Inverse problems for Jacobi operators III: Mass-spring perturbations of semi-infinite systems

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November 2012, 6(4): 599-621. doi: 10.3934/ipi.2012.6.599

Inverse problems for Jacobi operators III: Mass-spring perturbations of semi-infinite systems

Rafael del Rio ^1, , Mikhail Kudryavtsev ^2, and Luis O. Silva ^1,

1.	Departamento de Métodos Matemáticos y Numéricos, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, C.P. 04510, México D.F., Mexico, Mexico
2.	Department of Mathematics, Institute for Low Temperature Physics and Engineering, Lenin Av. 47, 61103, Kharkov, Ukraine

Received June 2011 Revised July 2012 Published November 2012

Abstract
Related Papers

Consider an infinite linear mass-spring system and a modification of it obtained by changing the first mass and spring of the system. We give results on the interplay of the spectra of such systems and on the reconstruction of the system from its spectrum and the one of the modified system. Furthermore, we provide necessary and sufficient conditions for two sequences to be the spectra of the mass-spring system and the perturbed one.

Keywords: Jacobi matrices, Infinite mass-spring system, Two-spectra inverse problem..

Mathematics Subject Classification: Primary: 34K29, 47A75, 47B36, 70F1.

Citation: Rafael del Rio, Mikhail Kudryavtsev, Luis O. Silva. Inverse problems for Jacobi operators III: Mass-spring perturbations of semi-infinite systems. Inverse Problems and Imaging, 2012, 6 (4) : 599-621. doi: 10.3934/ipi.2012.6.599

References:

[1]	N. I. Akhiezer, “The Classical Moment Problem and Some Related Questions in Analysis,'' Hafner Publishing Co., New York, 1965.
[2]	N. I. Akhiezer and I. M. Glazman, "Theory of Linear Operators in Hilbert Space,'' Dover Publications, Inc., New York, 1993.
[3]	Ju. M. Berezans'kiĭ, "Expansions in eigenfunctions of selfadjoint operators,'' Translations of Mathematical Monographs, 17, American Mathematical Society, Providence, R.I., 1968.
[4]	M. Sh. Birman and M. Z. Solomjak, "Spectral Theory of Selfadjoint Operators in Hilbert Space,'' Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, 1987.
[5]	M. T. Chu and G. H. Golub, "Inverse Eigenvalue Problems: Theory Algorithms and Applications,'' Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2005.
[6]	C. de Boor and G. H. Golub, The numerically stable reconstruction of a Jacobi matrix from spectral data, Linear Alg. Appl., 21 (1978), 245-260.
[7]	R. del Rio and M. Kudryavtsev, Inverse problems for Jacobi operators I: Interior mass-spring perturbations in finite systems, arXiv:1106.1691.
[8]	R. del Rio, M. Kudryavtsev and L. O. Silva, Inverse problems for Jacobi operators II: Mass perturbations of semi-infinite mass-spring systems, arXiv:1106.4598.
[9]	L. Fu and H. Hochstadt, Inverse theorems for Jacobi matrices, J. Math. Anal. Appl., 47 (1974), 162-168. doi: 10.1016/0022-247X(74)90044-4.
[10]	F. Gesztesy and B. Simon, $m$-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices, J. Anal. Math., 73 (1997), 267-297. doi: 10.1007/BF02788147.
[11]	G. M. L. Gladwell, "Inverse Problems in Vibration,'' Second edition, Solid Mechanics and its Applications, 119, Kluwer Academic Publishers, Dordrecht, 2004.
[12]	R. Z. Halilova, An inverse problem, (Russian), Izv. Akad. Nauk Azerbaĭ džan, SSR Ser. Fiz.-Tehn. Mat. Nauk, 1967 (1967), 169-175.
[13]	B. Ja. Levin, "Distribution of Zeros of Entire Functions,'' Translations of Mathematical Monographs, 5, American Mathematical Society, Providence, R.I., 1980.
[14]	V. A. Marchenko and T. V. Misyura, "Señalamientos Metodológicos y Didácticos al Tema: Problemas Inversos de la Teoría Espectral de Operadores de Dimensión Finita,'' Monografías IIMAS-UNAM, 12, No. 28, México, 2004.
[15]	Y. M. Ram, Inverse eigenvalue problem for a modified vibrating system, SIAM Appl. Math., 53 (1993), 1762-1775.
[16]	L. O. Silva and R. Weder, On the two spectra inverse problem for semi-infinite Jacobi matrices, Math. Phys. Anal. Geom., 9 (2006), 263-290.
[17]	B. Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math., 137 (1998), 82-203.
[18]	M. Spletzer, A. Raman, H. Sumali and J. P. Sullivan, Highly sensitive mass detection and identification using vibration localization in coupled microcantilever arrays, Applied Physics Letters, 92 (2008), 114102.
[19]	M. Spletzer, A. Raman, A. Q. Wu and X. Xu, Ultrasensitive mass sensing using mode localization in coupled microcantilevers, Applied Physics Letters, 88 (2006), 254102.
[20]	G. Teschl, Trace formulas and inverse spectral theory for Jacobi operators, Comm. Math. Phys., 196 (1998), 175-202.
[21]	G. Teschl, "Jacobi Operators and Completely Integrable Nonlinear Lattices,'' Mathematical Surveys and Monographs, 72, American Mathematical Society, Providence, RI, 2000.

