Continuous dependence of the transmission eigenvalues in one dimension

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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

Abstract
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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 and 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.
[2]	R. P. Boas, Entire Functions, Academic Press, New York, 1954.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	A. Kirsch, On the existence of transmission eigenvalues, Inverse Probl. Imaging, 3 (2009), 155-172. doi: 10.3934/ipi.2009.3.155.
[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.
[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.
[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.
[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.
[19]	J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, Boston, 1987.
[20]	J. Sylvester, Transmission eigenvalues in one dimension, Inverse Problems, 29 (2013), 104009, 11pp. doi: 10.1088/0266-5611/29/10/104009.
[21]	A. Zettl, Sturm-Liouville Theory, vol. 121 of Mathematical Surveys and Monographs, American Mathematical Society, 2005. doi: 10.1090/surv/121.

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.
[2]	R. P. Boas, Entire Functions, Academic Press, New York, 1954.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	A. Kirsch, On the existence of transmission eigenvalues, Inverse Probl. Imaging, 3 (2009), 155-172. doi: 10.3934/ipi.2009.3.155.
[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.
[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.
[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.
[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.
[19]	J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, Boston, 1987.
[20]	J. Sylvester, Transmission eigenvalues in one dimension, Inverse Problems, 29 (2013), 104009, 11pp. doi: 10.1088/0266-5611/29/10/104009.
[21]	A. Zettl, Sturm-Liouville Theory, vol. 121 of Mathematical Surveys and Monographs, American Mathematical Society, 2005. doi: 10.1090/surv/121.

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