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November  2011, 5(4): 843-858. doi: 10.3934/ipi.2011.5.843

## Spectral shift functions in the fixed energy inverse scattering

 1 Department of Mathematical Analysis, Institute of Mathematics, Budapest University of Technology and Economics, H 1111 Budapest, Műegyetem rkp. 3-9., Hungary

Received  June 2011 Revised  June 2011 Published  November 2011

In this paper the notion of the Krein spectral shift function is extended to the radial Schrödinger operator with fixed energy. Then we analyze the connections between the tail of the potential and the decay rate of the fixed-energy phase shifts. Finally we extend former results on the uniqueness of the fixed-energy inverse scattering problem to a general class of potentials.
Citation: Miklós Horváth. Spectral shift functions in the fixed energy inverse scattering. Inverse Problems and Imaging, 2011, 5 (4) : 843-858. doi: 10.3934/ipi.2011.5.843
##### References:
 [1] , Digital Library of Mathematical Functions, http://dlmf.nist.gov/idx/. [2] M. Abramowitz and I. Stegun, eds., "Handbook of Mathematical Functions," Dover Publications, New York, 1972. [3] V. Alfaro and T. Regge, "Potential Scattering," North-Holland, Amsterdam, 1965. [4] N. Aronszajn and W. F. Donoghue, On exponential representations of analytic functions in the upper half-plane with positive imaginary part, J. d'Analyse Math., 5 (1957), 321-388. doi: 10.1007/BF02937349. [5] A. Erdélyi, W. Magnus, F. Oberhettinger, F. Tricomi and G. Francesco, "Higher Transcendental Functions," Vol. II, Based on notes left by Harry Bateman, McGraw-Hill, 1953. [6] M. Sh. Birman and D. R. Yafaev, The spectral shift function. The papers of M. G. Kreĭn and their further development, St. Petersburg Math. J., 4 (1993), 833-870. [7] K. Chadan and P. C. Sabatier, "Inverse Problems in Quantum Scattering Theory," Second edition, With a foreword by R. G. Newton, Texts in Monographs in Physics, Springer-Verlag, New York, 1989. [8] W. H. J. Fuchs, A generalization of Carlson's theorem, J. London Math. Soc., 21 (1946), 106-110. doi: 10.1112/jlms/s1-21.2.106. [9] M. Giffon G. Burdet and E. Predazzi, On the inversion problem in the $\lambda$-plane, Nuovo Cimento (10), 36 (1965), 1337-1347. [10] F. Gesztesy and H. Holden, On new trace formulae for Schrödinger operators, Acta Appl. Math., 39 (1995), 315-333. doi: 10.1007/BF00994640. [11] F. Gesztesy and K. A. Makarov, The $\Xi$ operator and its relation to Krein’s spectral shift function, J. Anal. Math., 81 (2000), 139-183. doi: 10.1007/BF02788988. [12] F. Gesztesy and B. Simon, Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators, Trans. Amer. Math. Soc., 348 (1996), 349-373. doi: 10.1090/S0002-9947-96-01525-5. [13] _____, The xi function, Acta Mathematica, 176 (1996), 49-71. doi: 10.1007/BF02547335. [14] P. G. Grinevich, Rational solitons of the Veselov–Novikov equations and