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Dirichlet to Neumann maps for infinite quantum graphs
1. | Department of Mathematics, University of Colorado at Colorado Springs, Colorado Springs, CO 80933 |
References:
[1] |
G. Auchmuty, Steklov eigenproblems and the representation of solutions of elliptic boundary value problems,, Numerical Functional Analysis and Optimization, 25 (2004), 321.
doi: 10.1081/NFA-120039655. |
[2] |
S. Avdonin and P. Kurasov, Inverse problems for quantum trees,, Inverse Probl. Imaging, 2 (2008), 1.
doi: 10.3934/ipi.2008.2.1. |
[3] |
M. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method,, Inverse Problems, 20 (2004), 647.
doi: 10.1088/0266-5611/20/3/002. |
[4] |
M. Brown and R. Weikard, A Borg-Levinson theorem for trees,, Proc. R. Soc. London Ser. A, 461 (2005), 3231.
doi: 10.1098/rspa.2005.1513. |
[5] |
A. Calderon, On an inverse boundary value problem,, Computational and Applied Mathematics, 25 (2006), 133.
doi: 10.1590/S0101-82052006000200002. |
[6] |
R. Carlson, Linear network models related to blood flow,, in, 415 (2006), 65.
doi: 10.1090/conm/415/07860. |
[7] |
R. Carlson, Boundary value problems for infinite metric graphs,, in Analysis on Graphs and Its Applications, 77 (2008), 355.
|
[8] |
R. Carlson, After the explosion: Dirichlet forms and boundary problems for infinite graphs,, preprint, (). Google Scholar |
[9] |
E. Curtis, D. Ingerman and J. Morrow, Circular planar graphs and resistor networks,, Linear Algebra Appl., 283 (1998), 115.
doi: 10.1016/S0024-3795(98)10087-3. |
[10] |
P. Cartier, Fonctions harmoniques sur un arbre,, Sympos. Math, 9 (1972), 203.
|
[11] |
F. Chung, "Spectral Graph Theory,'', American Mathematical Society, (1997).
|
[12] |
J. Cohen, F. Colonna and D. Singman, Distributions and measures on the boundary of a tree,, Journal of Mathematical Analysis and Applications, 293 (2004), 89.
doi: 10.1016/j.jmaa.2003.12.015. |
[13] |
Y. Colin de Verdiere, "Spectres de Graphes,'', Societe Mathematique de France, (1998).
|
[14] |
Y. Colin de Verdiere, N. Torki-Hamza and F. Truc, Essential self-adjointness for combinatorial Schrödinger operators II-metrically noncomplete graphs,, Mathematical Physics, 14 (2011), 21.
|
[15] |
P. Doyle and J. L. Snell, "Random Walks and Electric Networks,'', MAA, (1984).
|
[16] |
P. Exner, J. Keating, P. Kuchment, T. Sunada and A. Teplaev, "Analysis on Graphs and Its Applications,'', American Mathematical Society, (2008).
|
[17] |
G. Folland, "Real Analysis,'', John Wiley and Sons, (1984).
|
[18] |
A. Georgakopoulos, Graph topologies induced by edge lengths,, Discrete Mathematics, 311 (2011), 1523.
doi: 10.1016/j.disc.2011.02.012. |
[19] |
J. Hocking and G. Young, "Topology,'', Addison-Wesley, (1961).
|
[20] |
P. E. T. Jorgensen and E. P. J. Pearse, Operator theory and analysis of infinite networks,, preprint, (). Google Scholar |
[21] |
T. Kato, "Perturbation Theory for Linear Operators,'', Springer-Verlag, (1995).
|
[22] |
M. Keller and D. Lenz, Unbounded laplacians on graphs: Basic spectral properties and the heat equation,, Math. Model. Nat. Phenom., 5 (2010), 198.
doi: 10.1051/mmnp/20105409. |
[23] |
P. Lax, "Functional Analysis,'', Wiley, (2002).
|
[24] |
R. Lyons and Y. Peres, "Probability on Trees and Networks,'', Cambridge University Press. In preparation. , (). Google Scholar |
[25] |
B. Maury, D. Salort and C. Vannier, Trace theorem for trees and application to the human lungs,, Networks and Heterogeneous Media, 4 (2009), 469.
|
[26] |
S. Nicaise, Some results on spectral theory over networks, applied to nerve impulse transmission,, Springer Lecture Notes in Mathematics, 1171 (1985), 532.
doi: 10.1007/BFb0076584. |
[27] |
H. Royden, "Real Analysis,'', Macmillan, (1988).
|
[28] |
J. Sylvester and G. Uhlmann, The Dirichlet to Neumann map and applications,, Inverse problems in partial differential equations (Arcata, (1989).
|
[29] |
W. Woess, "Denumerable Markov Chains,'', European Mathematical Society, (2009).
doi: 10.4171/071. |
[30] |
M. Picardello and W. Woess, Martin boundaries of random walks: ends of trees and groups,, Trans. American Math. Soc., 302 (1987), 185.
