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L2-lower bounds to solutions of one-body Schrödinger equations

Published online by Cambridge University Press:  14 November 2011

R. Froese ,
I. Herbst ,
M. Hoffmann-Ostenhof  and
T. Hoffmann-Ostenhof
R. Froese
Affiliation:
Institut Mittag-Leffler, Auravägen 17, S-182 62 Djursholm, Sweden
I. Herbst
Affiliation:
Institut Mittag-Leffler, Auravägen 17, S-182 62 Djursholm, Sweden
M. Hoffmann-Ostenhof
Affiliation:
Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, 1090 Wien, Austria
T. Hoffmann-Ostenhof
Affiliation:
Institut für Theoretische Chemie und Strahlenchemie, Universität Wien, Währingerstrasse 17, 1090 Wien, Austria
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Extract

The asymptotic behaviour of L2-solutions of one-body Schrödinger equations (–δ+V–E)ψ = 0 in ΩR = {x ∊ Rn||x|>R} is investigated. We show, for example, that if V tends to zero in a certain sense for |x|→∞, then either |x|γ exp for some γ>0 or ψ has compact support. Related results are given for potentials tending to infinity for |x|→∞.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 95 , Issue 1-2 , 1983 , pp. 25 - 38
Copyright
Copyright © Royal Society of Edinburgh 1983

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References

1Agmon, S.. Lectures on exponential decay of solutions of second order elliptic equations. Bounds on eigenfunctions of N-body Schrödinger operators, to be published.Google Scholar
2Agmon, S.. Lower bounds for solutions of Schrödinger equations. J. Analyse Math. 23 (1970), 1–25.CrossRefGoogle Scholar
3Ahlrichs, R., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T. and Morgan III, J.. Bounds on the decay of electron densities with screening. Phys. Rev. A 23 (1981), 2106–2117.CrossRefGoogle Scholar
4Amrein, W. O., Berthier, A. M. and Georgescu, V.. Lp-inequalities for the Laplacian and unique continuation. Ann. Inst. Fourier (Grenoble) 31 (1981), 153–168.CrossRefGoogle Scholar
5Bardos, C. and Merigot, M.. Asymptotic decay of solutions of a second order elliptic equation in an unbounded domain. Applications to the spectral properties of a Hamiltonian. Proc. Roy. Soc. Edinburgh Sect. A 76 (1977), 323–344.CrossRefGoogle Scholar
6Carmona, R. and Simon, B.. Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, V: lower bounds and path integrals. Comm. Math. Phys. 80 (1981), 59–98.Google Scholar
7Combes, J. M., Hoffmann-Ostenhof, M. and Hoffmann-Ostenhof, T.. Asymptotics of atomic groundstates: the relation between the groundstate of Helium and the groundstate of He+. J. Math. Phys. 22 (1981), 1299–1305.Google Scholar
8Davies, E. B.. JWKB and related bounds on Schrödinger eigenfunctions. Bull. London Math. Soc. 14 (1982), 273–284.CrossRefGoogle Scholar
9Deift, P., Hunziker, W., Simon, B. and Vock, E.. Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, IV. Comm. Math. Phys. 64 (1978), 1–34.CrossRefGoogle Scholar
10Froese, R., Herbst, I., Hoffmann-Ostenhof, M. and Hoffmann-Ostenhof, T.. L2-exponential lower bounds to solutions of the Schrödinger equation. Comm. Math. Phys. 87 (1982), 265–286.Google Scholar
11Froese, R., Herbst, I., Hoffmann-Ostenhof, M. and Hoffmann-Ostenhof, T.. On the absence of positive eigenvalues for one-body Schrödinger operators. J. Analyse Math. (1983), to appear.Google Scholar
12Georgescu, V.. On the unique continuation property for Schrödinger Hamiltonians. Helv. Phys. Acta 52 (1979), 655–670.Google Scholar
13Hörmander, L.. Théorie de la diffusion à courte portée pour des opérateurs a caractéristique simple. Sém. Goulaouic-Meyer-Schwartz, 1980–1981, Exp. XIV.Google Scholar
14Hörmander, L.. Uniqueness theorems for second order elliptic differential equations, preprint (1982).Google Scholar
15Lieb, E. H.. and Simon, B... Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, VI. Asymptotics in the two cluster region. Adv. Appl. Math. 1 (1980), 324–343.CrossRefGoogle Scholar
16Reed, M.. and Simon, B... Methods of modem mathematical physics, IV. Analysis of operators (New York: Academic Press, 1978).Google Scholar
17Reed, M.. and Simon, B... Methods of modem mathematical physics, II. Fourier analysis, self-adjointness (New York: Academic Press, 1975).Google Scholar
18Simon, B... Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447–526CrossRefGoogle Scholar