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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

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 & 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., (1965).   Google Scholar

[2]

N. I. Akhiezer and I. M. Glazman, "Theory of Linear Operators in Hilbert Space,'', Dover Publications, (1993).   Google Scholar

[3]

Ju. M. Berezans'kiĭ, "Expansions in eigenfunctions of selfadjoint operators,'', Translations of Mathematical Monographs, 17 (1968).   Google Scholar

[4]

M. Sh. Birman and M. Z. Solomjak, "Spectral Theory of Selfadjoint Operators in Hilbert Space,'', Mathematics and its Applications (Soviet Series), (1987).   Google Scholar

[5]

M. T. Chu and G. H. Golub, "Inverse Eigenvalue Problems: Theory Algorithms and Applications,'', Numerical Mathematics and Scientific Computation, (2005).   Google Scholar

[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.   Google Scholar

[7]

R. del Rio and M. Kudryavtsev, Inverse problems for Jacobi operators I: Interior mass-spring perturbations in finite systems,, , ().   Google Scholar

[8]

R. del Rio, M. Kudryavtsev and L. O. Silva, Inverse problems for Jacobi operators II: Mass perturbations of semi-infinite mass-spring systems,, , ().   Google Scholar

[9]

L. Fu and H. Hochstadt, Inverse theorems for Jacobi matrices,, J. Math. Anal. Appl., 47 (1974), 162.  doi: 10.1016/0022-247X(74)90044-4.  Google Scholar

[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.  doi: 10.1007/BF02788147.  Google Scholar

[11]

G. M. L. Gladwell, "Inverse Problems in Vibration,'', Second edition, 119 (2004).   Google Scholar

[12]

R. Z. Halilova, An inverse problem,, (Russian), 1967 (1967), 169.   Google Scholar

[13]

B. Ja. Levin, "Distribution of Zeros of Entire Functions,'', Translations of Mathematical Monographs, 5 (1980).   Google Scholar

[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 (2004).   Google Scholar

[15]

Y. M. Ram, Inverse eigenvalue problem for a modified vibrating system,, SIAM Appl. Math., 53 (1993), 1762.   Google Scholar

[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.   Google Scholar

[17]

B. Simon, The classical moment problem as a self-adjoint finite difference operator,, Adv. Math., 137 (1998), 82.   Google Scholar

[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).   Google Scholar

[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).   Google Scholar

[20]

G. Teschl, Trace formulas and inverse spectral theory for Jacobi operators,, Comm. Math. Phys., 196 (1998), 175.   Google Scholar

[21]

G. Teschl, "Jacobi Operators and Completely Integrable Nonlinear Lattices,'', Mathematical Surveys and Monographs, 72 (2000).   Google Scholar

show all references

References:
[1]

N. I. Akhiezer, “The Classical Moment Problem and Some Related Questions in Analysis,'', Hafner Publishing Co., (1965).   Google Scholar

[2]

N. I. Akhiezer and I. M. Glazman, "Theory of Linear Operators in Hilbert Space,'', Dover Publications, (1993).   Google Scholar

[3]

Ju. M. Berezans'kiĭ, "Expansions in eigenfunctions of selfadjoint operators,'', Translations of Mathematical Monographs, 17 (1968).   Google Scholar

[4]

M. Sh. Birman and M. Z. Solomjak, "Spectral Theory of Selfadjoint Operators in Hilbert Space,'', Mathematics and its Applications (Soviet Series), (1987).   Google Scholar

[5]

M. T. Chu and G. H. Golub, "Inverse Eigenvalue Problems: Theory Algorithms and Applications,'', Numerical Mathematics and Scientific Computation, (2005).   Google Scholar

[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.   Google Scholar

[7]

R. del Rio and M. Kudryavtsev, Inverse problems for Jacobi operators I: Interior mass-spring perturbations in finite systems,, , ().   Google Scholar

[8]

R. del Rio, M. Kudryavtsev and L. O. Silva, Inverse problems for Jacobi operators II: Mass perturbations of semi-infinite mass-spring systems,, , ().   Google Scholar

[9]

L. Fu and H. Hochstadt, Inverse theorems for Jacobi matrices,, J. Math. Anal. Appl., 47 (1974), 162.  doi: 10.1016/0022-247X(74)90044-4.  Google Scholar

[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.  doi: 10.1007/BF02788147.  Google Scholar

[11]

G. M. L. Gladwell, "Inverse Problems in Vibration,'', Second edition, 119 (2004).   Google Scholar

[12]

R. Z. Halilova, An inverse problem,, (Russian), 1967 (1967), 169.   Google Scholar

[13]

B. Ja. Levin, "Distribution of Zeros of Entire Functions,'', Translations of Mathematical Monographs, 5 (1980).   Google Scholar

[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 (2004).   Google Scholar

[15]

Y. M. Ram, Inverse eigenvalue problem for a modified vibrating system,, SIAM Appl. Math., 53 (1993), 1762.   Google Scholar

[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.   Google Scholar

[17]

B. Simon, The classical moment problem as a self-adjoint finite difference operator,, Adv. Math., 137 (1998), 82.   Google Scholar

[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).   Google Scholar

[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).   Google Scholar

[20]

G. Teschl, Trace formulas and inverse spectral theory for Jacobi operators,, Comm. Math. Phys., 196 (1998), 175.   Google Scholar

[21]

G. Teschl, "Jacobi Operators and Completely Integrable Nonlinear Lattices,'', Mathematical Surveys and Monographs, 72 (2000).   Google Scholar

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