Inverse spectral problem for quantum graphs

Journal of Physics A: Mathematical and General

Volumn 38, Issue 22, 2005, Pages 4901-4915

  Kurasov, Pavel ^a,b   Nowaczyk, Marlena ^a  

^a LUND INSTITUTE OF TECHNOLOGY   (Sweden)

^b ST PETERSBURG STATE UNIVERSITY   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 19944417603     PISSN: 03054470     EISSN: None     Source Type: Journal    
DOI: 10.1088/0305-4470/38/22/014     Document Type: Article

References (37)

1. Adamyan V 1992 Scattering matrices for microschemes Operator Theory: Adv. Appl. 59 1-10
2. Albeverio S and Kurasov P 2000 Singular Perturbations of Differential Operators (London Mathematical Society Lecture Notes N271) (Cambridge: Cambridge University Press)
3. Boman J and Kurasov P Symmetries of quantum graphs and the inverse scattering problem Adv. Appl. Math. submitted
4. Brown B M and Wiekard R A Borg-Levinson theorem on trees at press
5. Carlson R 1999 Inverse eigenvalue problems on directed graphs Trans. Am. Math. Soc. 351 4069-88
6. Evans W D and Saito Y 2000 Neumann Laplacians on domains and operators on associated trees Q. J. Math. 51 313-42
7. Evans W D and Harris D J 1993 Fractals, trees and the Neumann Laplacian Math. Ann. 296 493-527
8. Everitt W N and Zettl A 1992 Differential operators generated by a countable number of quasi-differential expressions on the real line Proc. Lond. Math. Soc. 64 524-44
9. Everitt W N, Shubin C, Stolz G and Zettl A 1997 Sturm-Liouville problems with an infinite number of interior singularities Spectral Theory and Computational Methods of Sturm-Liouville Problems (Knoxville, TN, 1996) (Lecture Notes in Pure and Appl. Math. vol 191) (New York: Dekker) pp 211-49
10. Everitt W N and Markus L 2001 Multi-interval linear ordinary boundary value problems and complex symplectic algebra Mem. Am. Math. Soc. 151 64
11. Everitt W N and Markus L 2003 Elliptic partial differential operators and symplectic algebra Mem. Am. Math. Soc. 162 111
12. Everitt W N and Markus L 2004 Infinite dimensional complex symplectic spaces Mem. Am. Math. Soc. 171 76
13. Exner P and Šeba P 1987 Quantum motion on a half-line connected to a plane J. Math. Phys. 28 386-91
14. Exner P and Šeba P 1989 Schrödinger operators, standard and nonstandard Papers from the Conference held in Dubna (6-10 Sept. 1988) (Teaneck, NJ: World Scientific) p 409
15. Exner P and Šeba P 1989 Applications of selfadjoint extensions in quantum physics Proc. Conference held in Dubna (29 Sept.-1 Oct. 1987) (Lecture Notes in Physics vol 324) (Berlin: Springer) p 273
16. Exner P and Šeba P 1989 Free quantum motion on a branching graph Rep. Math. Phys. 28 7-26
17. Gerasimenko N I and Pavlov B S 1988 Scattering problems on noncompact graphs Teor. Mat. Fiz. 74 345-59
18. Gerasimenko N I and Pavlov B S 1988 Theor. Math. Phys. 74 230-40 (Engl. Transl.)
19. Gerasimenko N I 1988 Inverse scattering problem on a noncompact graph Teor. Mat. Fiz. 75 187-200
20. Gerasimenko N I 1988 Theor. Math. Phys. 75 460-70 (Engl. Transl.)
21. 2 Adv. Math. 32 128-48
22. Guillemin V and Melrose R 1979 The Poisson summation formula for manifolds with boundary Adv. Math. 32 204-32
23. Gutkin B and Smilansky U 2001 Can one hear the shape of a graph? J. Phys. A: Math. Gen. 34 6061-8
24. Harmer M 2002 Inverse scattering for the matrix Schrödinger operator and Schrödinger operator on graphs with general self-adjoint boundary conditions ANZIAM J. 44 161-8
25. Kostrykin V and Schrader R 1999 Kirchoff's rule for quantum wires J. Phys A: Math. Gen. 32 595-630
26. Kottos T and Smilansky U 1999 Periodic orbit theory and spectral statistics for quantum graphs Ann. Phys., NY 274 76-124
27. Kuchment P 2003 Waves in periodic and random media Proc. AMS-IMS-SIAM Joint Summer Research Conference (Mount Holyoke College, South Hadley, MA, 22-28 June 2002) (Contemporary Mathematics vol 339) (Providence, RI: American Mathematical Society)
28. Kuchment P 2004 Quantum graphs. I. Some basic structures. Special section on quantum graphs Waves Random Media 14 S107-28
29. Kurasov P and Stenberg F 2002 On the inverse scattering problem on branching graphs J. Phys. A: Math. Gen. 35 101-21
30. Naimark K and Solomyak M 2000 Eigenvalue estimates for the weighted Laplacian on metric trees Proc. Lond. Math. Soc. 80 690-724
31. Naimark K and Solomyak M 2001 Geometry of Sobolev spaces on regular trees and the Hardy inequalities Russ. J. Math. Phys. 8 322-35
32. Pavlov B S 1987 The theory of extensions, and explicitly solvable models Usp. Mat. Nauk 42 99-131, 247 (in Russian)
33. Roth J-P 1984 Le spectre du laplacien sur un graphe Lect Notes Math: Theorie du Potentiel 1096 521-39
34. Sobolev A V and Solomyak M 2002 Schrödinger operators on homogeneous metric trees: spectrum in gaps Rev. Math. Phys. 14 421-67
35. Solomyak M 2003 Laplace and Schrödinger operators on regular metric trees: the discrete spectrum case Function Spaces, Differential Operators and Nonlinear Analysis (Teistungen, 2001) (Basel: Birkhäuser) pp 161-81
36. Solomyak M 2003 On approximation of functions from Sobolev spaces on metric graphs J. Approx. Theory 121 199-219
37. Solomyak M 2004 On the spectrum of the Laplacian on regular metric trees. Special section on quantum graphs Waves Random Media 14 S155-71