American Institute of Mathematical Sciences

Advanced
June  2011, 6(2): 257-277. doi: 10.3934/nhm.2011.6.257

Spectral theory for nonconservative transmission line networks

Robert Carlson 1,

1. 

Department of Mathematics, University of Colorado at Colorado Springs, Colorado Springs, CO 80933, United States

Received  August 2010 Revised  April 2011 Published  May 2011

The global theory of transmission line networks with nonconservative junction conditions is developed from a spectral theoretic viewpoint. The rather general junction conditions lead to spectral problems for nonnormal operators. The theory of analytic functions which are almost periodic in a strip is used to establish the existence of an infinite sequence of eigenvalues and the completeness of generalized eigenfunctions. Simple eigenvalues are generic. The asymptotic behavior of an eigenvalue counting function is determined. Specialized results are developed for rational graphs.
Keywords: Transmission line equations, quantum graphs, dissipative boundary conditions, boundary value problems on networks..
Mathematics Subject Classification: Primary: 34B4.
Citation: Robert Carlson. Spectral theory for nonconservative transmission line networks. Networks & Heterogeneous Media, 2011, 6 (2) : 257-277. doi: 10.3934/nhm.2011.6.257
References:
[1]

A. Agarwal, S. Das and D. Sen, Power dissipation for systems with junctions of multiple quantum wires,, Physical Review B, 81 (2010).   Google Scholar

[2]

L. Ahlfors, "Complex Analysis,", McGraw-Hill, (1966).   Google Scholar

[3]

F. Ali Mehmeti, "Nonlinear Waves in Networks,", Akademie Verlag, (1994).   Google Scholar

[4]

R. Carlson, Inverse eigenvalue problems on directed graphs,, Transactions of the American Mathematical Society, 351 (1999), 4069.  doi: 10.1090/S0002-9947-99-02175-3.  Google Scholar

[5]

R. Carlson, Linear network models related to blood flow,, in Quantum Graphs and Their Applications, 415 (2006), 65.   Google Scholar

[6]

C. Cattaneo and L. Fontana, D'Alembert formula on finite one-dimensional networks,, J. Math. Anal. Appl., 284 (2003), 403.  doi: 10.1016/S0022-247X(02)00392-X.  Google Scholar

[7]

G. Chen, S. Krantz, D. Russell, C. Wayne, H. West and M. Coleman, Analysis, designs, and behavior of dissipative joints for coupled beams,, SIAM Journal on Applied Mathematics., 49 (1989), 1665.  doi: 10.1137/0149101.  Google Scholar

[8]

S. Cox and E. Zuazua, The rate at which energy decays in a string damped at one end,, Indiana Univ. Math. J., 44 (1995), 545.  doi: 10.1512/iumj.1995.44.2001.  Google Scholar

[9]

E. B. Davies, Eigenvalues of an elliptic system,, Math. Z., 243 (2003), 719.  doi: 10.1007/s00209-002-0464-0.  Google Scholar

[10]

E. B. Davies, P. Exner and J. Lipovsky, Non-Weyl asymptotics for quantum graphs with general coupling conditions,, J. Phys. A, 43 (2010).   Google Scholar

[11]

Y. Fung, "Biomechanics,", Springer, (1997).   Google Scholar

[12]

I. Herstein, "Topics in Algebra,", Xerox College Publishing, (1964).   Google Scholar

[13]

B. Jessen and H. Tornhave, Mean motions and zeros of almost periodic functions,, Acta Math., 77 (1945), 137.  doi: 10.1007/BF02392225.  Google Scholar

[14]

T. Kato, "Perturbation Theory for Linear Operators,", Springer-Verlag, (1995).   Google Scholar

[15]

V. Kostrykin, J. Potthoff and R. Schrader, Contraction semigroups on metric graphs,, in Analysis on Graphs and Its Applications, 77 (2008), 423.   Google Scholar

[16]

T. Kottos and U. Smilansky, Periodic orbit theory and spectral statistics for quantum graphs,, Ann. Phys., 274 (1999), 76.  doi: 10.1006/aphy.1999.5904.  Google Scholar

