American Institute of Mathematical Sciences

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February  2015, 9(1): 273-287. doi: 10.3934/ipi.2015.9.273

Continuous dependence of the transmission eigenvalues in one dimension

Yalin Zhang 1, and  Guoliang Shi 1,

1. 

Department of Mathematics, Tianjin University, Tianjin, 300072, China, China

Received  May 2014 Revised  September 2014 Published  January 2015

The transmission eigenvalue problem on the interval $\left[ a,b\right] $ is considered. We show that the isolated transmission eigenvalues are continuous functions of the coefficients of the problem and that the transmission eigenfunctions can be normalized so that they depend continuously on all coefficients in the uniform norm. Throughout this work, our results are established without assumptions on the sign of the contrasts.
Keywords: inhomogeneous medium, transmission eigenfunctions, continuous dependence., transmission eigenvalues, Interior transmission problem.
Mathematics Subject Classification: Primary: 34B40, 34L25; Secondary: 78A2.
Citation: Yalin Zhang, Guoliang Shi. Continuous dependence of the transmission eigenvalues in one dimension. Inverse Problems & Imaging, 2015, 9 (1) : 273-287. doi: 10.3934/ipi.2015.9.273
References:
[1]

T. Aktosun, D. Gintides and V. G. Papanicolaou, The uniqueness in the inverse problem for transmission eigenvalues for the spherically symmetric variable-speed wave equation, Inverse Problems, 27 (2011), 115004, 17pp. doi: 10.1088/0266-5611/27/11/115004.  Google Scholar

[2]

R. P. Boas, Entire Functions, Academic Press, New York, 1954.  Google Scholar

[3]

A. S. Bonnet-BenDhia, L. Chesnel and H. Haddar, On the use of T-coercivity to study the interior transmission eigenvalue problem, C. R. Math. Acad. Sci., 349 (2011), 647-651. doi: 10.1016/j.crma.2011.05.008.  Google Scholar

[4]

F. Cakoni, D. Colton and H. Hadder, The interior transmission problem for regions with cavities, SIAM J. Math. Anal., 42 (2010), 145-162. doi: 10.1137/090754637.  Google Scholar

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F. Cakoni, A. Cossonnière and H. Haddar, Transmission eigenvalues for inhomogeneous media containing obstacles, Inverse Probl. Imaging, 6 (2012), 373-398. doi: 10.3934/ipi.2012.6.373.  Google Scholar

[6]

F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42 (2010), 237-255. doi: 10.1137/090769338.  Google Scholar

[7]

F. Cakoni and H. Hadder, On the existence of transmission eigenvalues in an inhomogeneous medium, Appl. Anal., 88 (2009), 475-493. doi: 10.1080/00036810802713966.  Google Scholar

[8]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, $3^{nd}$ edition, Springer-Verlag, Berlin, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[9]

F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem, Int. J. Comput. Sci. Math., 3 (2010), 142-167. doi: 10.1137/100793542.  Google Scholar

[10]

D. Colton, A. Kirsch and L. Päivärinta, Far-field patterns for acoustic waves in an inhomogeneous medium, SIAM J. Math. Anal., 20 (1989), 1472-1483. doi: 10.1137/0520096.  Google Scholar

[11]

D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem, Inverse Probl. Imaging, 1 (2007), 13-28. doi: 10.3934/ipi.2007.1.13.  Google Scholar

[12]

K. S. Hickmann, Interior transmission eigenvalue problem with refractive index having $C^{2}$-transition to the background medium, Appl. Anal., 91 (2012), 1675-1690. doi: 10.1080/00036811.2011.577741.  Google Scholar

[13]

M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients, SIAM J. Math. Anal., 42 (2010), 2965-2986. doi: 10.1137/100793748.  Google Scholar

[14]

A. Kirsch, On the existence of transmission eigenvalues, Inverse Probl. Imaging, 3 (2009), 155-172. doi: 10.3934/ipi.2009.3.155.  Google Scholar

[15]

Q. Kong and A. Zettl, Eigenvalues of regular Sturm-Liouville problems, J. Differential Equations, 131 (1996), 1-19. doi: 10.1006/jdeq.1996.0154.  Google Scholar

[16]

Q. Kong and A. Zettl, Linear ordinary differential equations, in Inequalities and Applications (eds. R. P. Agarwal), WSSIAA, 3 (1994), 381-397. doi: 10.1142/9789812798879_0031.  Google Scholar

[17]

Y. Leung and D. Colton, Complex transmission eigenvalues for spherically stratified media, Inverse Problems, 28 (2012), 075005, 9pp. doi: 10.1088/0266-5611/28/7/075005.  Google Scholar

[18]

