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

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 & 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

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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

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