[17]

M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks,, Math. Z., 249 (2005), 139.  doi: 10.1007/s00209-004-0695-3.  Google Scholar

[18]

M. Kramar Fijavz, D. Mugnolo and E. Sikolya, Variational and semigroup methods for waves and diffusions in networks,, Appl. Math. Optim., 55 (2007), 219.  doi: 10.1007/s00245-006-0887-9.  Google Scholar

[19]

M. Krein and A. Nudelman, Some spectral properties of a nonhomogeneous string with a dissipative boundary condition,, J. Operator Theory, 22 (1989), 369.   Google Scholar

[20]

B. Levin, "Distribution of Zeros of Entire Functions,", American Mathematical Society, (1980).   Google Scholar

[21]

S. Lang, "Algebra,", Addison-Wesley, (1984).   Google Scholar

[22]

G. Lumer, "Equations de Diffusion Generales sur des Reseaux Infinis,", Seminar Goulaouic-Schwartz, (1980).   Google Scholar

[23]

G. Lumer, "Connecting of Local Operators and Evolution Equations on Networks,", Lecture Notes in Math., 787 (1980).   Google Scholar

[24]

P. Magnusson, G. Alexander, V. Tripathi and A. Weisshaar, "Transmission Lines and Wave Propagation,", CRC Press, (2001).   Google Scholar

[25]

G. Miano and A. Maffucci, "Transmission Lines and Lumped Circuits,", Academic Press, (2001).   Google Scholar

[26]

L. J. Myers and W. L. Capper, A transmission line model of the human foetal circulatory system,, Medical Engineering and Physics, 24 (2002), 285.  doi: 10.1016/S1350-4533(02)00019-X.  Google Scholar

[27]

S. Nicaise, Spectre des reseaux topologiques finis,, Bulletin des sciences mathematique, 111 (1987), 401.   Google Scholar

[28]

J. Ottesen, M. Olufsen and J. Larsen, "Applied Mathematical Models in Human Physiology,", SIAM, (2004).  doi: 10.1137/1.9780898718287.  Google Scholar

[29]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer, (1983).   Google Scholar

[30]

S. Sherwin, V. Franke, J. Peiro and K. Parker, One-dimensional modeling of a vascular network in space-time variables,, Journal of Engineering Mathematics, 47 (2003), 217.  doi: 10.1023/B:ENGI.0000007979.32871.e2.  Google Scholar

[31]

J. von Below, A characteristic equation associated to an eigenvalue problem on C2 networks,, Lin. Alg. Appl., 71 (1985), 309.  doi: 10.1016/0024-3795(85)90258-7.  Google Scholar

[32]

L. Zhou and G. Kriegsmann, A simple derivation of microstrip transmission line equations,, SIAM J. Appl. Math., 70 (2009), 353.  doi: 10.1137/080737563.  Google Scholar

show all references

References:
[1]

A. Agarwal, S. Das and D. Sen, Power dissipation for systems with junctions of multiple quantum wires,, Physical Review B, 81 (2010).   Google Scholar

[2]

L. Ahlfors, "Complex Analysis,", McGraw-Hill, (1966).   Google Scholar

[3]

F. Ali Mehmeti, "Nonlinear Waves in Networks,", Akademie Verlag, (1994).   Google Scholar

[4]

R. Carlson, Inverse eigenvalue problems on directed graphs,, Transactions of the American Mathematical Society, 351 (1999), 4069.  doi: 10.1090/S0002-9947-99-02175-3.  Google Scholar

[5]

R. Carlson, Linear network models related to blood flow,, in Quantum Graphs and Their Applications, 415 (2006), 65.   Google Scholar

[6]

C. Cattaneo and L. Fontana, D'Alembert formula on finite one-dimensional networks,, J. Math. Anal. Appl., 284 (2003), 403.  doi: 10.1016/S0022-247X(02)00392-X.  Google Scholar

[7]

