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On the asymptotic decay of L2-solutions of one-body Schrödinger equations in unbounded domains*

Published online by Cambridge University Press:  14 November 2011

M. Hoffmann-Ostenhof
M. Hoffmann-Ostenhof
Affiliation:
Institut für Theoretische Physik, Universität Wien, Boltzmanng. 5, A-1090 Wien, Austria
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Synopsis

The asymptotic decay of L2-solutions of Schrödinger equations (-Δ+V)ψ=0 in ΔR= {x εRn∣∣x∣=r>R} is investigated, where V(x) = V1(r) + V2(x) with V1→ ∞ for r↑∞ and with some ε > 0 for large r. Under additional assumptions on the decay of V1, pointwise upper bounds to |ψ |and lower bounds to the spherical average of ψ are given showing the same asymptotics for r→ ∞. For the case V→ const. > 0 for r→ ℝ (investigated in [8] a simplified treatment is given.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 115 , Issue 1-2 , 1990 , pp. 65 - 86
Copyright
Copyright © Royal Society of Edinburgh 1990

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References

1Agmon, S.. On the asymptotic behaviour of solutions of Schrodinger type equations in unbounded domains. In Analyse Mathématique et Applications, pp. 1–22. (Paris: Gauthier-Villars, 1988).Google Scholar
2Amrein, W. O., Berthier, A. M. and Georgescu, V.. Lower bounds for zero energy eigenfunctions of Schrödinger operators. Helv. Phys. Acta 57 (1984), 301–306.Google Scholar
3Bardos, C. and Merigot, M.. Asymptotic decay of solutions of a second order elliptic equation in an unbounded domain. Proc. Roy. Soc. Edinburgh Sect. A 76 (1977), 323–344.CrossRefGoogle Scholar
4Davies, B. and Simon, B.. Ultracontractivity and the Heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. 59 (1984), 335–395.CrossRefGoogle Scholar
5Froese, R., Herbst, I., Hoffmann-Ostenhof, M. and Hoffmann-Ostenhof, T.. L 2-lower bounds to solutions of one-body Schrödinger equations. Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), 25–38.CrossRefGoogle Scholar
6Froese, R. and Herbst, I.. Exponential lower bounds to solutions of the Schrödinger equation: lower bounds for the spherical average. Comm. Math. Phys. 92 (1983), 71–80.CrossRefGoogle Scholar
7Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order (Berlin: Springer, 1983).Google Scholar
8Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T. and Swetina, J.. Pointwise bounds on the asymptotics of spherically averaged L 2-solutions of one-body Schrödinger equations. Ann. Inst. H Poincaré Anal. Physique théorique 42 (1985), 341–361.Google Scholar
9Hoffmann-Ostenhof, M. and Hoffmann-Ostenhof, T.. Asymptotics and continuity properties near infinity of solutions of Schrodinger equations in exterior domains. Ann. Inst. H. Poincaré Anal. Physique théorique 46 (1987), 247–280.Google Scholar
10Hoffmann-Ostenhof, M.. Asymptotics of the nodal lines of solutions of 2-dimensional Schrödinger equations. Math. Z. 198 (1988), 161–179.CrossRefGoogle Scholar
11Hoffmann-Ostenhof, M. and Hoffmann-Ostenhof, T.. On the asymptotics of nodes of L 3-solutions of Schrödinger equations in dimensions ≧ 3. Comm. Math. Phys. 117 (1988), 49–77.CrossRefGoogle Scholar
12Reed, M. and Simon, B.. Methods of mathematical physics II, Fourier analysis, selfadjointness (New York: Academic Press, 1975).Google Scholar
13Reed, M. and Simon, B.. Methods of mathematical physics IV, Analysis of operators (New York: Academic Press, 1978).Google Scholar
14Thoe, D. W.. Lower bounds for solutions of perturbed Helmholtz equations in exterior regions. J. Math. Anal. Appl. 102 (1984), 113–122.CrossRefGoogle Scholar
15Uchiyama, J.. Decay order of eigenfunctions of second order elliptic operators in an unbounded domain, and its application. Publ. RIMS. Kyoto Univ. 22 (1986), 1079–1104.CrossRefGoogle Scholar