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

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	O. H. Hald, Inverse eigenvalue problem for the mantle, Geophys. J. R. Astr. Soc., 62 (1980), 41-48.
[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.
[11]	B. J. Levin, Distribution of Zeros of Entire Functions, Trans. math. Mon. AMS, vol. 5, Springer-Verlag, Providence, R.I., 1964.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	O. H. Hald, Inverse eigenvalue problem for the mantle, Geophys. J. R. Astr. Soc., 62 (1980), 41-48.
[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.
[11]	B. J. Levin, Distribution of Zeros of Entire Functions, Trans. math. Mon. AMS, vol. 5, Springer-Verlag, Providence, R.I., 1964.
[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.
[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.

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