G. Chen, S. Krantz, D. Russell, C. Wayne, H. West and M. Coleman, Analysis, designs, and behavior of dissipative joints for coupled beams,, SIAM Journal on Applied Mathematics., 49 (1989), 1665.  doi: 10.1137/0149101.  Google Scholar

[8]

S. Cox and E. Zuazua, The rate at which energy decays in a string damped at one end,, Indiana Univ. Math. J., 44 (1995), 545.  doi: 10.1512/iumj.1995.44.2001.  Google Scholar

[9]

E. B. Davies, Eigenvalues of an elliptic system,, Math. Z., 243 (2003), 719.  doi: 10.1007/s00209-002-0464-0.  Google Scholar

[10]

E. B. Davies, P. Exner and J. Lipovsky, Non-Weyl asymptotics for quantum graphs with general coupling conditions,, J. Phys. A, 43 (2010).   Google Scholar

[11]

Y. Fung, "Biomechanics,", Springer, (1997).   Google Scholar

[12]

I. Herstein, "Topics in Algebra,", Xerox College Publishing, (1964).   Google Scholar

[13]

B. Jessen and H. Tornhave, Mean motions and zeros of almost periodic functions,, Acta Math., 77 (1945), 137.  doi: 10.1007/BF02392225.  Google Scholar

[14]

T. Kato, "Perturbation Theory for Linear Operators,", Springer-Verlag, (1995).   Google Scholar

[15]

V. Kostrykin, J. Potthoff and R. Schrader, Contraction semigroups on metric graphs,, in Analysis on Graphs and Its Applications, 77 (2008), 423.   Google Scholar

[16]

T. Kottos and U. Smilansky, Periodic orbit theory and spectral statistics for quantum graphs,, Ann. Phys., 274 (1999), 76.  doi: 10.1006/aphy.1999.5904.  Google Scholar

[17]

M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks,, Math. Z., 249 (2005), 139.  doi: 10.1007/s00209-004-0695-3.  Google Scholar

[18]

M. Kramar Fijavz, D. Mugnolo and E. Sikolya, Variational and semigroup methods for waves and diffusions in networks,, Appl. Math. Optim., 55 (2007), 219.  doi: 10.1007/s00245-006-0887-9.  Google Scholar

[19]

M. Krein and A. Nudelman, Some spectral properties of a nonhomogeneous string with a dissipative boundary condition,, J. Operator Theory, 22 (1989), 369.   Google Scholar

[20]

B. Levin, "Distribution of Zeros of Entire Functions,", American Mathematical Society, (1980).   Google Scholar

[21]

S. Lang, "Algebra,", Addison-Wesley, (1984).   Google Scholar

[22]

G. Lumer, "Equations de Diffusion Generales sur des Reseaux Infinis,", Seminar Goulaouic-Schwartz, (1980).   Google Scholar

[23]

G. Lumer, "Connecting of Local Operators and Evolution Equations on Networks,", Lecture Notes in Math., 787 (1980).   Google Scholar

[24]

P. Magnusson, G. Alexander, V. Tripathi and A. Weisshaar, "Transmission Lines and Wave Propagation,", CRC Press, (2001).   Google Scholar

[25]

G. Miano and A. Maffucci, "Transmission Lines and Lumped Circuits,", Academic Press, (2001).   Google Scholar

[26]

L. J. Myers and W. L. Capper, A transmission line model of the human foetal circulatory system,, Medical Engineering and Physics, 24 (2002), 285.  doi: 10.1016/S1350-4533(02)00019-X.  Google Scholar

[27]

S. Nicaise, Spectre des reseaux topologiques finis,, Bulletin des sciences mathematique, 111 (1987), 401.   Google Scholar

[28]

J. Ottesen, M. Olufsen and J. Larsen, "Applied Mathematical Models in Human Physiology,", SIAM, (2004).  doi: 10.1137/1.9780898718287.  Google Scholar

[29]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer, (1983).   Google Scholar

[30]