reflectionless two-dimensional potentials at fixed energy, Teoret. Mat. Fiz., 69 (1986), 307-310. [15] G. M. Khenkin and R. G. Novikov, The delta-bar equation in the multidimensional inverse scattering problem, Uspekhi Mat. Nauk, 42 (1987), 93-152, 255. [16] M. Horváth, Inverse scattering with fixed energy and an inverse eigenvalue problem on the half-line, Trans. Amer. Math. Soc., 358 (2006), 5161-5177. doi: 10.1090/S0002-9947-06-03996-1. [17] B. Ya. Levin, "Distribution of Zeros of Entire Functions," (in Russian), Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956. [18] J.-J. Loeffel, On an inverse problem in quantum scattering theory, Ann. Inst. H. Poincaré Sect. A (N.S.), 8 (1968), 339-447. [19] A. Martin and Gy. Targonski, On the uniqueness of a potential fitting a scattering amplitude at a given energy, Nuovo Cimento (10), 20 (1961), 1182-1190. doi: 10.1007/BF02732527. [20] R. G. Newton, "Scattering Theory of Waves and Particles," Reprint of the 1982 second edition, Dover Publications, Inc., Mineola, NY, 2002. [21] R. G. Novikov, The inverse scattering problem at fixed energy for the three-dimensional Schrödinger equation with an exponentially decreasing potential, Commun. Math. Phys., 161 (1994), 569-595. doi: 10.1007/BF02101933. [22] _____, Multidimensional inverse spectral problem for the equation $-\Delta\psi+(v(x)-Eu(x))\psi=0$, (in Russian), Funkts. Analiz i ego Prilozhenija, 22 (1988), 11-22. [23] A. G. Ramm, Recovery of the potential from fixed-energy scattering data, Inverse Problems, 4 (1988), 877-886. doi: 10.1088/0266-5611/4/3/020. [24] _____, An inverse scattering problem with part of the fixed-energy phase shifts, Comm. Math. Phys., 207 (1999), 231-247. [25] A. G. Ramm and P. D. Stefanov, Fixed energy inverse scattering for non-compactly supported potentials, Math. Comput. Modelling, 18 (1993), 57-64. doi: 10.1016/0895-7177(93)90079-E. [26] T. Regge, Introduction to complex orbital momenta, Nuovo Cimento (10), 14 (1959), 951-976. doi: 10.1007/BF02728177. [27] A. Rybkin, On the trace approach to the inverse scattering problem in dimension one, SIAM J. Math. Anal., 32 (2001), 1248-1264. doi: 10.1137/S0036141000365620. [28] B. Simon, Spectral analysis of rank one perturbations and applications, in "Mathematical Quantum Theory. II. Schrödinger Operators" (Vancouver, BC, 1993), CRM Proc. Lecture Notes, 8, Amer. Math. Soc., Providence, RI, (1995), 109-149. [29] G. N. Watson, "A Treatise on the Theory of Bessel Functions," Cambridge University Press, Cambridge; The Macmillan Company, New York, 1944. [30] R. Weder, Global uniqueness at fixed energy in multidimensional inverse scattering theory, Inverse Problems, 7 (1991), 927-938. doi: 10.1088/0266-5611/7/6/012. [31] R. Weder and D. Yafaev, On inverse scattering at a fixed energy for potentials with a regular behaviour at infinity, Inverse Problems, 21 (2005), 1937-1952. doi: 10.1088/0266-5611/21/6/009.