J. McLaughlin and P. L. Polyakov, On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues, J. Differential Equations, 107 (1994), 351-382. doi: 10.1006/jdeq.1994.1017.  Google Scholar

[19]

J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, Boston, 1987.  Google Scholar

[20]

J. Sylvester, Transmission eigenvalues in one dimension, Inverse Problems, 29 (2013), 104009, 11pp. doi: 10.1088/0266-5611/29/10/104009.  Google Scholar

[21]

A. Zettl, Sturm-Liouville Theory, vol. 121 of Mathematical Surveys and Monographs, American Mathematical Society, 2005. doi: 10.1090/surv/121.  Google Scholar

show all references

References:
[1]

T. Aktosun, D. Gintides and V. G. Papanicolaou, The uniqueness in the inverse problem for transmission eigenvalues for the spherically symmetric variable-speed wave equation, Inverse Problems, 27 (2011), 115004, 17pp. doi: 10.1088/0266-5611/27/11/115004.  Google Scholar

[2]

R. P. Boas, Entire Functions, Academic Press, New York, 1954.  Google Scholar

[3]

A. S. Bonnet-BenDhia, L. Chesnel and H. Haddar, On the use of T-coercivity to study the interior transmission eigenvalue problem, C. R. Math. Acad. Sci., 349 (2011), 647-651. doi: 10.1016/j.crma.2011.05.008.  Google Scholar

[4]

F. Cakoni, D. Colton and H. Hadder, The interior transmission problem for regions with cavities, SIAM J. Math. Anal., 42 (2010), 145-162. doi: 10.1137/090754637.  Google Scholar

[5]

F. Cakoni, A. Cossonnière and H. Haddar, Transmission eigenvalues for inhomogeneous media containing obstacles, Inverse Probl. Imaging, 6 (2012), 373-398. doi: 10.3934/ipi.2012.6.373.  Google Scholar

[6]

F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42 (2010), 237-255. doi: 10.1137/090769338.  Google Scholar

[7]

F. Cakoni and H. Hadder, On the existence of transmission eigenvalues in an inhomogeneous medium, Appl. Anal., 88 (2009), 475-493. doi: 10.1080/00036810802713966.  Google Scholar

[8]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, $3^{nd}$ edition, Springer-Verlag, Berlin, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[9]

F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem, Int. J. Comput. Sci. Math., 3 (2010), 142-167. doi: 10.1137/100793542.  Google Scholar

[10]

D. Colton, A. Kirsch and L. Päivärinta, Far-field patterns for acoustic waves in an inhomogeneous medium, SIAM J. Math. Anal., 20 (1989), 1472-1483. doi: 10.1137/0520096.  Google Scholar

[11]

D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem, Inverse Probl. Imaging, 1 (2007), 13-28. doi: 10.3934/ipi.2007.1.13.  Google Scholar

[12]

K. S. Hickmann, Interior transmission eigenvalue problem with refractive index having $C^{2}$-transition to the background medium, Appl. Anal., 91 (2012), 1675-1690. doi: 10.1080/00036811.2011.577741.  Google Scholar

[13]

M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients, SIAM J. Math. Anal., 42 (2010), 2965-2986. doi: 10.1137/100793748.  Google Scholar

[14]

A. Kirsch, On the existence of transmission eigenvalues, Inverse Probl. Imaging, 3 (2009), 155-172. doi: 10.3934/ipi.2009.3.155.  Google Scholar

[15]

Q. Kong and A. Zettl, Eigenvalues of regular Sturm-Liouville problems, J. Differential Equations, 131 (1996), 1-19. doi: 10.1006/jdeq.1996.0154.  Google Scholar

[16]

Q. Kong and A. Zettl, Linear ordinary differential equations, in Inequalities and Applications (eds. R. P. Agarwal), WSSIAA, 3 (1994), 381-397. doi: 10.1142/9789812798879_0031.  Google Scholar

[17]

Y. Leung and D. Colton, Complex transmission eigenvalues for spherically stratified media, Inverse Problems, 28 (2012), 075005, 9pp. doi: 10.1088/0266-5611/28/7/075005.  Google Scholar

[18]

J. McLaughlin and P. L. Polyakov, On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues, J. Differential Equations, 107 (1994), 351-382. doi: 10.1006/jdeq.1994.1017.  Google Scholar

[19]

J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, Boston, 1987.  Google Scholar

[20]

J. Sylvester, Transmission eigenvalues in one dimension, Inverse Problems, 29 (2013), 104009, 11pp. doi: 10.1088/0266-5611/29/10/104009.  Google Scholar

[21]

A. Zettl, Sturm-Liouville Theory, vol. 121 of Mathematical Surveys and Monographs, American Mathematical Society, 2005. doi: 10.1090/surv/121.  Google Scholar

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