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Analog of Levinson’s formula for a Schrödinger operator with long-range potential. (English. Russian original) Zbl 0625.35021
Theor. Math. Phys. 68, 801-808 (1986); translation from Teor. Mat. Fiz. 68, No. 2, 244-254 (1986).
Trace formulas of order zero are obtained for a radial Schrödinger operator with long-range potential V(x) that decreases as \(x\to \infty\) as the power \(x^{-\alpha}\) with \(1\leq \alpha \leq 2\). These formulas relate the increment of the phase shift in the continuum to the characteristics of the discrete spectrum and generalize Levinson’s theorem to the case of slowly decreasing potentials.

MSC:
35J10 Schrödinger operator, Schrödinger equation
35Q99 Partial differential equations of mathematical physics and other areas of application
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[1] N. Levinson, K. Dan. Vidensk. Selsk. Mat.-Fys. Medd.,25, 9 (1949).
[2] L. D. Faddeev, Dokl. Akad. Nauk SSSR,115, 878 (1957); V. S. Buslaev and L. D. Faddeev, Dokl. Akad. Nauk SSSR,132, 13 (1960).
[3] H. M. Nussenzveig, Causality and Dispersion Relations, New York (1972). · Zbl 0093.44202
[4] P. G. Burke, Potential Scattering in Atomic Physics, New York (1977).
[5] M. J. Seaton, C. R. Acad. Sci.,240, 1317 (1955).
[6] D. R. Yafaev, Teor. Mat. Fiz.,11, 78 (1972).
[7] D. R. Yafaev, Commun. Math. Phys.,85, 177 (1982). · Zbl 0509.35065 · doi:10.1007/BF01254456
[8] R. G. Newton, Scattering Theory of Waves and Particles, McGraw-Hill, New York (1966). · Zbl 0178.52703
[9] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover (1964). · Zbl 0171.38503
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