show all references

References:

[1]	N. I. Akhiezer, “The Classical Moment Problem and Some Related Questions in Analysis,'' Hafner Publishing Co., New York, 1965.
[2]	N. I. Akhiezer and I. M. Glazman, "Theory of Linear Operators in Hilbert Space,'' Dover Publications, Inc., New York, 1993.
[3]	Ju. M. Berezans'kiĭ, "Expansions in eigenfunctions of selfadjoint operators,'' Translations of Mathematical Monographs, 17, American Mathematical Society, Providence, R.I., 1968.
[4]	M. Sh. Birman and M. Z. Solomjak, "Spectral Theory of Selfadjoint Operators in Hilbert Space,'' Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, 1987.
[5]	M. T. Chu and G. H. Golub, "Inverse Eigenvalue Problems: Theory Algorithms and Applications,'' Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2005.
[6]	C. de Boor and G. H. Golub, The numerically stable reconstruction of a Jacobi matrix from spectral data, Linear Alg. Appl., 21 (1978), 245-260.
[7]	R. del Rio and M. Kudryavtsev, Inverse problems for Jacobi operators I: Interior mass-spring perturbations in finite systems, arXiv:1106.1691.
[8]	R. del Rio, M. Kudryavtsev and L. O. Silva, Inverse problems for Jacobi operators II: Mass perturbations of semi-infinite mass-spring systems, arXiv:1106.4598.
[9]	L. Fu and H. Hochstadt, Inverse theorems for Jacobi matrices, J. Math. Anal. Appl., 47 (1974), 162-168. doi: 10.1016/0022-247X(74)90044-4.
[10]	F. Gesztesy and B. Simon, $m$-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices, J. Anal. Math., 73 (1997), 267-297. doi: 10.1007/BF02788147.
[11]	G. M. L. Gladwell, "Inverse Problems in Vibration,'' Second edition, Solid Mechanics and its Applications, 119, Kluwer Academic Publishers, Dordrecht, 2004.
[12]	R. Z. Halilova, An inverse problem, (Russian), Izv. Akad. Nauk Azerbaĭ džan, SSR Ser. Fiz.-Tehn. Mat. Nauk, 1967 (1967), 169-175.
[13]	B. Ja. Levin, "Distribution of Zeros of Entire Functions,'' Translations of Mathematical Monographs, 5, American Mathematical Society, Providence, R.I., 1980.
[14]	V. A. Marchenko and T. V. Misyura, "Señalamientos Metodológicos y Didácticos al Tema: Problemas Inversos de la Teoría Espectral de Operadores de Dimensión Finita,'' Monografías IIMAS-UNAM, 12, No. 28, México, 2004.
[15]	Y. M. Ram, Inverse eigenvalue problem for a modified vibrating system, SIAM Appl. Math., 53 (1993), 1762-1775.
[16]	L. O. Silva and R. Weder, On the two spectra inverse problem for semi-infinite Jacobi matrices, Math. Phys. Anal. Geom., 9 (2006), 263-290.
[17]	B. Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math., 137 (1998), 82-203.
[18]	M. Spletzer, A. Raman, H. Sumali and J. P. Sullivan, Highly sensitive mass detection and identification using vibration localization in coupled microcantilever arrays, Applied Physics Letters, 92 (2008), 114102.
[19]	M. Spletzer, A. Raman, A. Q. Wu and X. Xu, Ultrasensitive mass sensing using mode localization in coupled microcantilevers, Applied Physics Letters, 88 (2006), 254102.
[20]	G. Teschl, Trace formulas and inverse spectral theory for Jacobi operators, Comm. Math. Phys., 196 (1998), 175-202.
[21]	G. Teschl, "Jacobi Operators and Completely Integrable Nonlinear Lattices,'' Mathematical Surveys and Monographs, 72, American Mathematical Society, Providence, RI, 2000.

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