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##### References:
 [1] , Digital Library of Mathematical Functions, http://dlmf.nist.gov/idx/. [2] M. Abramowitz and I. Stegun, eds., "Handbook of Mathematical Functions," Dover Publications, New York, 1972. [3] V. Alfaro and T. Regge, "Potential Scattering," North-Holland, Amsterdam, 1965. [4] N. Aronszajn and W. F. Donoghue, On exponential representations of analytic functions in the upper half-plane with positive imaginary part, J. d'Analyse Math., 5 (1957), 321-388. doi: 10.1007/BF02937349. [5] A. Erdélyi, W. Magnus, F. Oberhettinger, F. Tricomi and G. Francesco, "Higher Transcendental Functions," Vol. II, Based on notes left by Harry Bateman, McGraw-Hill, 1953. [6] M. Sh. Birman and D. R. Yafaev, The spectral shift function. The papers of M. G. Kreĭn and their further development, St. Petersburg Math. J., 4 (1993), 833-870. [7] K. Chadan and P. C. Sabatier, "Inverse Problems in Quantum Scattering Theory," Second edition, With a foreword by R. G. Newton, Texts in Monographs in Physics, Springer-Verlag, New York, 1989. [8] W. H. J. Fuchs, A generalization of Carlson's theorem, J. London Math. Soc., 21 (1946), 106-110. doi: 10.1112/jlms/s1-21.2.106. [9] M. Giffon G. Burdet and E. Predazzi, On the inversion problem in the $\lambda$-plane, Nuovo Cimento (10), 36 (1965), 1337-1347. [10] F. Gesztesy and H. Holden, On new trace formulae for Schrödinger operators, Acta Appl. Math., 39 (1995), 315-333. doi: 10.1007/BF00994640. [11] F. Gesztesy and K. A. Makarov, The $\Xi$ operator and its relation to Krein’s spectral shift function, J. Anal. Math., 81 (2000), 139-183. doi: 10.1007/BF02788988. [12] F. Gesztesy and B. Simon, Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators, Trans. Amer. Math. Soc., 348 (1996), 349-373. doi: 10.1090/S0002-9947-96-01525-5. [13] _____, The xi function, Acta Mathematica, 176 (1996), 49-71. doi: 10.1007/BF02547335. [14] P. G. Grinevich, Rational solitons of the Veselov–Novikov equations and reflectionless two-dimensional potentials at fixed energy, Teoret. Mat. Fiz., 69 (1986), 307-310. [15] G. M. Khenkin and R. G. Novikov, The delta-bar equation in the multidimensional inverse scattering problem, Uspekhi Mat. Nauk, 42 (1987), 93-152, 255. [16] M. Horváth, Inverse scattering with fixed energy and an inverse eigenvalue problem on the half-line, Trans. Amer. Math. Soc., 358 (2006), 5161-5177. doi: 10.1090/S0002-9947-06-03996-1. [17] B. Ya. Levin, "Distribution of Zeros of Entire Functions," (in Russian), Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956. [18] J.-J. Loeffel, On an inverse problem in quantum scattering theory, Ann. Inst. H. Poincaré Sect. A (N.S.), 8 (1968), 339-447. [19] A. Martin and Gy. Targonski, On the uniqueness of a potential fitting a scattering amplitude at a given energy, Nuovo Cimento (10), 20 (1961), 1182-1190. doi: 10.1007/BF02732527. [20] R. G. Newton, "Scattering Theory of Waves and Particles," Reprint of the 1982 second edition, Dover Publications, Inc., Mineola, NY, 2002. [21] R. G. Novikov, The inverse scattering problem at fixed energy for the three-dimensional Schrödinger equation with an exponentially decreasing potential, Commun. Math. Phys., 161 (1994), 569-595. doi: 10.1007/BF02101933. [22] _____, Multidimensional inverse spectral problem for the equation $-\Delta\psi+(v(x)-Eu(x))\psi=0$, (in Russian), Funkts. Analiz i ego Prilozhenija, 22 (1988), 11-22. [23] A. G. Ramm, Recovery of the potential from fixed-energy scattering data, Inverse Problems, 4 (1988), 877-886. doi: 10.1088/0266-5611/4/3/020. [24] _____, An inverse scattering problem with part of the fixed-energy phase shifts, Comm. Math. Phys., 207 (1999), 231-247. [25] A. G. Ramm and P. D. Stefanov, Fixed energy inverse scattering for non-compactly supported potentials, Math. Comput. Modelling, 18 (1993), 57-64. doi: 10.1016/0895-7177(93)90079-E. [26] T. Regge, Introduction to complex orbital momenta, Nuovo Cimento (10), 14 (1959), 951-976. doi: 10.1007/BF02728177. [27] A. Rybkin, On the trace approach to the inverse scattering problem in dimension one, SIAM J. Math. Anal., 32 (2001), 1248-1264. doi: 10.1137/S0036141000365620. [28] B. Simon, Spectral analysis of rank one perturbations and applications, in "Mathematical Quantum Theory. II. Schrödinger Operators" (Vancouver, BC, 1993), CRM Proc. Lecture Notes, 8, Amer. Math. Soc., Providence, RI, (1995), 109-149. [29] G. N. Watson, "A Treatise on the Theory of Bessel Functions," Cambridge University Press, Cambridge; The Macmillan Company, New York, 1944. [30] R. Weder, Global uniqueness at fixed energy in multidimensional inverse scattering theory, Inverse Problems, 7 (1991), 927-938. doi: 10.1088/0266-5611/7/6/012. [31] R. Weder and D. Yafaev, On inverse scattering at a fixed energy for potentials with a regular behaviour at infinity, Inverse Problems, 21 (2005), 1937-1952. doi: 10.1088/0266-5611/21/6/009.
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