Quantum-graph vertex couplings: some old and new approximations

Stepan Manko

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 2, page 259-267
  • ISSN: 0862-7959

Abstract

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In 1986 P. Šeba in the classic paper considered one-dimensional pseudo-Hamiltonians containing the first derivative of the Dirac delta function. Although the paper contained some inaccuracy, it was one of the starting points in approximating one-dimension self-adjoint couplings. In the present paper we develop the above results to the case of quantum systems with complex geometry.

How to cite

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Manko, Stepan. "Quantum-graph vertex couplings: some old and new approximations." Mathematica Bohemica 139.2 (2014): 259-267. <http://eudml.org/doc/261946>.

@article{Manko2014,
abstract = {In 1986 P. Šeba in the classic paper considered one-dimensional pseudo-Hamiltonians containing the first derivative of the Dirac delta function. Although the paper contained some inaccuracy, it was one of the starting points in approximating one-dimension self-adjoint couplings. In the present paper we develop the above results to the case of quantum systems with complex geometry.},
author = {Manko, Stepan},
journal = {Mathematica Bohemica},
keywords = {quantum graph; vertex coupling; singularly scaled potential; quantum graph; vertex coupling; singularly scaled potential},
language = {eng},
number = {2},
pages = {259-267},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Quantum-graph vertex couplings: some old and new approximations},
url = {http://eudml.org/doc/261946},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Manko, Stepan
TI - Quantum-graph vertex couplings: some old and new approximations
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 2
SP - 259
EP - 267
AB - In 1986 P. Šeba in the classic paper considered one-dimensional pseudo-Hamiltonians containing the first derivative of the Dirac delta function. Although the paper contained some inaccuracy, it was one of the starting points in approximating one-dimension self-adjoint couplings. In the present paper we develop the above results to the case of quantum systems with complex geometry.
LA - eng
KW - quantum graph; vertex coupling; singularly scaled potential; quantum graph; vertex coupling; singularly scaled potential
UR - http://eudml.org/doc/261946
ER -

References

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  1. Albeverio, S., Cacciapuoti, C., Finco, D., 10.1063/1.2710197, J. Math. Phys. 48 (2007), Article ID 032103, 21 pages. (2007) Zbl1137.81330MR2314483DOI10.1063/1.2710197
  2. Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H., Solvable Models in Quantum Mechanics, AMS Chelsea, Providence (2005). (2005) Zbl1078.81003MR2105735
  3. Albeverio, S., Koshmanenko, V., Kurasov, P., Nizhnik, L., 10.1090/S0002-9939-02-06694-7, Proc. Amer. Math. Soc. 131 (2003), 1443-1452. (2003) MR1949874DOI10.1090/S0002-9939-02-06694-7
  4. Berkolaiko, G., Kuchment, P., Introduction to Quantum Graphs, Mathematical Surveys and Monographs 186 American Mathematical Society, Providence (2013). (2013) MR3013208
  5. Bollé, D., Gesztesy, F., Wilk, S. F. J., A complete treatment of low-energy scattering in one dimension, J. Oper. Theory 13 (1985), 3-32. (1985) Zbl0567.47008MR0768299
  6. Cacciapuoti, C., Exner, P., 10.1088/1751-8113/40/26/F02, J. Phys. A, Math. Theor. 40 (2007), F511--F523. (2007) Zbl1136.81442MR2344437DOI10.1088/1751-8113/40/26/F02
  7. Christiansen, P. L., Arnbak, H. C., Zolotaryuk, A. V., Ermakov, V. N., Gaididei, Y. B., 10.1088/0305-4470/36/27/311, J. Phys. A, Math. Gen. 36 (2003), 7589-7600. (2003) Zbl1047.81567MR2006513DOI10.1088/0305-4470/36/27/311
  8. Exner, P., Manko, S. S., 10.1088/1751-8113/46/34/345202, J. Phys. A, Math. Theor. 46 (2013), Article ID 345202, 17 pages. (2013) Zbl1278.81095MR3101681DOI10.1088/1751-8113/46/34/345202
  9. Golovaty, Yu., 10.1007/s00020-012-2027-z, Integral Equations Oper. Theory 75 (2013), 341-362. (2013) Zbl1270.34198MR3019277DOI10.1007/s00020-012-2027-z
  10. Golovaty, Yu. D., Hryniv, R. O., Norm resolvent convergence of singularly scaled Schrö-dinger operators and δ ' -potentials, Proc. R. Soc. Edinb., Sect. A, Math. 143 (2013), 791-816. (2013) MR3082301
  11. Golovaty, Yu. D., Man'ko, S. S., Solvable models for the Schrödinger operators with δ ' -like potentials, Ukr. Math. Bulletin 6 (2009), 169-203. (2009) MR2768971
  12. Jensen, A., Nenciu, G., 10.1142/S0129055X01000843, Rev. Math. Phys. 13 (2001), 717-754. (2001) Zbl1029.81067MR1841744DOI10.1142/S0129055X01000843
  13. Kurasov, P., Scrinzi, A., Elander, N., 10.1103/PhysRevA.49.5095, Phys. Rev. A 49 (1994), 5095-5097. (1994) DOI10.1103/PhysRevA.49.5095
  14. Man'ko, S. S., 10.1088/1751-8113/43/44/445304, J. Phys. A, Math. Theor. 43 (2010), Article ID 445304, 14 pages. (2010) Zbl1202.81201MR2733833DOI10.1088/1751-8113/43/44/445304
  15. Man'ko, S. S., 10.1063/1.4769425, J. Math. Phys. 53 (2012), Article ID 123521, 13 pages. (2012) Zbl1278.81097MR3405911DOI10.1063/1.4769425
  16. Šeba, P., 10.1016/0034-4877(86)90045-5, Rep. Math. Phys. 24 (1986), 111-120. (1986) Zbl0638.70016MR0932938DOI10.1016/0034-4877(86)90045-5

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