Operators with Point Interactions

Letters in Mathematical Physics

Volumn 45, Issue 1, 1998, Pages 33-47

  Albeverio, S ^a,b   Dabrowski L ^a,c   Kurasov, P ^a,b,d,e  

^a UNIVERSITY OF BONN   (Germany)

^b RUHR UNIVERSITY BOCHUM   (Germany)

^c SISSA   (Italy)

^d LULEÅ UNIVERSITY OF TECHNOLOGY   (Sweden)

^e ST PETERSBURG STATE UNIVERSITY   (Russian Federation)

Author keywords

Exactly solvable models; Extension theory; Point interactions; Schr dinger operators; Symmetries

Indexed keywords

EID: 0002623985     PISSN: 03779017     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1007493325970     Document Type: Article

References (30)

1. Albeverio, S., Brzeźniak, Z. and Da̧browski, L.: Time dependent propagator for point interactions. J. Phys. A 27 (1994), 4933-4943.
2. Albeverio, S., Breźniak, Z. and Da̧abrowski, L.: Fundamental solution of the heat and Schrödinger equation with point interactions, J. Funct. Anal. 128 (1995), 220-254.
3. Alberio, S., Gesztesy, F., Høegh-Krohn, R. and Holden, H.: Solvable Models in Quantum Mechanics, Springer-Verlag, Berlin, 1988.
4. Albeverio, S., Gesztesy, F. and Holden, H.: Comment on a recent note on the Schrödinger equation with a δ′-interaction, J. Phys. A: Math. Gen. 26 (1993), 3903-3904.
5. Albeverio, S. and Kurasov, P.: Rank one perturbations, approximations and selfadjoint extensions, J. Funct. Anal. 148 (1997), 152-169.
6. Albeverio, S. and Kurasov, P.: Rank one perturbations of not semibounded operators, Integral Equations Operator Theory 27 (1997), 349-400.
7. Berezin, F. A. and Faddeev, L. D.: A remark on Schrödinger equation with a singular potential, Soviet Math. Dokl. 2 1961, 372-375 (Math. USSR Dokl., 137 (1961), 1011-1014).
8. Berezin, F. A. and Faddeev, L. D.: A remark on Schrödinger equation with a singular potential, Soviet Math. Dokl. 2 1961, 372-375 (Math. USSR Dokl., 137 (1961), 1011-1014).
9. Cheon, T. and Shigehara, T.: Realizing discontinuous wave function with renormalized short-range potentials, quant-ph/9709035.
10. Chernoff, P. and Hughes, R.: A new class of point interactions in one dimension, J. Funct. Anal. 111 (1993), 97-117.
11. Coutinho, F. A. B., Nogami, Y. and Fernando Perez, J.: Generalized point interactions in one-dimensional quantum mechanics, J. Phys. A: Math. Gen. 30 (1997), 3937-3945
12. Demkov, Yu. N., Kurasov, P. B. and Ostrovsky, V. N.: Doubly periodical in time and energy exactly soluble system with two interacting systems of states, J. Phys. A: Math. Gen. 28 (1995), 4361-4380.
13. Demkov, Yu. N. and Ostrovskii, V. N.: Zero-range Potentials and their Applications in Atomic Physics, Leningrad Univ. Press, Leningrad, 1975; English translation: Plenum Press, New York-London, 1988.
14. Gesztesy, F. and Holden, H.: A new class of solvable models in quantum mechanics describing point interactions on the line, J. Phys. A: Math. Gen. 20 (1987), 5157.
15. Gesztesy, F. and Kirsch, W.: One-dimensional Schrödinger operators with interactions singular on a discrete set, J. Reine Angew. Math. 362 (1985), 28-50.
16. Griffiths, D. J.: Boundary conditions at the derivative of a delta function, J. Phys. A: Math. Gen. 26 (1993), 2265-2267.
17. Hassi, S., Langer, H. and de Snoo, H.: Selfadjoint extensions for a class of symmetric operators with defect numbers (1,1), 15th OT Conference Proc., 1995, pp. 115-145.
18. Hassi, S. and de Snoo, H.: On rank one perturbations of selfadjoint operators, Integral Equations Operator Theory 29 (1997), 288-300.
19. Kiselev, A. and Simon, B.: Rank one perturbations with infinitesimal coupling, J. Funct. Anal. 130 (1995), 345-356.
20. Kurasov, P.: Distribution theory for the discontinuous test functions and differential operators with the generalized coefficients, J. Math. Anal. Appl. 201 (1996), 297-323.
21. Kurasov, P. and Boman, J.: Finite rank singular perturbations and distributions with discontinuous test functions, accepted for publication in Proc. Amer. Math. Soc.
22. Kurasov, P. and Elander, N.: On the δ′-potential in one dimension, Preprint MSI 93-7, ISSN-1100-214X (1993).
23. Kurasov, P., Scrinzi, A. and Elander, N.: On the δ′-potential arising in exterior complex scaling, Phys. Rev. A 49 (1994) 5095-5097.
24. Krein, M. G.: On the resolvents of a Hermitian operator with deficiency indices (m, m), Dokl. Akad. Nauk SSSR 52 (1946), 657-660 (in Russian).
25. Pavlov, B. S.: The theory of extensions and explicitly-soluble models, Russian Math. Surveys 42(6) (1987), 127-168.
26. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, vol. IV, Academic Press, New York, 1975.
27. Seba, P.: The generalized point interaction in one dimension, Czech. J. Phys. B 36 (1986), 667-673.
28. Seba, P.: Some remarks on the δ′-interaction in one dimension, Rep. Math. Phys. 24 (1986), 111-120.
29. Shubin Christ, C. and Stolz, G.: Spectral theory of one-dimensional Schrödinger operators with point interactions, J. Math. Anal. Appl. 184 (1994), 491-516.
30. Simon, B.: Spectral analysis of rank one perturbations and applications, in: CRM Proc. Lectures Notes 8, Amer. Math. Soc., Providence, 1995, pp. 109-149.