An extension of maximum and antimaximum principle to a Schrödinger equation in ℝ^{2}

J. Fleckinger, UMR MIP and Ceremath Université Toulouse 1 pl. A. France 31042 Toulouse Cedex

                  jfleck@univ-tlse1.fr

Strong maximum and antimaximum principles are extended to weak solutions in ℝ^{2} of the Schrödinger equation Lu := −Δu + qu = αu + f in ℝ^{2}. For positive potentials q, tending to +∞ at infinity L is with discrete spectrum. Denote by (λ_{1}, φ_{1}) its principal eigenpair with φ_{1} > 0. For a “nearly radial” potential, growing faster than |x|^{2}, we show that for 0 ≤ f/φ_{1} ≤ C there exists a positive number δ such that, for every α ∈ (−∞, λ_{1} + δ), the inequality (λ_{1} − α)u ≥ cφ_{1}) holds a.e. in ℝ^{2}.

Joint work with B. Alziary and P. Takáč.