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

Spectral theory for nonconservative transmission line networks

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.
Citation: Robert Carlson. Spectral theory for nonconservative transmission line networks. Networks and 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).

[2]

L. Ahlfors, "Complex Analysis," McGraw-Hill, New York, 1966.

[3]

F. Ali Mehmeti, "Nonlinear Waves in Networks," Akademie Verlag, Berlin, 1994.

[4]

R. Carlson, Inverse eigenvalue problems on directed graphs, Transactions of the American Mathematical Society, 351 (1999), 4069-4088. doi: 10.1090/S0002-9947-99-02175-3.

[5]

R. Carlson, Linear network models related to blood flow, in Quantum Graphs and Their Applications, Contemporary Mathematics, 415 (2006), 65-80.

[6]

C. Cattaneo and L. Fontana, D'Alembert formula on finite one-dimensional networks, J. Math. Anal. Appl., 284 (2003), 403-424. doi: 10.1016/S0022-247X(02)00392-X.

[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), 1665-1693. doi: 10.1137/0149101.

[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), 545-573. doi: 10.1512/iumj.1995.44.2001.

[9]

E. B. Davies, Eigenvalues of an elliptic system, Math. Z., 243 (2003), 719-743. doi: 10.1007/s00209-002-0464-0.

[10]

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

[11]

Y. Fung, "Biomechanics," Springer, New York, 1997.

[12]

I. Herstein, "Topics in Algebra," Xerox College Publishing, Waltham, 1964.

[13]

B. Jessen and H. Tornhave, Mean motions and zeros of almost periodic functions, Acta Math., 77 (1945), 137-279. doi: 10.1007/BF02392225.

[14]

T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, New York, 1995.

[15]

V. Kostrykin, J. Potthoff and R. Schrader, Contraction semigroups on metric graphs, in Analysis on Graphs and Its Applications, PSUM, 77 (2008), 423-458.

[16]

T. Kottos and U. Smilansky, Periodic orbit theory and spectral statistics for quantum graphs, Ann. Phys., 274 (1999), 76-124. doi: 10.1006/aphy.1999.5904.

[17]

M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Math. Z., 249 (2005), 139-162. doi: 10.1007/s00209-004-0695-3.

[18]

M. Kramar Fijavz, D. Mugnolo and E. Sikolya, Variational and semigroup methods for waves and diffusions in networks, Appl. Math. Optim., 55 (2007), 219-240. doi: 10.1007/s00245-006-0887-9.

[19]

M. Krein and A. Nudelman, Some spectral properties of a nonhomogeneous string with a dissipative boundary condition, J. Operator Theory, 22 (1989), 369-395.

[20]

B. Levin, "Distribution of Zeros of Entire Functions," American Mathematical Society, Providence, 1980.

[21]

S. Lang, "Algebra," Addison-Wesley, 1984.

[22]

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

[23]

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

[24]

P. Magnusson, G. Alexander, V. Tripathi and A. Weisshaar, "Transmission Lines and Wave Propagation," CRC Press, Boca Raton, 2001.

[25]

G. Miano and A. Maffucci, "Transmission Lines and Lumped Circuits," Academic Press, San Diego, 2001.

[26]

L. J. Myers and W. L. Capper, A transmission line model of the human foetal circulatory system, Medical Engineering and Physics, 24 (2002), 285-294. doi: 10.1016/S1350-4533(02)00019-X.

[27]

S. Nicaise, Spectre des reseaux topologiques finis, Bulletin des sciences mathematique, 111 (1987), 401-413.

[28]

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

[29]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer, New York, 1983.

[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), 217-250. doi: 10.1023/B:ENGI.0000007979.32871.e2.

[31]

J. von Below, A characteristic equation associated to an eigenvalue problem on C2 networks, Lin. Alg. Appl., 71 (1985), 309-325. doi: 10.1016/0024-3795(85)90258-7.

[32]

L. Zhou and G. Kriegsmann, A simple derivation of microstrip transmission line equations, SIAM J. Appl. Math., 70 (2009), 353-367. doi: 10.1137/080737563.

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).

[2]

L. Ahlfors, "Complex Analysis," McGraw-Hill, New York, 1966.

[3]

F. Ali Mehmeti, "Nonlinear Waves in Networks," Akademie Verlag, Berlin, 1994.

[4]

R. Carlson, Inverse eigenvalue problems on directed graphs, Transactions of the American Mathematical Society, 351 (1999), 4069-4088. doi: 10.1090/S0002-9947-99-02175-3.

[5]

R. Carlson, Linear network models related to blood flow, in Quantum Graphs and Their Applications, Contemporary Mathematics, 415 (2006), 65-80.

