# Positivity and Negativity of Solutions to a Schrödinger Equation in ℝ N

Positivity and Negativity of Solutions to a Schrödinger Equation in ℝ N Weak L 2 $$(\mathbb{R}^N )$$ -solutions u of the Schrödinger equation, −Δu + q(x) u − λu = f(x) in L 2 $$(\mathbb{R}^N )$$ , are represented by a Fourier series using spherical harmonics in order to prove the following strong maximum and anti-maximum principles in $$\mathbb{R}^N$$ (N ≥ 2): Let ϕ1 denote the positive eigenfunction associated with the principal eigenvalue λ1 of the Schrödinger operator $$A = - \Delta + q(x) \bullet {\text{ in }}L^2 (\mathbb{R}^N )$$ . Assume that the potential q(x) is radially symmetric and grows fast enough near infinity, and f is a sufficiently smooth' perturbation of a radially symmetric function, f ≢ 0 and 0 ≤ f / ϕ ≤ C ≡ const a.e. in $$\mathbb{R}^N$$ . Then u is ϕ1-positive for -∞ < λ < λ1 (i.e., u ≥ c ϕ1 with c ≡ const > 0) and ϕ1-negative for λ1 < λ < λ1 +δ (i.e., u ≤ −cϕ1 with c ≡ const > 0), where δ > 0 is a number depending on f. The constant c > 0 depends on both λ and f. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Positivity and Negativity of Solutions to a Schrödinger Equation in ℝ N

, Volume 5 (4) – Oct 3, 2004
24 pages

/lp/springer_journal/positivity-and-negativity-of-solutions-to-a-schr-dinger-equation-in-n-ZRDA2cUxAn
Publisher
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1011899206325
Publisher site
See Article on Publisher Site

### Abstract

Weak L 2 $$(\mathbb{R}^N )$$ -solutions u of the Schrödinger equation, −Δu + q(x) u − λu = f(x) in L 2 $$(\mathbb{R}^N )$$ , are represented by a Fourier series using spherical harmonics in order to prove the following strong maximum and anti-maximum principles in $$\mathbb{R}^N$$ (N ≥ 2): Let ϕ1 denote the positive eigenfunction associated with the principal eigenvalue λ1 of the Schrödinger operator $$A = - \Delta + q(x) \bullet {\text{ in }}L^2 (\mathbb{R}^N )$$ . Assume that the potential q(x) is radially symmetric and grows fast enough near infinity, and f is a sufficiently smooth' perturbation of a radially symmetric function, f ≢ 0 and 0 ≤ f / ϕ ≤ C ≡ const a.e. in $$\mathbb{R}^N$$ . Then u is ϕ1-positive for -∞ < λ < λ1 (i.e., u ≥ c ϕ1 with c ≡ const > 0) and ϕ1-negative for λ1 < λ < λ1 +δ (i.e., u ≤ −cϕ1 with c ≡ const > 0), where δ > 0 is a number depending on f. The constant c > 0 depends on both λ and f.

### Journal

PositivitySpringer Journals

Published: Oct 3, 2004

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