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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 & Imaging, 2011, 5 (4) : 843-858. doi: 10.3934/ipi.2011.5.843
##### References:
 [1] , Digital Library of Mathematical Functions,, \url{http://dlmf.nist.gov/idx/}., (). Google Scholar [2] M. Abramowitz and I. Stegun, eds., "Handbook of Mathematical Functions,", Dover Publications, (1972). Google Scholar [3] V. Alfaro and T. Regge, "Potential Scattering,", North-Holland, (1965). Google Scholar [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. doi: 10.1007/BF02937349. Google Scholar [5] A. Erdélyi, W. Magnus, F. Oberhettinger, F. Tricomi and G. Francesco, "Higher Transcendental Functions,", Vol. II, (1953). Google Scholar [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. Google Scholar [7] K. Chadan and P. C. Sabatier, "Inverse Problems in Quantum Scattering Theory,", Second edition, (1989). Google Scholar [8] W. H. J. Fuchs, A generalization of Carlson's theorem,, J. London Math. Soc., 21 (1946), 106. doi: 10.1112/jlms/s1-21.2.106. Google Scholar [9] M. Giffon G. Burdet and E. Predazzi, On the inversion problem in the $\lambda$-plane,, Nuovo Cimento (10), 36 (1965), 1337. Google Scholar [10] F. Gesztesy and H. Holden, On new trace formulae for Schrödinger operators,, Acta Appl. Math., 39 (1995), 315. doi: 10.1007/BF00994640. Google Scholar [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. doi: 10.1007/BF02788988. Google Scholar [12] F. Gesztesy and B. Simon, Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators,, Trans. Amer. Math. Soc., 348 (1996), 349. doi: 10.1090/S0002-9947-96-01525-5. Google Scholar [13] _____, The xi function,, Acta Mathematica, 176 (1996), 49. doi: 10.1007/BF02547335. Google Scholar [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. Google Scholar [15] G. M. Khenkin and R. G. Novikov, The delta-bar equation in the multidimensional inverse scattering problem,, Uspekhi Mat. Nauk, 42 (1987), 93. Google Scholar [16] M. Horváth, Inverse scattering with fixed energy and an inverse eigenvalue problem on the half-line,, Trans. Amer. Math. Soc., 358 (2006), 5161. doi: 10.1090/S0002-9947-06-03996-1. Google Scholar [17] B. Ya. Levin, "Distribution of Zeros of Entire Functions,", (in Russian), (1956). Google Scholar [18] J.-J. Loeffel, On an inverse problem in quantum scattering theory,, Ann. Inst. H. Poincaré Sect. A (N.S.), 8 (1968), 339. Google Scholar [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. doi: 10.1007/BF02732527. Google Scholar [20] R. G. Newton, "Scattering Theory of Waves and Particles,", Reprint of the 1982 second edition, (1982). Google Scholar [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. doi: 10.1007/BF02101933. Google Scholar [22] _____, Multidimensional inverse spectral problem for the equation $-\Delta\psi+(v(x)-Eu(x))\psi=0$,, (in Russian), 22 (1988), 11. Google Scholar [23] A. G. Ramm, Recovery of the potential from fixed-energy scattering data,, Inverse Problems, 4 (1988), 877. doi: 10.1088/0266-5611/4/3/020. Google Scholar [24] _____, An inverse scattering problem with part of the fixed-energy phase shifts,, Comm. Math. Phys., 207 (1999), 231. Google Scholar [25] A. G. Ramm and P. D. Stefanov, Fixed energy inverse scattering for non-compactly supported potentials,, Math. Comput. Modelling, 18 (1993), 57. doi: 10.1016/0895-7177(93)90079-E. Google Scholar [26] T. Regge, Introduction to complex orbital momenta,, Nuovo Cimento (10), 14 (1959), 951. doi: 10.1007/BF02728177. Google Scholar [27] A. Rybkin, On the trace approach to the inverse scattering problem in dimension one,, SIAM J. Math. Anal., 32 (2001), 1248. doi: 10.1137/S0036141000365620. Google Scholar [28] B. Simon, Spectral analysis of rank one perturbations and applications,, in, 8 (1995), 109. Google Scholar [29] G. N. Watson, "A Treatise on the Theory of Bessel Functions,", Cambridge University Press, (1944). Google Scholar [30] R. Weder, Global uniqueness at fixed energy in multidimensional inverse scattering theory,, Inverse Problems, 7 (1991), 927. doi: 10.1088/0266-5611/7/6/012. Google Scholar [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. doi: 10.1088/0266-5611/21/6/009. Google Scholar

