Lower bounds for solutions of Schrödinger equations

1970 ◽  
Vol 23 (1) ◽  
pp. 1-25 ◽  
Author(s):  
Shmuel Agmon
Keyword(s):  
Lower Bounds ◽  
Schrodinger Equations ◽  
Schrödinger Equations

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

1983 ◽  
Vol 95 (1-2) ◽  
pp. 25-38 ◽  
Author(s):  
R. Froese ◽  
I. Herbst ◽  
M. Hoffmann-Ostenhof ◽  
T. Hoffmann-Ostenhof
Keyword(s):  
Asymptotic Behaviour ◽  
Lower Bounds ◽  
Compact Support ◽  
Schrodinger Equations ◽  
Schrödinger Equations

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|→∞.

Lower bounds for derivatives of solutions for nonlinear Schrödinger equations

2009 ◽  
Vol 139 (2) ◽  
pp. 237-251 ◽  
Author(s):  
Andrei Biryuk
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Small Viscosity ◽  
Nonlinear Schrodinger Equations ◽  
Image Position ◽  
Derivatives Of

The main goal of this paper is to obtain the lower boundsfor solutions uν of nonlinear Schrödinger equations with small viscosity ν. We also discuss the application of the above estimate to the theory of turbulence. Namely, we are interested in time-averaged lower bounds, which are important in establishing upper bounds for the turbulent space-scale.

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

1990 ◽  
Vol 115 (1-2) ◽  
pp. 65-86 ◽  
Author(s):  
M. Hoffmann-Ostenhof
Keyword(s):  
Lower Bounds ◽  
Unbounded Domains ◽  
Upper Bounds ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Spherical Average ◽  
Asymptotic Decay ◽  
Additional Assumptions

SynopsisThe 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.

scholarly journals Unique continuation at infinity of solutions to Schrödinger equations with complex-valued potentials

1999 ◽  
Vol 42 (1) ◽  
pp. 143-153 ◽  
Author(s):  
J. Cruz-Sampedro
Keyword(s):  
Lower Bounds ◽  
Compact Support ◽  
Unique Continuation ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Complex Valued

We obtain optimal L2-lower bounds for nonzero solutions to – ΔΨ + VΔ = EΨ in Rn, n ≥ 2, E ∈ R where V is a measurable complex-valued potential with V(x) = 0(|x|-c) as |x|→∞, for some ε∈ R. We show that if 3δ = max{0, 1 – 2ε} and exp (τ|x|1+δ)Ψ ∈ L2(Rn)for all τ > 0, then Ψ; has compact support. This result is new for 0 < ε ½ and generalizes similar results obtained by Meshkov for = 0, and by Froese, Herbst, M. Hoffmann-Ostenhof, and T. Hoffmann-Ostenhof for both ε≤O and ε≥½. These L2-lower bounds are well known to be optimal for ε ≥ ½ while for ε < ½ this last is only known for ε = O in view of an example of Meshkov. We generalize Meshkov's example for ε< ½ and thus show that for complex-valued potentials our result is optimal for all ε ∈ R.

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