On a complete analysis of high-energy scattering matrix asymptotics for one dimensional Schrödinger operators with integrable potentials

Rybkin, Alexei

doi:10.1090/S0002-9939-01-06014-2

On a complete analysis of high-energy scattering matrix asymptotics for one dimensional Schrödinger operators with integrable potentials
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by Alexei Rybkin PDF

Proc. Amer. Math. Soc. 130 (2002), 59-67 Request permission

Abstract:

For the general one dimensional Schrödinger operator $-\frac {d^{2}}{dx^{2}}+q\left (x\right )$ with real $q\in L_{1} \left (\mathbb {R}\right )$ we present a complete streamlined treatment of large spectral parameter power asymptotics of Jost solutions and the scattering matrix. We find simple necessary and sufficient conditions relating the number of exact terms in the asymptotics with the smoothness of $q$. These conditions are expressed in terms of the Fourier transform of some functions related to $q$. In particular, under the usual conditions $q^{\left ( N\right ) }\in L_{1}\left (\mathbb {R}\right ), N\in \mathbb {N}_{0},$ we derive up to two extra terms in the asymptotic expansion of the Jost solution and for the transmission coefficient we derive twice as many terms. Our main results are complete.

References

V. S. Buslaev and L. D. Faddeev, Formulas for traces for a singular Sturm-Liouville differential operator, Soviet Math. Dokl. 1 (1960), 451–454. MR 0120417
S. Clark and F. Gesztesy, “Weyl-Titchmarsh $M$-function asymptotics for matrix-valued Schrödinger operators", Proc. London Math. Soc.,
S. Clark, F. Gesztesy, H. Holden, and B. Levitan, “Borg-type theorems for matrix-valued Schrödinger operators", J. Diff. Eq. 167 (2000), 181–210.
L.D. Faddeev and V.E. Zakharov, “Korteweg-De Vries equation: a completely integrable Hamiltonian system", Funct. Anal. Appl. 5, 4 (1971), 280-287.
L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
F. Gesztesy, H. Holden, B. Simon, and Z. Zhao, Higher order trace relations for Schrödinger operators, Rev. Math. Phys. 7 (1995), no. 6, 893–922. MR 1348829, DOI 10.1142/S0129055X95000347
F. Gesztesy, H. Holden, and B. Simon, Absolute summability of the trace relation for certain Schrödinger operators, Comm. Math. Phys. 168 (1995), no. 1, 137–161. MR 1324393
F. Gesztesy and B. Simon, “A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure", Ann. Math. 152 (2000), 593–643.
Fritz Gesztesy and Barry Simon, On local Borg-Marchenko uniqueness results, Comm. Math. Phys. 211 (2000), no. 2, 273–287. MR 1754515, DOI 10.1007/s002200050812
D. B. Hinton, M. Klaus, and J. K. Shaw, High-energy asymptotics for the scattering matrix on the line, Inverse Problems 5 (1989), no. 6, 1049–1056. MR 1027284
D. B. Hinton, M. Klaus, and J. K. Shaw, Series representation and asymptotics for Titchmarsh-Weyl $m$-functions, Differential Integral Equations 2 (1989), no. 4, 419–429. MR 996750
T. Miyazawa, “Boson representations of one-dimensional scattering", J. Phys. A: Math. Gen. 33 (2000), 191-225.
A.V. Rybkin, “On the trace approach to the inverse scattering in dimension one", SIAM J. Math. Anal. 32 (2001), no 6, 1248–1264.
A.V. Rybkin, “A Weyl $m$-function approach to trace formulas for varies Schrödinger operators", in preparation.
A.V. Rybkin, “Some new and old asymptotic representations of the Jost solution and Weyl $m$-function for Schrödinger operators on the line", to appear in Bulletin of LMS.
B. Simon, “A new approach to inverse spectral theory, I. Fundamental formalism", Annals of Math. 150 (1999), 1029-1057.
E.C. Titchmarsh, “On eigenfunction expansions associated with second-order differential equations", Oxford University Press, 1950.
E. C. Titchmarsh, Introduction to the theory of Fourier integrals, 3rd ed., Chelsea Publishing Co., New York, 1986. MR 942661