show all references

##### References:
 [1] , Digital Library of Mathematical Functions,, \url{http://dlmf.nist.gov/idx/}., (). Google Scholar [2] M. Abramowitz and I. Stegun, eds., "Handbook of Mathematical Functions,", Dover Publications, (1972). Google Scholar [3] V. Alfaro and T. Regge, "Potential Scattering,", North-Holland, (1965). Google Scholar [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. doi: 10.1007/BF02937349. Google Scholar [5] A. Erdélyi, W. Magnus, F. Oberhettinger, F. Tricomi and G. Francesco, "Higher Transcendental Functions,", Vol. II, (1953). Google Scholar [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. Google Scholar [7] K. Chadan and P. C. Sabatier, "Inverse Problems in Quantum Scattering Theory,", Second edition, (1989). Google Scholar [8] W. H. J. Fuchs, A generalization of Carlson's theorem,, J. London Math. Soc., 21 (1946), 106. doi: 10.1112/jlms/s1-21.2.106. Google Scholar [9] M. Giffon G. Burdet and E. Predazzi, On the inversion problem in the $\lambda$-plane,, Nuovo Cimento (10), 36 (1965), 1337. Google Scholar [10] F. Gesztesy and H. Holden, On new trace formulae for Schrödinger operators,, Acta Appl. Math., 39 (1995), 315. doi: 10.1007/BF00994640. Google Scholar [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. doi: 10.1007/BF02788988. Google Scholar [12] F. Gesztesy and B. Simon, Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators,, Trans. Amer. Math. Soc., 348 (1996), 349. doi: 10.1090/S0002-9947-96-01525-5. Google Scholar [13] _____, The xi function,, Acta Mathematica, 176 (1996), 49. doi: 10.1007/BF02547335. Google Scholar [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. Google Scholar [15] G. M. Khenkin and R. G. Novikov, The delta-bar equation in the multidimensional inverse scattering problem,, Uspekhi Mat. Nauk, 42 (1987), 93. Google Scholar [16] M. Horváth, Inverse scattering with fixed energy and an inverse eigenvalue problem on the half-line,, Trans. Amer. Math. Soc., 358 (2006), 5161. doi: 10.1090/S0002-9947-06-03996-1. Google Scholar [17] B. Ya. Levin, "Distribution of Zeros of Entire Functions,", (in Russian), (1956). Google Scholar [18] J.-J. Loeffel, On an inverse problem in quantum scattering theory,, Ann. Inst. H. Poincaré Sect. A (N.S.), 8 (1968), 339. Google Scholar [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. doi: 10.1007/BF02732527. Google Scholar [20] R. G. Newton, "Scattering Theory of Waves and Particles,", Reprint of the 1982 second edition, (1982). Google Scholar [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. doi: 10.1007/BF02101933. Google Scholar [22] _____, Multidimensional inverse spectral problem for the equation $-\Delta\psi+(v(x)-Eu(x))\psi=0$,, (in Russian), 22 (1988), 11. Google Scholar [23] A. G. Ramm, Recovery of the potential from fixed-energy scattering data,, Inverse Problems, 4 (1988), 877. doi: 10.1088/0266-5611/4/3/020. Google Scholar [24] _____, An inverse scattering problem with part of the fixed-energy phase shifts,, Comm. Math. Phys., 207 (1999), 231. Google Scholar [25] A. G. Ramm and P. D. Stefanov, Fixed energy inverse scattering for non-compactly supported potentials,, Math. Comput. Modelling, 18 (1993), 57. doi: 10.1016/0895-7177(93)90079-E. Google Scholar [26] T. Regge, Introduction to complex orbital momenta,, Nuovo Cimento (10), 14 (1959), 951. doi: 10.1007/BF02728177. Google Scholar [27] A. Rybkin, On the trace approach to the inverse scattering problem in dimension one,, SIAM J. Math. Anal., 32 (2001), 1248. doi: 10.1137/S0036141000365620. Google Scholar [28] B. Simon, Spectral analysis of rank one perturbations and applications,, in, 8 (1995), 109. Google Scholar [29] G. N. Watson, "A Treatise on the Theory of Bessel Functions,", Cambridge University Press, (1944). Google Scholar [30] R. Weder, Global uniqueness at fixed energy in multidimensional inverse scattering theory,, Inverse Problems, 7 (1991), 927. doi: 10.1088/0266-5611/7/6/012. Google Scholar [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. doi: 10.1088/0266-5611/21/6/009. Google Scholar
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