Geometric properties of quantum graphs and vertex scattering matrices

• Autorzy: Kurasov P. , Nowaczyk M.
• Języki publikacji: EN

Abstrakty

EN

Differential operators on metric graphs are investigated. It is proven that vertex matching (boundary) conditions can be successfully parameterized by the vertex scattering matrix. Two new families of matching conditions are investigated: hyperplanar Neumann and hyperplanar Dirichlet conditions. Using trace formula it is shown that the spectrum of the Laplace operator determines certain geometric properties of the underlying graph.

Słowa kluczowe

EN

scattering theory; quantum graphs; matching (boundary) conditions

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2010
• Tom: Vol. 30, no. 3
• Strony: 295--309
• Opis fizyczny: Bibliogr. 18 poz.

Twórcy

autor

Kurasov P.

autor

Nowaczyk M.

• Institute of Mathematics, PAN ul. Św.Tomasza 30, 31-027 Kraków, Poland [Marlena Nowaczyk],

Bibliografia

• [1] R. Band, T. Shapira, U. Smilansky, Nodal domains on isospectral quantum graphs: The resolution of isospectrality?, J. Phys. A: Math. Theor. 39 (2006), 13999–14014.
• [2] R. Band, O. Parzanchevski, G. Ben-Shach, The isospectral fruits of representation theory: quantum graphs and drums, J. Phys. A: Math. Theor. 42 (2009), 175202, 42 pp.
• [3] J. Bolte, S. Endres, The trace formula for quantum graphs with general self adjoint boundary conditions, Ann. Henri Poincaré 10 (2009), 189–223.
• [4] P. Exner, Contact interactions on graph superlattices, J. Phys. A: Math. Gen. 29 (1996), 87–102.
• [5] P. Exner, P. Šeba, Free quantum motion on a branching graph, Rep. Math. Phys. 28 (1989), 7–26.
• [6] S.A. Fulling, P. Kuchment, J.H. Wilson, Index theorems for quantum graphs, J. Phys. A: Math. Theor. 40 (2007), 14165–14180.
• [7] N. Gerasimenko, B. Pavlov, Scattering problems on noncompact graphs, Teoret. Mat. Fiz. 74 (1988), 345–359 (Eng. transl.: Theoret. and Math. Phys. 74 (1988), 230–240).
• [8] B. Gutkin, U. Smilansky, Can one hear the shape of a graph?, J. Phys. A: Math. Gen. 34 (2001), 6061–6068.
• [9] M. Harmer, Hermitian symplectic geometry and extension theory, J. Phys. A: Math. Gen. 33 (2000), 9193–9203.
• [10] V. Kostrykin, R. Schrader, Kirchhoff’s rule for quantum wires, J. Phys A: Math. Gen. 32 (1999), 595–630.
• [11] V. Kostrykin, R. Schrader, Kirchhoff’s rule for quantum wires. II: The inverse problem with possible applications to quantum computers, Fortschr. Phys. 48 (2000), 703–716.
• [12] T. Kottos, U. Smilansky, Periodic orbit theory and spectral statistics for quantum graphs, Ann. Physics 274 (1999), 76–124.
• [13] P. Kuchment, Quantum graphs. I. Some basic structures. Special section on quantum graphs, Waves Random Media 14 (2004), S107–S128.
• [14] P. Kurasov, Graph Laplacians and topology, Ark. Mat. 46 (2008), 95–111.
• [15] P. Kurasov, Schrödinger operators on graphs and geometry I: Essentially bounded potentials, J. Funct. Anal. 254 (2008), 934–953.
• [16] P. Kurasov, M. Nowaczyk, Inverse spectral problem for quantum graphs, J. Phys. A: Math. Gen. 38 (2005), 4901–4915, corrigendum: 39 (2006), 993.
• [17] P. Kurasov, F. Stenberg, On the inverse scattering problem on branching graphs, J. Phys. A: Math. Gen. 35 (2002), 101–121.
• [18] J.-P. Roth, Le spectre du laplacien sur un graphe, Lecture Notes in Math. 1096 (1984), 521–539.