Nous considérons un problème spectral inverse pour des équations de Sturm–Liouville sur l'intervalle unité avec une singularité explicite , . Un tel problème survient après décomposition de l'opérateur de Schrödinger à potentiel radial agissant sur la boule unité de . Notre but est la paramétrisation globale des potentiels par des données spectrales, notées et des constantes de normalisation, notées . Pour et 1, il est déjà connu que forme un système de coordonnées globales sur . Nous étendons cela à tout entier positif a. Un résultat similaire est obtenu pour un opérateur de type AKNS singulier.
We consider an inverse spectral problem for singular Sturm–Liouville equations on the unit interval with explicit singularity , . This problem arises by splitting of the Schrödinger operator with radial potential acting on the unit ball of . Our goal is the global parametrization of potentials by spectral data noted by , and some norming constants noted by . For and 1, was already known to be a global coordinate system on . We extend this to any non-negative integer a. Similar result is obtained for singular AKNS operator.
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Frédéric Serier 1
@article{CRMATH_2005__340_9_671_0, author = {Fr\'ed\'eric Serier}, title = {Inverse spectral problem for singular {AKNS} and {Schr\"odinger} operators on $ [0,1]$}, journal = {Comptes Rendus. Math\'ematique}, pages = {671--676}, publisher = {Elsevier}, volume = {340}, number = {9}, year = {2005}, doi = {10.1016/j.crma.2005.03.025}, language = {en}, }
Frédéric Serier. Inverse spectral problem for singular AKNS and Schrödinger operators on $ [0,1]$. Comptes Rendus. Mathématique, Volume 340 (2005) no. 9, pp. 671-676. doi : 10.1016/j.crma.2005.03.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.03.025/
[1] High- and low-energy estimates for the Dirac equation, J. Math. Phys., Volume 36 (1995) no. 3, pp. 991-1015
[2] Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math., Volume 78 (1946), pp. 1-96
[3] Inverse spectral theory for some singular Sturm–Liouville problems, J. Differential Equations, Volume 106 (1993) no. 1, pp. 121-140
[4] A Borg–Levinson theorem for Bessel operators, Pacific J. Math., Volume 177 (1997) no. 1, pp. 1-26
[5] Spectral rigidity for radial Schrödinger operators, J. Differential Equations, Volume 113 (1994) no. 2, pp. 338-354
[6] Gaps of one-dimensional periodic AKNS systems, Forum Math., Volume 5 (1993) no. 5, pp. 459-504
[7] Estimates on periodic and Dirichlet eigenvalues for the Zakharov–Shabat system, Asymptotic Anal., Volume 25 (2001) no. 3–4, pp. 201-237
[8] Inverse spectral theory for a singular Sturm–Liouville operator on , J. Differential Equations, Volume 76 (1988) no. 2, pp. 353-373
[9] The inverse Sturm–Liouville problem, Mat. Tidsskr. B., Volume 1949 (1949), pp. 25-30
[10] Inverse Spectral Theory, Academic Press, Boston, 1987
[11] Methods of Modern Mathematical Physics II, Academic Press, New York, 1975
[12] Reconstruction of a radially symmetric potential from two spectral sequences, J. Math. Anal. Appl., Volume 264 (2001) no. 2, pp. 354-381
[13] F. Serier, Inverse Spectral Problem for Singular AKNS Operator with a Radial Potential, in preparation
[14] Inverse eigenvalue problems for a singular Sturm–Liouville operator on , Inverse Problems, Volume 10 (1994) no. 4, pp. 975-987
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