doi: 10.1090/S0002-9947-1987-0887505-2. |
[31] |
D. Zelig, "Properties of Solutions of Partial Differential Equations Defined on Human Lung Shaped Domains,'', Ph.D. Thesis, (2005). Google Scholar |
show all references
References:
[1] |
G. Auchmuty, Steklov eigenproblems and the representation of solutions of elliptic boundary value problems,, Numerical Functional Analysis and Optimization, 25 (2004), 321.
doi: 10.1081/NFA-120039655. |
[2] |
S. Avdonin and P. Kurasov, Inverse problems for quantum trees,, Inverse Probl. Imaging, 2 (2008), 1.
doi: 10.3934/ipi.2008.2.1. |
[3] |
M. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method,, Inverse Problems, 20 (2004), 647.
doi: 10.1088/0266-5611/20/3/002. |
[4] |
M. Brown and R. Weikard, A Borg-Levinson theorem for trees,, Proc. R. Soc. London Ser. A, 461 (2005), 3231.
doi: 10.1098/rspa.2005.1513. |
[5] |
A. Calderon, On an inverse boundary value problem,, Computational and Applied Mathematics, 25 (2006), 133.
doi: 10.1590/S0101-82052006000200002. |
[6] |
R. Carlson, Linear network models related to blood flow,, in, 415 (2006), 65.
doi: 10.1090/conm/415/07860. |
[7] |
R. Carlson, Boundary value problems for infinite metric graphs,, in Analysis on Graphs and Its Applications, 77 (2008), 355.
|
[8] |
R. Carlson, After the explosion: Dirichlet forms and boundary problems for infinite graphs,, preprint, (). Google Scholar |
[9] |
E. Curtis, D. Ingerman and J. Morrow, Circular planar graphs and resistor networks,, Linear Algebra Appl., 283 (1998), 115.
doi: 10.1016/S0024-3795(98)10087-3. |
[10] |
P. Cartier, Fonctions harmoniques sur un arbre,, Sympos. Math, 9 (1972), 203.
|
[11] |
F. Chung, "Spectral Graph Theory,'', American Mathematical Society, (1997).
|
[12] |
J. Cohen, F. Colonna and D. Singman, Distributions and measures on the boundary of a tree,, Journal of Mathematical Analysis and Applications, 293 (2004), 89.
doi: 10.1016/j.jmaa.2003.12.015. |
[13] |
Y. Colin de Verdiere, "Spectres de Graphes,'', Societe Mathematique de France, (1998).
|
[14] |
Y. Colin de Verdiere, N. Torki-Hamza and F. Truc, Essential self-adjointness for combinatorial Schrödinger operators II-metrically noncomplete graphs,, Mathematical Physics, 14 (2011), 21.
|
[15] |
P. Doyle and J. L. Snell, "Random Walks and Electric Networks,'', MAA, (1984).
|
[16] |
P. Exner, J. Keating, P. Kuchment, T. Sunada and A. Teplaev, "Analysis on Graphs and Its Applications,'', American Mathematical Society, (2008).
|
[17] |
G. Folland, "Real Analysis,'', John Wiley and Sons, (1984).
|
[18] |
A. Georgakopoulos, Graph topologies induced by edge lengths,, Discrete Mathematics, 311 (2011), 1523.
doi: 10.1016/j.disc.2011.02.012. |
[19] |
J. Hocking and G. Young, "Topology,'', Addison-Wesley, (1961).
|
[20] |
P. E. T. Jorgensen and E. P. J. Pearse, Operator theory and analysis of infinite networks,, preprint, (). Google Scholar |
[21] |
T. Kato, "Perturbation Theory for Linear Operators,'', Springer-Verlag, (1995).
|
[22] |
M. Keller and D. Lenz, Unbounded laplacians on graphs: Basic spectral properties and the heat equation,, Math. Model. Nat. Phenom., 5 (2010), 198.
doi: 10.1051/mmnp/20105409. |
[23] |
P. Lax, "Functional Analysis,'', Wiley, (2002).
|
[24] |
R. Lyons and Y. Peres, "Probability on Trees and Networks,'', Cambridge University Press. In preparation. , (). Google Scholar |
[25] |
B. Maury, D. Salort and C. Vannier, Trace theorem for trees and application to the human lungs,, Networks and Heterogeneous Media, 4 (2009), 469.
|
[26] |
S. Nicaise, Some results on spectral theory over networks, applied to nerve impulse transmission,, Springer Lecture Notes in Mathematics, 1171 (1985), 532.
doi: 10.1007/BFb0076584. |
[27] |
H. Royden, "Real Analysis,'', Macmillan, (1988).
|
[28] |
J. Sylvester and G. Uhlmann, The Dirichlet to Neumann map and applications,, Inverse problems in partial differential equations (Arcata, (1989).
|
[29] |
W. Woess, "Denumerable Markov Chains,'', European Mathematical Society, (2009).
doi: 10.4171/071. |
[30] |
M. Picardello and W. Woess, Martin boundaries of random walks: ends of trees and groups,, Trans. American Math. Soc., 302 (1987), 185.
doi: 10.1090/S0002-9947-1987-0887505-2. |
[31] |
D. Zelig, "Properties of Solutions of Partial Differential Equations Defined on Human Lung Shaped Domains,'', Ph.D. Thesis, (2005). Google Scholar |
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