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Asymptotic decay of the solution of a second-order elliptic equation in an unbounded domain. Applications to the spectral properties of a Hamiltonian

Published online by Cambridge University Press:  14 February 2012

C. Bardos
Affiliation:
I.M.S.P., Département de Mathématiques, Université de Nice
M. Merigot
Affiliation:
I.M.S.P., Département de Mathématiques, Université de Nice

Synopsis

In an unbounded domain Ω we study the asymptotic decay (for | x |→∞) of functions uL2(Ω) which are solutions of the following problem –Δu + cu = 0. c denotes a strictly positive function. Upper bounds are easily found via the maximum principle. When c is rotationally invariant lower bounds are obtained via asymptotic expansion. In the general case we use a method of ‘commutation’ of operators. In particular we consider the case where . Applications to the asymptotic decay of the bound states of a Hamiltonian are given.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1977

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References

1Agmon, S.. Lower bounds for solutions of Schrödinger Equation. J. Analyse Math. 23 (1970), 125.CrossRefGoogle Scholar
2Aronszajn, N.. A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pures Appl. 36 (1957), 235240.Google Scholar
3Bell, R. P.. Eigenvalues and eigenfunctions for the operator d2/dx2-x. Philos. Mag. 35 (1944), 582588.CrossRefGoogle Scholar
4Calderon, A. P.. Uniqueness in the Cauchy problem for partial differential equations. Amer. J. Math. 80 (1958), 1636.Google Scholar
5Courrège, P., Renouard, P., Priouret, P. and Yor, M.. Oscillateur anharmonique processus de diffusion et mesures quasi-invariantes. Astérisque (Soc. Math. France), 22–23 (1975).Google Scholar
6Dieudonne, J.. Calcul infinitésimal (Paris: Hermann, 1968).Google Scholar
7Hsieh, P. F. and Sibuya, Y.. On the asymptotic integration of the second order linear ordinary differential equation with polynomial coefficients. J. Math. Anal. Appl. 16 (1965), 84103.Google Scholar
8Kato, T.. Perturbation theory for linear operators (Berlin: Springer, 1966).Google Scholar
9Kato, T.. Schrödinger Operators with singular potentials. Israel. Math. 13 (1972), 135148.Google Scholar
10Landis, E. M.. Some problems of the qualitative theory of second order elliptic equations (case of several independent variables). Russian Math. Surveys 18 (1963), 162.Google Scholar
11Lions, J. L. and Magenes, E.. Problèmes aux limites non homogènes et applications (Paris: Dunod, 1970).Google Scholar
12Schechter, M.. Spectra of partial differential operators (London: North Holland, 1971).Google Scholar
13Simon, B.. Quantum mechanics for hamiltonians defined as quadratic forms. Princeton Series in Physics (Princeton: Univ. Press, 1971).Google Scholar
14Simon, B.. Resonances in n-body quantum system with dilatation analytic potential and the foundation of time dependant perturbation theory. Ann. of Math. 97 (1973), 247274.CrossRefGoogle Scholar
15Trèves, F.. Relations entre operateurs differentiels. Ada Math. 101 (1959), 1139.Google Scholar
16Weinstein, A. and Stenger, W.. Methods of intermediate problem for eigenvalues (New York: Academic Press, 1972).Google Scholar
17Zerner, M.. Inégalités du type Harnack, principe de Phragmen Lindelöf, diverses conséquences d'une formule de Green. Séminaire Schwartz (Paris: Faculté des Sciences, 1960).Google Scholar