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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 |
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. |
show all references
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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