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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,, \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. |
[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.
|
[7] |
K. Chadan and P. C. Sabatier, "Inverse Problems in Quantum Scattering Theory,", Second edition, (1989).
|
[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. |
[9] |
M. Giffon G. Burdet and E. Predazzi, On the inversion problem in the $\lambda$-plane,, Nuovo Cimento (10), 36 (1965), 1337.
|
[10] |
F. Gesztesy and H. Holden, On new trace formulae for Schrödinger operators,, Acta Appl. Math., 39 (1995), 315.
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.
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.
doi: 10.1090/S0002-9947-96-01525-5. |
[13] |
_____, The xi function,, Acta Mathematica, 176 (1996), 49.
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.
|
[15] |
G. M. Khenkin and R. G. Novikov, The delta-bar equation in the multidimensional inverse scattering problem,, Uspekhi Mat. Nauk, 42 (1987), 93.
|
[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. |
[17] |
B. Ya. Levin, "Distribution of Zeros of Entire Functions,", (in Russian), (1956).
|
[18] |
J.-J. Loeffel, On an inverse problem in quantum scattering theory,, Ann. Inst. H. Poincaré Sect. A (N.S.), 8 (1968), 339.
|
[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. |
[20] |
R. G. Newton, "Scattering Theory of Waves and Particles,", Reprint of the 1982 second edition, (1982).
|
[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. |
[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. |
[24] |
_____, An inverse scattering problem with part of the fixed-energy phase shifts,, Comm. Math. Phys., 207 (1999), 231.
|
[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. |
[26] |
T. Regge, Introduction to complex orbital momenta,, Nuovo Cimento (10), 14 (1959), 951.
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.
doi: 10.1137/S0036141000365620. |
[28] |
B. Simon, Spectral analysis of rank one perturbations and applications,, in, 8 (1995), 109.
|
[29] |
G. N. Watson, "A Treatise on the Theory of Bessel Functions,", Cambridge University Press, (1944).
|
[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. |
[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. |
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. |
[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.
|
[7] |
K. Chadan and P. C. Sabatier, "Inverse Problems in Quantum Scattering Theory,", Second edition, (1989).
|
[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. |
[9] |
M. Giffon G. Burdet and E. Predazzi, On the inversion problem in the $\lambda$-plane,, Nuovo Cimento (10), 36 (1965), 1337.
|
[10] |
F. Gesztesy and H. Holden, On new trace formulae for Schrödinger operators,, Acta Appl. Math., 39 (1995), 315.
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.
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.
doi: 10.1090/S0002-9947-96-01525-5. |
[13] |
_____, The xi function,, Acta Mathematica, 176 (1996), 49.
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.
|
[15] |
G. M. Khenkin and R. G. Novikov, The delta-bar equation in the multidimensional inverse scattering problem,, Uspekhi Mat. Nauk, 42 (1987), 93.
|
[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. |
[17] |
B. Ya. Levin, "Distribution of Zeros of Entire Functions,", (in Russian), (1956).
|
[18] |
J.-J. Loeffel, On an inverse problem in quantum scattering theory,, Ann. Inst. H. Poincaré Sect. A (N.S.), 8 (1968), 339.
|
[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. |
[20] |
R. G. Newton, "Scattering Theory of Waves and Particles,", Reprint of the 1982 second edition, (1982).
|
[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. |
[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. |
[24] |
_____, An inverse scattering problem with part of the fixed-energy phase shifts,, Comm. Math. Phys., 207 (1999), 231.
|
[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. |
[26] |
T. Regge, Introduction to complex orbital momenta,, Nuovo Cimento (10), 14 (1959), 951.
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.
doi: 10.1137/S0036141000365620. |
[28] |
B. Simon, Spectral analysis of rank one perturbations and applications,, in, 8 (1995), 109.
|
[29] |
G. N. Watson, "A Treatise on the Theory of Bessel Functions,", Cambridge University Press, (1944).
|
[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. |
[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. |
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