ISSN:
1432-0916
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract Let Ω R =ℝ n ∖B R, wheren≧3 andB R={x∈ℝ n :|x|≦R}. We investigate the asymptotics of real valued solutions ψ∈L 2(Ω R ) of the Schrödinger equation (−Δ+V−E)ψ=0, whereE〈0 andV(x)→0 for |x|→∞: LetD denote an unbounded nodal domain of ψ (i.e. a component of Ω R ∖{x:ψ(x)=0}), and letS(r)={y∈S n−1:ry∈D} withS n−1 the unit sphere in ℝ n . Under suitable assumptions onV it is shown that for some γ〉0, $$\begin{gathered} \mathop {\lim \inf }\limits_{r \to \infty } r^\gamma \int\limits_{S(r)} {\psi ^2 d\sigma } /\int\limits_{S(r)} {\psi ^2 d\sigma } 〉 0 and \hfill \\ \mathop {\lim \inf }\limits_{r \to \infty } \ln (Volume(D \cap B_r ) )/\ln r \geqq (n + 1)/2. \hfill \\ \end{gathered} $$ Results of this type are already non-trivial for radial problems with ψ satisfying non-radial boundary conditions on ∂Ω R or for excited states of the Hydrogen atom if one considers linear combinations of differentl-waves.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01228411
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