[6]

C. Cattaneo and L. Fontana, D'Alembert formula on finite one-dimensional networks, J. Math. Anal. Appl., 284 (2003), 403-424. doi: 10.1016/S0022-247X(02)00392-X.

[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), 1665-1693. doi: 10.1137/0149101.

[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), 545-573. doi: 10.1512/iumj.1995.44.2001.

[9]

E. B. Davies, Eigenvalues of an elliptic system, Math. Z., 243 (2003), 719-743. doi: 10.1007/s00209-002-0464-0.

[10]

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

[11]

Y. Fung, "Biomechanics," Springer, New York, 1997.

[12]

I. Herstein, "Topics in Algebra," Xerox College Publishing, Waltham, 1964.

[13]

B. Jessen and H. Tornhave, Mean motions and zeros of almost periodic functions, Acta Math., 77 (1945), 137-279. doi: 10.1007/BF02392225.

[14]

T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, New York, 1995.

[15]

V. Kostrykin, J. Potthoff and R. Schrader, Contraction semigroups on metric graphs, in Analysis on Graphs and Its Applications, PSUM, 77 (2008), 423-458.

[16]

T. Kottos and U. Smilansky, Periodic orbit theory and spectral statistics for quantum graphs, Ann. Phys., 274 (1999), 76-124. doi: 10.1006/aphy.1999.5904.

[17]

M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Math. Z., 249 (2005), 139-162. doi: 10.1007/s00209-004-0695-3.

[18]

M. Kramar Fijavz, D. Mugnolo and E. Sikolya, Variational and semigroup methods for waves and diffusions in networks, Appl. Math. Optim., 55 (2007), 219-240. doi: 10.1007/s00245-006-0887-9.

[19]

M. Krein and A. Nudelman, Some spectral properties of a nonhomogeneous string with a dissipative boundary condition, J. Operator Theory, 22 (1989), 369-395.

[20]

B. Levin, "Distribution of Zeros of Entire Functions," American Mathematical Society, Providence, 1980.

[21]

S. Lang, "Algebra," Addison-Wesley, 1984.

[22]

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

[23]

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

[24]

P. Magnusson, G. Alexander, V. Tripathi and A. Weisshaar, "Transmission Lines and Wave Propagation," CRC Press, Boca Raton, 2001.

[25]

G. Miano and A. Maffucci, "Transmission Lines and Lumped Circuits," Academic Press, San Diego, 2001.

[26]

L. J. Myers and W. L. Capper, A transmission line model of the human foetal circulatory system, Medical Engineering and Physics, 24 (2002), 285-294. doi: 10.1016/S1350-4533(02)00019-X.

[27]

S. Nicaise, Spectre des reseaux topologiques finis, Bulletin des sciences mathematique, 111 (1987), 401-413.

[28]

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

[29]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer, New York, 1983.

[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), 217-250. doi: 10.1023/B:ENGI.0000007979.32871.e2.

[31]

J. von Below, A characteristic equation associated to an eigenvalue problem on C2 networks, Lin. Alg. Appl., 71 (1985), 309-325. doi: 10.1016/0024-3795(85)90258-7.

[32]

L. Zhou and G. Kriegsmann, A simple derivation of microstrip transmission line equations, SIAM J. Appl. Math., 70 (2009), 353-367. doi: 10.1137/080737563.

[1]

Barbara Bianconi, Francesca Papalini. Non-autonomous boundary value problems on the real line. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2020, 40 (2) : 1131-1157. doi: 10.3934/dcds.2020073

[3]

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 and Applied Analysis, 2014, 13 (2) : 881-901. doi: 10.3934/cpaa.2014.13.881

[4]

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

[5]

Vesselin Petkov. Location of eigenvalues for the wave equation with dissipative boundary conditions. Inverse Problems and 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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177

[10]

Iñigo U. Erneta. Well-posedness for boundary value problems for coagulation-fragmentation equations. Kinetic and Related Models, 2020, 13 (4) : 815-835. doi: 10.3934/krm.2020028

[11]

Oleksandr Boichuk, Victor Feruk. Boundary-value problems for weakly singular integral equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1379-1395. doi: 10.3934/dcdsb.2021094

[12]

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

[13]

Youngmok Jeon, Dongwook Shin. Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions. Electronic Research Archive, 2021, 29 (5) : 3361-3382. doi: 10.3934/era.2021043

[14]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3063-3092. doi: 10.3934/dcds.2020398

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

2020 Impact Factor: 1.213

Metrics

  • PDF downloads (85)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]