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November  2013, 7(4): 1115-1122. doi: 10.3934/ipi.2013.7.1115

Inverse spectral results in Sobolev spaces for the AKNS operator with partial informations on the potentials

Laurent Amour 1, and  Jérémy Faupin 2,

1. 

Laboratoire de Mathématiques de Reims, EA 4535 and FR CNRS 3399, Université de Reims Champagne-Ardenne, BP 1039, 51687 REIMS Cedex 2, France
2. 

Institut Elie Cartan de Lorraine, UMR CNRS 7502, Université de Lorraine, Ile du Saulcy, 57045 Metz Cedex 1, France

Received  October 2012 Revised  June 2013 Published  November 2013

We consider the AKNS (Ablowitz-Kaup-Newell-Segur) operator on the unit interval with potentials belonging to Sobolev spaces in the framework of inverse spectral theory. Precise sets of eigenvalues are given in order that they, together with the knowledge of the potentials on the side $(a,1)$ and partial informations on the potential on $(a-\varepsilon,a)$ for some arbitrary small $\varepsilon>0$, determine the potentials entirely on $(0,1)$. Naturally, the smaller is $a$ and the more partial informations are known, the less is the number of the needed eigenvalues.
Keywords: determination of potentials., AKNS operator, inverse spectral theory.
Mathematics Subject Classification: Primary: 34A55; Secondary: 34L40, 47E0.
Citation: Laurent Amour, Jérémy Faupin. Inverse spectral results in Sobolev spaces for the AKNS operator with partial informations on the potentials. Inverse Problems & Imaging, 2013, 7 (4) : 1115-1122. doi: 10.3934/ipi.2013.7.1115
References:
[1]

L. Amour, Inverse spectral theory for the AKNS system with separated boundary conditions, Inv. Problems, 5 (1993), 507-523. doi: 10.1088/0266-5611/9/5/001.  Google Scholar

[2]

L. Amour, The coordinates system $\kappa \times \mu$ on $L^2(0,1) \times L^2(0,1) $for the AKNS operators on the unit interval, Preprint Hal 00526898. Google Scholar

[3]

L. Amour and J. Faupin, Inverse spectral results for Schrödinger operators in Sobolev spaces, Int. Math. Res. Notes, 22 (2010), 4319-4333. doi: 10.1093/imrn/rnq040.  Google Scholar

[4]

L. Amour, J. Faupin and T. Raoux, Inverse spectral results for Schrödinger operators on the unit interval with partial informations given on the potentials, Journal of Mathematical Physics, 50 (2009), 033505. doi: 10.1063/1.3087426.  Google Scholar

[5]

R. del Rio, F. Gesztesy and B. Simon, Inverse spectral analysis with partial information on the potential, III. Updating boundary conditions, Int. Math. Res. Notices, 15 (1997), 751-758. doi: 10.1155/S1073792897000494.  Google Scholar

[6]

R. del Rio, F. Gesztesy and B. Simon, Corrections and addendum to Inverse spectral analysis with partial information on the potential, III. Updating boundary conditions, Int. Math. Res. Notices, 11 (1999), 623-625. doi: 10.1155/S107379289900032X.  Google Scholar

[7]

R. del Rio and B. Grébert, Inverse spectral results for the AKNS systems with partial information on the potentials, Math. Phys. Anal. Geom., 4 (2001), 229-244. doi: 10.1023/A:1012981630059.  Google Scholar

[8]

F. Gesztesy and B. Simon, Inverse spectral analysis with partial information on the potential. II. The case of discrete spectrum, Trans. Amer. Math. Soc., 352 (2000), 2765-2787. doi: 10.1090/S0002-9947-99-02544-1.  Google Scholar

[9]

O. H. Hald, Inverse eigenvalue problem for the mantle, Geophys. J. R. Astr. Soc., 62 (1980), 41-48. Google Scholar

[10]

H. Hochstadt and B. Lieberman, An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math., 34 (1978), 676-680. doi: 10.1137/0134054.  Google Scholar

[11]

B. J. Levin, Distribution of Zeros of Entire Functions, Trans. math. Mon. AMS, vol. 5, Springer-Verlag, Providence, R.I., 1964.  Google Scholar

[12]

B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac Operators, Mathematics and its Applications (Soviet Series), vol. 59, Kluwer Academic Publishers Group, Dordrecht 1991.  Google Scholar

[13]

F. Sérier, Inverse spectral problem for singular Ablowitz-Kaup-Newell-Segur operators, Inverse Problems, 22 (2006), 1457-1484. doi: 10.1088/0266-5611/22/4/018.  Google Scholar

show all references

References:
[1]

L. Amour, Inverse spectral theory for the AKNS system with separated boundary conditions, Inv. Problems, 5 (1993), 507-523. doi: 10.1088/0266-5611/9/5/001.  Google Scholar

[2]

L. Amour, The coordinates system $\kappa \times \mu$ on $L^2(0,1) \times L^2(0,1) $for the AKNS operators on the unit interval, Preprint Hal 00526898. Google Scholar

[3]

L. Amour and J. Faupin, Inverse spectral results for Schrödinger operators in Sobolev spaces, Int. Math. Res. Notes, 22 (2010), 4319-4333. doi: 10.1093/imrn/rnq040.  Google Scholar

[4]

L. Amour, J. Faupin and T. Raoux, Inverse spectral results for Schrödinger operators on the unit interval with partial informations given on the potentials, Journal of Mathematical Physics, 50 (2009), 033505. doi: 10.1063/1.3087426.  Google Scholar

[5]

R. del Rio, F. Gesztesy and B. Simon, Inverse spectral analysis with partial information on the potential, III. Updating boundary conditions, Int. Math. Res. Notices, 15 (1997), 751-758. doi: 10.1155/S1073792897000494.  Google Scholar

[6]

R. del Rio, F. Gesztesy and B. Simon, Corrections and addendum to Inverse spectral analysis with partial information on the potential, III. Updating boundary conditions, Int. Math. Res. Notices, 11 (1999), 623-625. doi: 10.1155/S107379289900032X.  Google Scholar

[7]

R. del Rio and B. Grébert, Inverse spectral results for the AKNS systems with partial information on the potentials, Math. Phys. Anal. Geom., 4 (2001), 229-244. doi: 10.1023/A:1012981630059.  Google Scholar

[8]

F. Gesztesy and B. Simon, Inverse spectral analysis with partial information on the potential. II. The case of discrete spectrum, Trans. Amer. Math. Soc., 352 (2000), 2765-2787. doi: 10.1090/S0002-9947-99-02544-1.  Google Scholar

[9]

O. H. Hald, Inverse eigenvalue problem for the mantle, Geophys. J. R. Astr. Soc., 62 (1980), 41-48. Google Scholar

[10]

H. Hochstadt and B. Lieberman, An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math., 34 (1978), 676-680. doi: 10.1137/0134054.  Google Scholar

[11]

B. J. Levin, Distribution of Zeros of Entire Functions, Trans. math. Mon. AMS, vol. 5, Springer-Verlag, Providence, R.I., 1964.  Google Scholar

[12]

B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac Operators, Mathematics and its Applications (Soviet Series), vol. 59, Kluwer Academic Publishers Group, Dordrecht 1991.  Google Scholar

[13]

F. Sérier, Inverse spectral problem for singular Ablowitz-Kaup-Newell-Segur operators, Inverse Problems, 22 (2006), 1457-1484. doi: 10.1088/0266-5611/22/4/018.  Google Scholar

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