Abstract
In this paper we prove the main step in establishing a limiting absorption principle for von Neumann-Wigner type Schrödinger Hamiltonians of the form −Δ+csinb|x|/|x|+V(x), whereV(x) is a short range potential. The first fundamental step is to obtain a limiting absorption principal for the “free” operator −Δ+csinb.|x|/|x|. The free operator is unitarily equivalent to a direct sum of ordinary differential operators. We obtain uniform estimates for the resolvents of these ordinary differential operators. by obtaining uniform estimates for the Weyl-Green kernels of these resolvents. In turn, these latter estimates require uniform estimates on the Wronskians of certain generalized eigen-solutions of these differential operators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agmon, S.,Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Sci. Mat. (1975) 151–218.
Ben-Artzi, M. and Devinatz, A.,Spectal and scattering theory for the adiabatic oscillator and related potentials, J. Math. Phys. 111 (1979), 594–607.
Ben-Artzi,The Limiting Absorption Principle for Partial Differential Operators, Amer. Math. Soc. Memoir 364, Providence, RI, 1987.
Devinatz, A. and Rejto, P.,A limiting absorption for Schrödinger operators with oscillating potentials I, J. Diff. Equations 49 (1983), 85–104.
—,A limiting absorption principle for Schrödinger operators with ocillating potentials II, J. Diff. Equations 49 (1983), 85–104.
Enz, C., P., (Ed),Wave Mechanics, Pauli Lectures on Physics, Vol. 5, Sect. 27, The WKB-method., MIT Press, Cambridge, MA, 1977.
Erdelyi, A.,Asymptotic Expansions, Dover, New York, 1956.
—,Asymptotic solutions of differential equations with transition points or singularities, J. Math. Phys. 1 (1960), 16–26. See Section 4.
Erdelyi, A.,The integral equations of asymptotic theory, in “Asymptotic Solutions of Differential Equations,” Wilcox, C. H. ed., John Wiley 1964, pp. 211–229. See Section 2 and the references given there.
Harris, W.A. and Lutz, D.A.,Asymptotic integration of adiabatic oscillators, J. Math. Anal. Appl. 51 (1975), 76–93.
—,A unified theory of asymptotic integration, J. Math. Anal. Appl. 57 (1977), 571–586.
Jaeger, W. and Rejto, P.,On the absolute continuity of the spectrum of Schrödinger operators with long range potentials, Oberwolfach Tagungsbericht, 17, 7–23.7, 1977.
—,Limiting absorption principle for some Schrödinger operators with exploding potentials I, J. Math. Anal. Appl. 91 (1983), 192–228.
—,Limiting absorption principle for some Schrödinger operators with exploding potentials II, J. Math. Anal. Appl. 95 (1983), 169–194.
Kato, T.,Perturbation theory for linear operators, Springer-Verlag, 1973.
Langer, R.E.,The asymptotic solutions of ordinary linear differential equations of the second order with special reference to a turning point, Trans. Amer. Math. Soc. 67 (1949), 461–490.
Love, C.E.,Singular integral equations of the Volterra type, Trans. Amer. Math. Soc. 15 (1914), 467–476.
Mochizuki, M., and Uchiyama, J.,Radiation conditions and spectral theory for 2-body Schrödinger operators with “oscillating” long range potentials I, J. Math. Kyoto Univ. 18(2) (1978), 377–408.
Olver, F.T.W.,Asymptotics and Special Functions, Academic Press, 1974.
Reed, M. and Simon, B.,Analysis of operators, Methods of Modern Mathematical Physics, Vol. IV, Section XIII.13, Example 1, Academic Press, 1978.
Rejto, P. and Taboada, M.,Weighted resolvent estimates for Volterra operators on unbounded intervals, to appear in J. Math. Anal. Appl.
Saito, Y.,On the asymptotic behavior of the solutions of the Schrödinger equation, Osaka. J. Math. 14 (1977), 11–35.
—,Schrödinger Operators with a nonspherical radiation condition, Pacific J. Math. 126 (1987), 331–359.
von Neumann, J. and Wigner, E.,Ueber merkwuerdige diskrete Eigenwerte, Phys. Z. 30 (1929), 465–467.
Wasow, W.,Asymptotic expansions for ordinary differential equations, Wiley-Interscience, 1965.
Weidmann, J.,Spectral Theory of Ordinary Differential Operators, Springer Verlag Lecture Notes in Mathematics #1258, 1987.
White, D.A.W.,Schroedinger operators with rapidly oscillating central potentials, Trans. Amer. Math. Soc. 275 (1983), 641–677.
Author information
Authors and Affiliations
Additional information
This paper is dedicated to the memory of the late Professor Charles C. Conley.
Rights and permissions
About this article
Cite this article
Devinatz, A., Moeckel, R. & Rejto, P. A limiting absorption principle for Schrödinger operators with Von Neumann-Wigner type potentials. Integr equ oper theory 14, 13–68 (1991). https://doi.org/10.1007/BF01194926
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01194926