S. Sherwin, V. Franke, J. Peiro and K. Parker, One-dimensional modeling of a vascular network in space-time variables,, Journal of Engineering Mathematics, 47 (2003), 217.  doi: 10.1023/B:ENGI.0000007979.32871.e2.  Google Scholar

[31]

J. von Below, A characteristic equation associated to an eigenvalue problem on C2 networks,, Lin. Alg. Appl., 71 (1985), 309.  doi: 10.1016/0024-3795(85)90258-7.  Google Scholar

[32]

L. Zhou and G. Kriegsmann, A simple derivation of microstrip transmission line equations,, SIAM J. Appl. Math., 70 (2009), 353.  doi: 10.1137/080737563.  Google Scholar

[1]

Barbara Bianconi, Francesca Papalini. Non-autonomous boundary value problems on the real line. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 759-776. doi: 10.3934/dcds.2006.15.759
[2]

Stefano Biagi, Teresa Isernia. On the solvability of singular boundary value problems on the real line in the critical growth case. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1131-1157. doi: 10.3934/dcds.2020073
[3]

Sergei Avdonin, Pavel Kurasov, Marlena Nowaczyk. Inverse problems for quantum trees II: Recovering matching conditions for star graphs. Inverse Problems & Imaging, 2010, 4 (4) : 579-598. doi: 10.3934/ipi.2010.4.579
[4]

Long Hu, Tatsien Li, Bopeng Rao. Exact boundary synchronization for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type. Communications on Pure & Applied Analysis, 2014, 13 (2) : 881-901. doi: 10.3934/cpaa.2014.13.881
[5]

Vesselin Petkov. Location of eigenvalues for the wave equation with dissipative boundary conditions. Inverse Problems & Imaging, 2016, 10 (4) : 1111-1139. doi: 10.3934/ipi.2016034
[6]

C. Bourdarias, M. Gisclon, A. Omrane. Transmission boundary conditions in a model-kinetic decomposition. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 69-94. doi: 10.3934/dcdsb.2002.2.69
[7]

Bruno Fornet, O. Guès. Penalization approach to semi-linear symmetric hyperbolic problems with dissipative boundary conditions. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 827-845. doi: 10.3934/dcds.2009.23.827
[8]

Colin J. Cotter, Darryl D. Holm. Geodesic boundary value problems with symmetry. Journal of Geometric Mechanics, 2010, 2 (1) : 51-68. doi: 10.3934/jgm.2010.2.51
[9]

Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177
[10]

Hung Le. Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3357-3385. doi: 10.3934/dcds.2018144
[11]

Laurence Halpern, Jeffrey Rauch. Hyperbolic boundary value problems with trihedral corners. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4403-4450. doi: 10.3934/dcds.2016.36.4403
[12]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760
[13]

J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177
[14]

Sergei Avdonin, Fritz Gesztesy, Konstantin A. Makarov. Spectral estimation and inverse initial boundary value problems. Inverse Problems & Imaging, 2010, 4 (1) : 1-9. doi: 10.3934/ipi.2010.4.1
[15]

G. Infante. Positive solutions of nonlocal boundary value problems with singularities. Conference Publications, 2009, 2009 (Special) : 377-384. doi: 10.3934/proc.2009.2009.377
[16]

Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43
[17]

John R. Graef, Lingju Kong, Qingkai Kong, Min Wang. Positive solutions of nonlocal fractional boundary value problems. Conference Publications, 2013, 2013 (special) : 283-290. doi: 10.3934/proc.2013.2013.283
[18]

Feliz Minhós, Rui Carapinha. On higher order nonlinear impulsive boundary value problems. Conference Publications, 2015, 2015 (special) : 851-860. doi: 10.3934/proc.2015.0851
[19]

John V. Baxley, Philip T. Carroll. Nonlinear boundary value problems with multiple positive solutions. Conference Publications, 2003, 2003 (Special) : 83-90. doi: 10.3934/proc.2003.2003.83
[20]

K. Q. Lan, G. C. Yang. Optimal constants for two point boundary value problems. Conference Publications, 2007, 2007 (Special) : 624-633. doi: 10.3934/proc.2007.2007.624

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (23)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences