Spectral theory for nonconservative transmission line networks

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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

Abstract
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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 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.

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