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A trace formula for schrödinger operators with step potentials

Published online by Cambridge University Press:  17 February 2009

D. M. O'Brien
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Abstract

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This paper shows how to compute the trace of G(T) – G(T0), where G is an infinitely differentiable function with compact support, and where T and T0 are one-dimensional Schrödinger operators on (−∞, ∞) with potentials q and q0. It is assumed that q0 is a simple step potential and that q decays exponentially to q0. The trace is expressed in terms of the reflection and transmission coefficients for the scattering of plane waves by the potential q.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 24 , Issue 2 , October 1982 , pp. 138 - 159
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Alsholm, P. and Kato, T., “Scattering with long range potentials”, Symp. Pure Math. 23 (1973), 393–399.CrossRefGoogle Scholar
[2]Bargmann, V., “On the connection between phase shifts and scattering potential”, Rev. Modern Phys. 21 (1949), 488–493.CrossRefGoogle Scholar
[3]Buslaev, V. S., “Spectral identities and the trace formula in the Friedrichs model”, Topics in Math. Phys. 4 (1971), 43–54.Google Scholar
[4]Buslaev, V. S. and Faddeev, L. D., “On trace formulas for Sturm-Liouville differential operators”, Dokl. Akad. Nauk SSSR 132 (1960), 13–16.Google Scholar
[5]Davies, E. B. and Simon, B., “Scattering theory for systems with different spatial asymptotics on the left and right”, Comm. Math. Phys. 63 (1978), 277–301.CrossRefGoogle Scholar
[6]de Alfaro, V. and Regge, T., Potential scattering (North-Holland, Amsterdam, 1965).Google Scholar
[7]Dunford, N. and Schwartz, J. T., Linear operators. Part II (Wiley, New York, 1963).Google Scholar
[8]Faddeev, L. D., “The inverse problem in the quantum theory of scattering”, J. Math. Phys. 4 (1963), 72–104.CrossRefGoogle Scholar
[9]Faddeev, L. D. and Korepin, V. E., “Quantum theory of solitons”, Phys. Rep. 42C (1978), 1–87.CrossRefGoogle Scholar
[10]Jost, R., “Ūber die falschen Nullstellen der Eigenwerte der S-Matrix”, Helv. Phys. Acta 20 (1947), 256–266.Google Scholar
[11]Lohe, M. A., “Soliton structures in p(φ)2”, Phys. Rev. D20 (1979), 3120–3130.Google Scholar
[12]Lohe, M. A. and O'Brien, D. M., “Soliton mass corrections and explicit models in two dimensions”, Phys. Rev D23 (1981), 1771–1780.Google Scholar
[13]O'Brien, D. M., “A simple test for nuclearity of integral operators on L2(Rn)”, J. Austral. Math. Soc. A (in press).Google Scholar
[14]Ruijsenaars, S. N. M. and Bongaarts, P. J. M., ‘Scattering theory for one-dimensional step potentials’, Ann. Inst. H. Poincaré A. 26 (1977), 1–17.Google Scholar