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

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.
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. 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., (0052).

[3]

L. Amour and J. Faupin, Inverse spectral results for Schrödinger operators in Sobolev spaces,, Int. Math. Res. Notes, 22 (2010), 4319. 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). 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. 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. 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. 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. 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.

[10]

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

[11]

B. J. Levin, Distribution of Zeros of Entire Functions,, Trans. math. Mon. AMS, (1964).

[12]

B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac Operators,, Mathematics and its Applications (Soviet Series), (1991).

[13]

F. Sérier, Inverse spectral problem for singular Ablowitz-Kaup-Newell-Segur operators,, Inverse Problems, 22 (2006), 1457. 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. 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., (0052).

[3]

L. Amour and J. Faupin, Inverse spectral results for Schrödinger operators in Sobolev spaces,, Int. Math. Res. Notes, 22 (2010), 4319. 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). 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. 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. 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. 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. 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.

[10]

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

[11]

B. J. Levin, Distribution of Zeros of Entire Functions,, Trans. math. Mon. AMS, (1964).

[12]

B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac Operators,, Mathematics and its Applications (Soviet Series), (1991).

[13]

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

[1]

J. Douglas Wright. On the spectrum of the superposition of separated potentials.. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 273-281. doi: 10.3934/dcdsb.2013.18.273

[2]

Leonid Golinskii, Mikhail Kudryavtsev. An inverse spectral theory for finite CMV matrices. Inverse Problems & Imaging, 2010, 4 (1) : 93-110. doi: 10.3934/ipi.2010.4.93

[3]

Cornelis van der Mee. Direct scattering of AKNS systems with $L^2$ potentials. Conference Publications, 2015, 2015 (special) : 1089-1097. doi: 10.3934/proc.2015.1089

[4]

Alexei Rybkin. On the boundary control approach to inverse spectral and scattering theory for Schrödinger operators. Inverse Problems & Imaging, 2009, 3 (1) : 139-149. doi: 10.3934/ipi.2009.3.139

[5]

O. A. Veliev. Essential spectral singularities and the spectral expansion for the Hill operator. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2227-2251. doi: 10.3934/cpaa.2017110

[6]

Rémi Leclercq. Spectral invariants in Lagrangian Floer theory. Journal of Modern Dynamics, 2008, 2 (2) : 249-286. doi: 10.3934/jmd.2008.2.249

[7]

Barry Simon. Equilibrium measures and capacities in spectral theory. Inverse Problems & Imaging, 2007, 1 (4) : 713-772. doi: 10.3934/ipi.2007.1.713

[8]

Eduardo Lara, Rodolfo Rodríguez, Pablo Venegas. Spectral approximation of the curl operator in multiply connected domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 235-253. doi: 10.3934/dcdss.2016.9.235

[9]

Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665

[10]

Mario Ahues, Filomena D. d'Almeida, Alain Largillier, Paulo B. Vasconcelos. Defect correction for spectral computations for a singular integral operator. Communications on Pure & Applied Analysis, 2006, 5 (2) : 241-250. doi: 10.3934/cpaa.2006.5.241

[11]

Sergei Avdonin, Fritz Gesztesy, Konstantin A. Makarov. Spectral estimation and inverse initial boundary value problems. Inverse Problems & Imaging, 2010, 4 (1) : 1-9. doi: 10.3934/ipi.2010.4.1

[12]

Miklós Horváth. Spectral shift functions in the fixed energy inverse scattering. Inverse Problems & Imaging, 2011, 5 (4) : 843-858. doi: 10.3934/ipi.2011.5.843

[13]

Simona Fornaro, Abdelaziz Rhandi. On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5049-5058. doi: 10.3934/dcds.2013.33.5049

[14]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703

[15]

Robert Carlson. Spectral theory for nonconservative transmission line networks. Networks & Heterogeneous Media, 2011, 6 (2) : 257-277. doi: 10.3934/nhm.2011.6.257

[16]

Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22

[17]

Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325

[18]

Guillaume Bal, Alexandre Jollivet. Generalized stability estimates in inverse transport theory. Inverse Problems & Imaging, 2018, 12 (1) : 59-90. doi: 10.3934/ipi.2018003

[19]

Veronica Felli, Ana Primo. Classification of local asymptotics for solutions to heat equations with inverse-square potentials. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 65-107. doi: 10.3934/dcds.2011.31.65

[20]

Ricardo Weder, Dimitri Yafaev. Inverse scattering at a fixed energy for long-range potentials. Inverse Problems & Imaging, 2007, 1 (1) : 217-224. doi: 10.3934/ipi.2007.1.217

2017 Impact Factor: 1.465

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]