Inverse spectral-scattering problem with two sets of discrete spectra for the radial Schrödinger equation

Inverse Problems

Volumn 22, Issue 1, 2006, Pages 89-114

  Aktosun, Tuncay ^a   Weder, Ricardo ^b,c  

^a UNIVERSITY OF TEXAS AT ARLINGTON   (United States)

^b Ciudad Universitaria   (Mexico)

^c Unidad Durango   (Mexico)

Author keywords

[No Author keywords available]

Indexed keywords

FINITE FIRST MOMENT; SCHRÖDINGER EQUATION; SPECTRAL MEASURE; THEOREM OF BORG AND MARCHENKO;

BOUNDARY CONDITIONS; DIFFERENTIAL EQUATIONS; EIGENVALUES AND EIGENFUNCTIONS; SPECTRUM ANALYSIS; THEOREM PROVING;

INVERSE PROBLEMS;

EID: 31844441080     PISSN: 02665611     EISSN: 13616420     Source Type: Journal    
DOI: 10.1088/0266-5611/22/1/006     Document Type: Article

References (47)

1. Ambartsumyan V A 1929 Über eine Frage der Eigenwerttheorie Z. Phys. 53 690-5
2. Borg G 1946 Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe Acta Math. 78 1-96
3. Borg G 1952 Uniqueness theorems in the spectral theory of y″ + (λ - q(x})y = 0 Proc. 11th Scandinavian Congress of Mathematicians (Johan Grundt Tanums Forlag, Oslo) pp 276-87
4. Marchenko V A 1952 Some questions in the theory of one-dimensional linear differential operators of the second order. I Tr. Monk. Mat. Obšč. 1 327-420
5. Marchenko V A 1973 Am. Math. Soc. Transl. (2) 101 1-104 (Engl. Transl.)
6. Gesztesy F and Simon B 1996 Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators Trans. Am. Math. Soc. 348 349-73
7. Levitan B M and Gasymov M G 1964 Determination of a differential equation by two of its spectra Usp. Mat. Nauk 19 3-63
8. Levitan B M and Gasymov M G 1964 Russ. Math. Surv. 19 1-63 (Engl. Transl.)
9. Marchenko V A 1986 Stunn-Liouville Operators and Applications (Basel: Birkhauser)
10. Levitan B M 1987 Inverse Sturm-Liouville Problems (Utrecht: VNU Science Press)
11. Weidmann J 1987 Spectral Theory of Ordinary Differential Operators (Lecture Notes in Mathematics vol 1258) (Berlin: Springer)
12. Chadan K and Sabatier P C 1989 Inverse Problems in Quantum Scattering Theory 2nd edn (New York: Springer)
13. Gel'fand IM and Levitan B M 1951 On the determination of a differential equation from its spectral function Izv. Akad. Nauk SSSR Ser. Mat. 15 309-60
14. Gel'fand I M and Levitan B M 1955 Am. Math. Soc: Transl. (2) 1 253-304 (Engl. Transl.)
15. Marchenko V A 1955 On reconstruction of the potential energy from phases of the scattered waves Dokl. Akad. Nauk SSSR 104 695-8
16. Novikova N N and Markushevich V M 1985 Uniqueness of the solution of the one-dimensional problem of scattering for potentials located on the positive semiaxis Vyrhisl. Seismol. 18 176-84
17. Novikova N N and Markushevich V M 1987 Comput. Seismol. 18 164-72 (Engl. Transl.)
18. Aktosun T, Klaus M and van der Mee C 1993 On the Riemann-Hilbert problem for the one-dimensional Schrödinger equation J. Math. Phys. 34 2651-90
19. Aktosun T 1994 Bound states and inverse scattering for the Schrödinger equation in one dimension J. Math. Phys. 356231-6
20. Grebert B and Weder R 1995 Reconstruction of a potential on the line that is a priori known on the half line SIAM J. Appl. Math. 55 242-54
21. Aktosun T 1996 Inverse Schrödinger scattering on the line with partial knowledge of the potential SIAM J. Appl. Math. 56219-31
22. Gesztesy F and Simon B 1997 Inverse spectral analysis with partial information on the potential: I. The case of an a.c. component in the spectrum Helv. Phys. Acta 70 66-71
23. Faddeev L D 1964 Properties of the 5-matrix of the one-dimensional Schrödinger equation Tr. Mat. Inst. Steklov. 73314-36
24. Faddeev L D 1967 Am. Math. Soc. Transl. (2) 65 139-66 (Engl. Transl.)
25. Melin A 1985 Operator methods for inverse scattering on the real line Commun. Partial Differ. Eqns. 10 677-766
26. Aktosun T and Klaus M 2001 Inverse theory: problem on the line Scattering ed E R Pike and P C Sabatier (London: Academic) chapter 2.2.4, pp 770-85
27. Deift P and Trubowitz E 1979 Inverse scattering on the line Commun. Pure Appl. Math. 32 121-251
28. Aktosun T 2004 Construction of the half-line potential from the Jost function Inverse Problems 20 859-76
29. Forbes B J, Pike E R and Sharp D B 2003 The acoustical Klein-Gordon equation: the wave-mechanical step and barrier potential functions J. Acoust. Soc. Am. 114 1291-302
30. Stein E M and Weiss G 1971 Introduction to Fourier Analysis on Euclidean Spaces (Princeton, NJ: Princeton University Press)
31. Rosenblum M and Rovnyak J 1994 Topics in Hardy Classes and Univalent Functions (Basel: Birkhäuser)
32. Titchmarsh E C 1962 Introduction to the Theory of Fourier Integral 2nd edn (London: Oxford University Press)
33. Ahlfors L 1966 Complex Analysis 2nd edn (New York: McGraw-Hill)
34. Churchill R V and Brown J W 1984 Complex Variables and Applications 4th edn (New York: McGraw-Hill)
35. Faddeev L D 1959 The inverse problem in the quantum theory of scattering Usp. Mat. Nauk 14 57-119
36. Faddeev L D 1963 J. Math. Phys. 4 72-104 (Engl. Transl.)
37. Newton R G 1982 Scattering Theory of Waves and Particles 2nd edn (New York: Springer)
38. Chadan K 2001 Potential scattering: the radial Schrödinger equation Scattering ed E R Pike and P C Sabatier (London: Academic) chapter 2.2.1, pp 669-85
39. Chadan K and Sabatier P C 2001 Radial inverse scattering problems Scattering ed E R Pike and P C Sabatier (London: Academic) chapter 2.2.1, pp 726-41
40. Aktosun T and Klaus M 2001 Small-energy asymptotics for the Schrödinger equation on the line Inverse Problems 17 619-32
41. Reed M and Simon B 1980 Methods of Modern Mathematical Physics, vol I, Functional Analysis (New York: Academic)
42. Kato T 1976 Perturbation Theory for Linear Operators 2nd edn (Berlin: Springer)
43. Adams R A 1975 Soholev Spaces (New York: Academic)
44. Coddington E A and Levinson N 1955 Theory of Ordinary Differential Equations (New York: McGraw-Hill)
45. Titchmarsh E C 1962 Eigenfunction Expansions Associated with Second-Order Differential Equations Part I, 2nd edn (Oxford: Clarendon)
46. Levitan B M and Sargsjan I S 1975 Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators (Providence, RI: American Mathematical Society)
47. Aktosun T and Papanicolaou V G 2003 Recovery of a potential from the ratio of reflection and transmission coefficients J. Math. Phys. 44 4875-83